IRMA Lectures in Mathematics and Theoretical Physics 20

IRMA Lectures in Mathematics and Theoretical Physics 20

Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique AvancéeCNRS et Université de Strasbourg

7 rue René-Descartes67084 Strasbourg Cedex

France

Chalk drawing by Tatsuo Suwa

Singularities in Geometry andTopology Strasbourg 2009

Vincent BlanlœilToru Ohmoto

Editors

Editors:

Vincent BlanloeilIRMA, UFR Mathématiques et InformatiqueUniversité de Strasbourg7, rue René Descartes67084 Strasbourg, France

E-mail: [email protected]

2010 Mathematical Subject Classification: 13A35, 14A22, 14B05, 14B07, 14B15, 14C17, 14D06, 14E15, 14E18,14F99, 14H25, 14J17, 14M25, 14N99, 18F30, 19K10, 32A27, 32C37, 32F75, 32G15, 32SXX, 35Q75, 52C35,53C05, 55N35, 55R40, 57N05, 57R18, 57R20, 58A30, 58K10, 58K40, 58K60, 60D05, 83C57

Key words: singularity theory, singularities, characteristic classes, Milnor fiber, jet schemes, equisingularity,intersection homology, knot theory, Hodge theory, Fulton–MacPherson bivariant theory, mixed weightedhomogeneous, nearby cycles, vanishing cycles, affine toric variety, (versal) deformation of surface singularities,noncommutative resolution, cyclic quotient surface singularity, splice quotient singularity, F-regular singulari-ties, semiquasihomogeneous isolated singularities, general relativity, statistical learning theory, singular distri-butions, localization of characteristic classes, Frobenius morphism, b-function, motivic Grothendieck group,motivic Hirzebruch class, monodromy covering, algebraic local cohomology, Riemann–Roch theorem forembeddings, birational invariant, Riemann surface, stable reduction, Teichmüller space, moduli space, orbifold

ISBN 978-3-03719-118-7

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and thedetailed bibliographic data are available on the Internet at http://www.helveticat.ch.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2012 European Mathematical Society

Contact address:

European Mathematical Society Publishing HouseETH-Zentrum SEW A27CH-8092 ZürichSwitzerland

Phone: +41 (0)44 632 34 36Email: [email protected]: www.ems-ph.org

Typeset using the authors’ TEX files: M. Zunino, StuttgartPrinting and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany∞ Printed on acid free paper9 8 7 6 5 4 3 2 1

Toru OhmotoDepartment of MathematicsFaculty of ScienceHokkaido UniversitySapporo 060-0810, Japan

E-mail: [email protected]

Preface

InAugust 2009 we organized the fifth Franco–Japanese Symposium on Singularities atthe Department of Mathematics of Strasbourg University. This symposium followedthe fourth one held in Toyama, Japan, two years before. The first day we scheduled aJSPS Forum on Singularities and Applications, and some applications of singularitytheory in physics, medicine and statistics were presented. The following days we hada conference; there were advanced talks in topology, algebraic geometry and complexgeometry, and recent results on singularities were discussed.

In this volume we collected some research papers from participants of the con-ference and surveys of some talks in the JSPS Forum. Moreover we add two lecturenotes of T. Suwa and S. Yokura. All papers in this volume have been refereed andare in final form. We hope that this book will give an opportunity to readers to get adeeper understanding of the marvelous field of Singularities.

On behalf of the editors of this proceedings, we would like to express our thanksto Strasbourg University, JSPS, CNRS, Region Alsace and CEEJA, for their support,and to all contributors for the proceedings and the participants of the symposium.

Vincent Blanlœil, Strasbourg

Toru Ohmoto, Sapporo

The participants of the Conference in front of the Opéra de Strasbourg

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Alain Joets

Optical caustics and their modelling as singularities (JSPS Forum) . . . . . . . . . . . . . 1

Helmut A. Hamm

On local equisingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Shihoko Ishii, Akiyoshi Sannai, and Kei-ichi Watanabe

Jet schemes of homogeneous hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Tatsuhiko Koike

Singularities in relativity (JSPS Forum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51

Yukio Matsumoto

On the universal degenerating family of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 71

Yayoi Nakamura and Shinichi Tajima

Algebraic local cohomologies and local b-functions attachedto semiquasihomogeneous singularities with L.f / D 2 . . . . . . . . . . . . . . . . . . . . . . . 103

T. Ohmoto,

A note on the Chern–Schwartz–MacPherson class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Mutsuo Oka

On mixed projective curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Tomohiro Okuma

Invariants of splice quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Oswald Riemenschneider

A note on the toric duality between An;q and An;n�q . . . . . . . . . . . . . . . . . . . . . . . . . 161

Jörg Schürmann

Nearby cycles and characteristic classes of singular spaces . . . . . . . . . . . . . . . . . . . . 181

Tatsuo Suwa

Residues of singular holomorphic distributions (lecture) . . . . . . . . . . . . . . . . . . . . 207

viii Contents

Sumio Watanabe

Two birational invariants in statistical learning theory (JSPS Forum). . . . . . . . . .249

Takehiko Yasuda

Frobenius morphisms of noncommutative blowups . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Shoji Yokura

Bivariant motivic Hirzebruch classand a zeta function of motivic Hirzebruch class (lecture) . . . . . . . . . . . . . . . . . . . . 285

Masahiko Yoshinaga

Minimality of hyperplane arrangements and basis of local system cohomology . . 345

Optical caustics and their modelling as singularities

Alain Joets

Laboratoire de Physique des Solides, Bât. 510Université Paris-Sud, 91405 Orsay cedex, France

e-mail: [email protected]

Abstract. Optical caustics are bright patterns, formed by the local focalization of light rays. Theyare caused, for instance, by the reflection or the refraction of the sun rays through a wavy watersurface. In the absence of an appropriate mathematical frame, their main characteristics haveremained unrecognized for a long time and the caustics appeared in the literature under differentnames: evolutes, envelopes, focals, etc. The creation of the singularity theory in the middleof the XX century radically changed the situation. Caustics are now understood as physicalrealizations of Lagrangian singularities. In this modelling, one predicts their local classificationinto five stable types (R. Thom, V. Arnold): folds, cusps, swallowtails, elliptic umbilics andhyperbolic umbilics. This local classification is indeed observed in experiments. However theglobal properties of the caustics are only partially taken into account by the Lagrangian model. Infact, it has been proved byYu. Chekanov that the special form of the eikonal equation governingthe propagation of the optical wave fronts imposes the existence of a topological constraint onthe singular set (representing the caustic in the phase space) and restricts the number of possiblebifurcations. Our experiments on caustics produced by bi-periodic structures in liquid crystalsconfirm the existence of the topological constraint, and validate the modelling of the caustics asspecial types of Lagrangian singularities.

Caustics constitute a phenomenon of light focalization, usually studied in the frame ofgeometrical optics or of wave optics. It is remarkable that they now constitute also apurely mathematical object, expressed in terms of singularities. These two notions arenot uncorrelated. The mathematical notion is the final outcome of a long process ofsuccessive modellings of the physical phenomenon, that we will call hereafter opticalcaustics. The aim of this paper is to show how the singularity theory drasticallychanged our viewpoint about the optical caustics. We will show that some problems,for instance the local classification of caustic points, may be solved only with the helpof the singularity theory, and that, conversely, the singularity theory is at the origin ofnew problems and new experiments on optical caustics.

mailto:[email protected]

2 Alain Joets

1 Physical aspects

1.1 The physical phenomenon of optical caustics

There are different ways to present caustics, according to whether one considers lightas composed of rays, or of scalar waves or of electromagnetic waves. However, in viewof our purpose, we shall mainly consider the geometrical description in which lightis composed of rays, or equivalently of wave fronts. In other words, the wavelengthof the light will be supposed to be 0, or very small with respect to the characteristicdimensions of the system.

Given a set of rays (a congruence of rays), a caustic point is a point (of our physicalspace) where the rays are locally focusing, i.e. where two finitely close rays intersect(see Figure 1). At a non caustic point, that is to say at a regular point of the congruence,the rays form a local beam (or the superposition of a finite number of local beams). Incontrast, the light beam is shrunk at a caustic point and the energy density becomesinfinite (at least in the frame considered here). This is the reason for the name “caustic”,that comes from the Greek root “kausticos” meaning “burning”. From the geometricalviewpoint, the caustic is the envelope of the ray congruence. This means that the raysare tangent to the caustic (at the corresponding caustic point). In the usual case ofstraight rays in our physical 3D-space, each ray contributes to 2 caustic points and thecaustic is composed of 2 sheets.

A very simple example of caustics is provided by the bright moving lines one seeson the bottom of a swimming pool. Another example is provided by a perfect focus.However, this example is a somewhat misleading, since a focus is a fully unstablecaustic point disappearing under any small perturbation of the congruence. Such anunstable situation must be excluded from the general study of caustics.

In the plane, the caustic points constitute curves (Figure 1). In the physical3D-space, they constitute surfaces. These geometrical objects are generally not reg-ular. They may possess special points: regression points for the caustic curves, andregression edges for the caustic surfaces. The regression edges themselves may pos-sess more particular points. In brief, caustics are structured objects and an importantproblem is to understand their structure into different types of points.

There is no special condition for producing optical caustics. Every congruence ofrays generates a caustic, more or less intricate. Even in the case of a beam of parallelrays, one may consider that a caustic point is generated at infinity. The caustics thenconstitute an optical phenomenon of great generality.

1.2 Observation of caustics

As (singular) surfaces in our physical space R3, the caustics cannot be directly ob-served, since they are not material surfaces. However they are easily visualizedby interposing some screen transversely to the rays. In other words, one sees only2D-sections of a caustic surface and the whole caustic itself necessitates a (tedious)

Optical caustics and their modelling as singularities 3

W

caustic

Figure 1. In the plane, a congruence of rays (represented here by arrows) has an envelope curve:its caustic. The wave font W propagates (normally to the rays in the case of a homogeneousand isotropic medium) and its regression point glides along the caustic.

work of reconstruction section by section (see Figure 2). Inside the screen, the trace ofthe caustic forms a set of bright curves called folds (symbolA2). These curves may benot regular at some points forming there a tip called cusp (symbolA3). Figure 2 showspairs of cusps forming “lips” (section 1-left) and a “beak-to-beak” (section 3-right).One has to remember that these curves and points are in fact the traces of fold surfacesA2 and of cusp linesA3. In addition to self-intersections, there is generically no othertype of caustic points in a 2D-section. However, for special positions of the screen,one may observe other types of bright points. They are associated with three types ofcaustic points: the swallowtails (symbol A4), the elliptic umbilics (symbol D�

4 ) andthe hyperbolic umbilics (symbol DC

4 ). Examples of these three types may be foundin Figure 5 (simulation) and Figure 8 (photo). The five types A2, A3, A4, D�

4 , andDC4 constitute the complete list of the generic caustic points of the physical space.The description of a caustic given here, in terms of different types of caustic points,

corresponds to a modern presentation, using the results, the names and symbols comingfrom the singularity theory. However, the usual presentation in textbooks on optics ismuch more elementary, very often limited to a formal definition of a caustic point andto some elementary properties. A reason for that is perhaps the high mathematical levelof the singularity theory. In fact, a more fundamental reason is that the traditional aimof optics is the lens design forcing light beams to be concentrated at well defined focalpoints. For that reason, optical systems have special symmetries. The caustics theyproduce are not generic and very special techniques have been developed to their study,for instance the geometrical theory of optical aberrations [1]. In a sense, instrumentaloptics are interested by degenerate caustics, not by the generic ones.

On the other hand, the light focusing by natural systems always produces genericcaustics [2]. Interesting examples include the optics of the eye [3], the electronicoptics [4] and [5], the gravitational lensing, [6], the visualization techniques calledshadowgraph methods [7] and [8].

4 Alain Joets

1)1)

1)1)

2)2)

2)2)

3)3)

3)3)

A2

A3A3

Figure 2. Two examples of the reconstruction of a cusp line A3 and its twofold surfaces A2:lips (left) and beak-to-beak (right).

2 Modelling caustics

The history of the mathematical notion of caustic is long and intricate. Our aim is notto present a detailed picture of it, but rather to give some elements which allow one toestimate the role of the notion of singularity in the recent revival of the subject.

2.1 First modellings

Surprisingly, the first study about caustics seems to be due to a Greek mathematicianof the 3rd century B.C., Apollonius of Perga, who considered the problem of findingsegments of extremal length linking an arbitrary point in the plane to a given conicsection [9] and [10]. The conic section plays the role of an initial wave front and the


extremal segments, normal to the conic, play the role of the rays. In this analogy, thelocus where the number of extremals changes represents the caustic. Apollonius founda geometrical construction for determining this locus. In the modern terminology, hestudied the generic Lagrangian singularities of the plane (A2 and A3).

One observes no substantial progress until the introduction of the name “caustica”by Tschirnhausen, who studied the reflection of sun rays in a circular mirror [11].He observes that the concentration of the rays occurs along an “entire curved line,which is produced by the intersections of reflected rays” (see Figure 3-top left). Thename itself appears in 1690, in the Tschirnhausen paper [12], in the Latin expression“caustica curva”, quickly abbreviated to “caustica”.

In fact, a few years before, C. Huygens had obtained more accurate results aboutcaustics by reflection or by refraction, including the propagation of the wave frontalong the caustic (see Figure 3-top right). However, his book, Traité de la lumière,appeared later, in 1690 [13]. Caustics (in the plane) appear in the L’Hospital’s book“Traité des infiniment petits” (1696), the first book on differential calculus [14]. Theyillustrate the power of the new Calculus.

Caustics in the 3D-space appeared latter, after the introduction of the notions ofcurvature, lines of curvature, principal curvatures, etc. We have to recall the creationof the word “umbilic” by G. Monge (1795), a point of a surface at which the twoprincipal curvatures are equal [15]. In 1873, A. Cayley studies the congruence formedby the normals to an ellipsoid [16]. He shows that the “centro-surface”, i.e. the causticassociated with the normals, possesses four special points, called by him “umbilicarcentres” or “omphaloi” (see Figure 3-bottom). In the modern terminology, they arenamed “hyperbolic umbilics” and denoted by DC

4 . The study of the umbilics hasbeen continued by Darboux [17], in 1896. More precisely, the author analyses thelines of curvature in the vicinity of an umbilic of a given surface, and not of a causticsurface. The link with the caustics and their umbilics exists only if one considersthat the surface represents a wave front propagating in an homogeneous and isotropicmedium. Darboux succeeded in classifying these umbilics into 3 types. Nevertheless,let us note that the Darbouxian classification is different from – although not unrelatedto – the modern classification of umbilics of a caustic into hyperbolic and elliptic typesand also from the classification according to their index (see [18] for the details).

Since Darboux’s work and until the singularity theory, the subject of caustics seemsto have been neglected, and at best considered as a source of academic exercises forstudents.

To sum up, all these partial results show that the caustics have been recognizedfrom the very beginning as complex objects, presenting a rich structure. However, theusual direct approach in the frame of the Euclidean space proved to be too restrictedand inadequate to obtain general results. The situation radically changed in 1955 withthe creation of the singularity theory by H. Whitney [19] and R. Thom [20].

6 Alain Joets

Figure 3. In his 1682 publication [11], W. Tschirnhausen defines a caustic line as the curved lineproduced by the “intersections des rayons réfléchis” [top left]. In his treatise [13] (written fouryears before, but published only in 1690), C. Huygens obtains, for the same problem, a moreaccurate description including the propagation of the wave front, called by him “onde repliée”[top right]. In 1871 [16], A. Cayley shows that the caustic of an ellipsoid possesses umbilicspoints, that is to say meeting points of the two sheets of the caustic (here in the plane x-z)[bottom].

2.2 Caustics as singularities of maps

The modelling has made an important progress thanks to the singularity theory. Thisprogress is based on the distinction between two spaces where the rays are represented:

• Our physical space R3 D fx1; x2; x3g, in which lie the rays and the caustic. Inthis space the rays may intersect.

• An abstract “ray space” R D fr1; r2; r3g above the physical space R3, whereeach ray is represented by some curve (also called “ray”). R is only composedof rays. It is a smooth 3D-manifold and it is constructed in such a way as the“rays” cannot intersect.

A simple way for constructing R is the following. One considers a surface W � R3transverse to the rays, for instance an initial wavefront. Each ray is thus parametrizedby the two parameters ofW, say r1 and r2. In order to specify the position of the currentpoint along a ray .r1; r2/, one needs a third coordinate r3, for example its distance toW along the ray. The space R is then parametrized by these three coordinates r1, r2,and r3. It is clear that the intersections of rays cannot occur in this space R, sincedifferent rays have different values for .r1; r2/.


We now recover the observed congruence by associating with each point of R itsposition .x1; x2; x3/ in the physical space. This defines a mapping p W R ! R3 by.x1; x2; x3/ D p.r1; r2; r3/. The mapping p is called projection. Let us note that thesource space R and the target space R3 have the same dimension, equal to 3.

In this frame, the local ray focalization at a caustic point is expressed by sayingthat the rank of the derivative dp of p is less than its maximal possible value 3. Sucha point in R is called singular or critical. The set † � R of the critical points iscalled the singular set. Finally, the caustic C is the projection of the singular set:C D p.†/. In practice, the equation for † is obtained by cancelling the Jacobiandeterminant associated with p, det @.x1; x2; x3/[email protected]; r2; r3/ D 0. By solving thisequation, one obtains one of the ri ’s as a function of the others, say r3 D r3.r1; r2/.Thus the caustic is found in a parametric form: x1 D x1.r1; r2; r3.r1; r2//, etc.

At this stage, we merely have a mathematical definition for the physical notion ofcaustic point. The singularity theory allows us to go farther and to find the nature ofthe caustic point. More precisely, let us recall that one defines the Thom–Boardmanset †i of p as the set of points of R where dp has a kernel of dimension i [20]. Thenones defines inductively the set†i;:::;j;k as the set†k of the restriction of p to†i;:::;j .Thus,†0 represents the regular points of the congruence,†1;0 the fold-surface,†1;1;0

the cusp-lines, †1;1;1;0 the swallowtails, and †2 the umbilics (hyperbolic or elliptic).By definition, each Thom–Boardman set is obtained by cancelling some functional

determinants associated with p or with the restriction of p to some other Thom–Boardman set. Therefore the effective calculation of the sets †I can always be per-formed at least numerically. However, this classification “by the rank” is not totallysatisfactory. First, it does not distinguish between the hyperbolic umbilicsDC

4 and theelliptic umbilics D�

4 . Worse, as singularities of a (general) map, the umbilics, havinga codimension (4) higher than the dimension of the space (3), are not stable. The factthat they are experimentally observed shows that the modelling of caustics as singu-larities of a map is incomplete. In fact, it ignores an important element, namely theFermat principle or, in mathematical terms, the symplectic structure of the problem.

2.3 Caustics as Lagrangian singularities

The wave propagation along a ray is described by the wave vector, or “momentum”,Ep [1]. The local ray direction is along Ep. One has the fundamental relation

Ep D rS; (2.1)

where S is the optical lengthRnds and n the (local) refractive index. S follows the

eikonal equation:.rS/2 D n2: (2.2)

Relation (2.1) shows that the “function” S must be considered as a multi-valued func-tion, since several local beams may be passing through a given point. This suggestsa new representation of the rays in a bigger space including at once the spatial coor-

8 Alain Joets

dinates xi and the vectorial coordinates pi . More precisely, one considers the phasespace T �R3 D fpi ; xig. The phase space is characterized by its symplectic structure,that is, the differential 2-form ! DP dpi ^ dxi , which is nondegenerate and closed(d! D 0). One sees immediately that ! cancels at the points for which Ep D rS . Oneis thus led to keep the cancellation of ! as the characteristic property of a congruenceof rays represented in the big space T �R3. One says that the submanifoldL � T �R3of dimension 3 (half of the dimension of the phase space) is a Lagrangian submani-fold if !jL D 0. The base space fx1; x2; x3g is called the configuration space. Everycongruence of rays is described by a Lagrangian submanifold.

In this frame, the role of the projection p is played by the natural projection �into the configuration space, �.p; x/ D x, or more precisely by its restriction to L,p D �jL (see Figure 4). As in the previous section, one defines the singular set† � L as the set of points where p has a non trivial kernel. The caustic C is �.†/.By reference to the name forL, the singular points are called Lagrangian singularities.

The advantage of the new definition comes from the properties attached to theLagrangian submanifolds. Indeed these submanifolds are constructed by startingfrom functions or from families of functions, rather than from maps. There are twoimportant formulations [21].

The first formulation is the generalization of the relation Ep D rS , valid also forthe singular points of L. It takes the (local) form

p˛ D @S

@x˛; xˇ D � @S

@pˇ: (2.3)

The generating function S is no more defined on the configuration space, but ratheron the Lagrangian submanifoldL (locally parametrized by the coordinates x˛ and pˇof (2.3)).

In the second formulation, the Lagrangian submanifold is given by a generatingfamily, i.e. a function F defined on the configuration space R3 D fx1; x2; x3g anddepending on some parameter s:

L Dn.p; x/ W there exists s such that

@F

@sD 0 and p D @F

@x

o: (2.4)

The first equation @F=@s D 0 determines the rays passing through .x1; x2; x3/,whereas the second one distinguishes these rays according to their wave vector Ep.So the generating-family technique links the caustics to the theory of singularitiesof functions depending on some parameters, that is to say to catastrophe theory [22]and [23].

2.4 Caustics and wave front singularities

Figure 3-middle, extracted from the pioneering work of Huygens [13] recalls that,in the case of the plane, the propagating wave front presents a singular point glidingalong the caustic curve. In fact, the entire caustic curve results from the sweeping


x1

x1

x2

x2

p1

p2

caustic C

configuration space

initial wave front W

Lagrangian projector �

singular set †

Lagrangian submanifold L

Figure 4. When represented in the phase space (here the space fp1; p2; x1; x2g), the raysconstitute a regular surface L called the Lagrangian submanifold. The points of the singular set† � L are characterized by a vertical tangent plane to L. The caustic C is the projection of thesingular set †: C D �.†/.

motion of the singularities ofW. This remarkable duality linking rays and wave frontsremains valid in the general case of caustics of the 3D-space. However, in this case,a typical instantaneous wave front W has more singularities: it may possess cuspidalcurves and swallowtails points. During the motion of W, governed by the eikonalequation (2.2), the cuspidal curves generate surfaces and the swallowtails generatecurves. The generated surfaces are exactly the fold surfaces A2 of the caustic C,whereas and the generated curves are the cusp lines of C. To obtain the other caustictypes, i.e. the swallowtailsA4 and the umbilicsD4, one has to consider the bifurcationsof the wave front, at some times of its motion.

3 Local and global aspects of caustics

3.1 Local types

In order to distinguish different types of singularities, one defines an equivalencerelation between Lagrangian projections, called Lagrange equivalence. This is a dif-feomorphism between the two phase spaces, preserving simultaneously the symplecticand the fiber structures and sending the first Lagrangian submanifoldL1 to the secondLagrangian submanifoldL2 (see [21] for the details). In fact one considers rather local

10 Alain Joets

situations, expressed in terms of germs. A Lagrangian singularity is then a Lagrangeequivalence class of a germ at a critical point. The same equivalence relation allowsone to define the stability of a singularity. A singularity is stable if its equivalenceclass constitutes a neighborhood of it.

A2

A3

A4 D�4

DC4

A2A2

Figure 5. The five generic types of Lagrangian singularities: the fold type A2 constitutessurfaces; the cusp type A3 constitutes edges of regression; the three other types are pointsingularities: the swallowtail typeA4 (at the meeting point of twe linesA3 and a self-intersectionline A2A2), the elliptic umbilic type D�

4(at the meeting point of three cusp lines A3) and the

hyperbolic umbilic type DC4

(at the meeting point of a cusp line A3 and a self-intersection lineA2A2).

The fundamental result of the Lagrangian singularity theory is the local classifi-cation of Lagrangian singularities: every stable Lagrangian singularity is equivalentto one of the five following types: A2, A3, A4, D�

4 , DC4 (see Figure 5). In terms of

generating families, the list is given by [21]:

A2 W F D s3 C q1s;A3 W F D ˙s4 C q1s2 C q2s;A4 W F D s5 C q1s3 C q2s2 C q3s;D4 W F D s21s2 ˙ s32 C q1s22 C q2s2 C q3s1:

These polynomial functions are called normal forms. They constitute a local modeldescribing the fine structure of every caustic type. The stability means that everysingularity of the above list survives the action of infinitely small perturbations. Con-versely, any other singularity type not pertaining to the above list is destroyed byperturbations and is replaced by singularities belonging to the list.


It is important to note that the vector Ep describes the phase behavior of the opticalwave. As a consequence, the normal forms describe at once the shape of the causticsurface (via the Lagrangian projection) and the amplitude of the interference patternaround it (via the Fresnel–Kirchhoff integral [1]); see Figure 6.

In addition to its normal form, one may associate some numbers with any caustictype. We have already seen that each type forms a set of some dimension d, or betterof some codimension d 0 D 3 � d . We have d 0 D 1 for the folds, d 0 D 2 for thecusps, and d 0 D 3 for the swallowtails and the umbilics. Another important numberis the rank r of the projection p, or equivalently the corank c D 3 � r . Folds, cuspsand swallowtails have a corank equal to 1, and a tangent plane is defined at the causticpoint, despite the fact that the caustic is not a regular surface at the points A3 and A4.In contrast, the umbilics have a corank equal to 2, and the caustic has no tangent planethere, but rather a local direction corresponding to the direction of the ray passingthrough the umbilic. There is also a “singularity index” governing the asymptoticincrease of the amplitude of the diffraction pattern in the limit of wavelengths tendingtowards 0; see [24]. The values of this index show that cusps must appear brighterthan the folds, and the swallowtails and the umbilics brighter than the cusps.

The local classification accounts for all of the observed (non-degenerate) causticsand for their diffraction patterns; see [2], [24], and [22]. It is the basis of a fine studyof each caustic type and the agreement between theory and experiment is found to beexcellent (see for example [25] for the case of the D�

4 ).

3.2 Bifurcation of caustics

Another local classification concerns the bifurcations of caustics themselves. Whenthe system of rays depends on some control parameter (for example a temperatureor a magnetic field), the caustics produced may undergo a topological transformationfor some value of the parameter. This transformation is called bifurcation. There areeleven possible bifurcations of Lagrangian singularities [21]. Some of them describehow point singularities appear or disappear by pairs.

To our knowledge, these caustic bifurcations have not yet been experimentallystudied in detail.

3.3 Global aspects

The global properties of caustics are less understood than the local ones. However, thegeneralization of the notion of Maslov’s index to spaces of higher dimensions has ledto the discovery of new invariants [26]. These invariants control the number of sometypes of singularities. For instance, in dimension n D 4, the number of butterflies A5(taking account of sign) is equal to zero.

For the 3D space, the case mainly considered here, there exists in addition a re-markable theorem due to Yu. Chekanov.

12 Alain Joets

A2 A3 A4

D�4

DC4

Figure 6. Diffraction patterns associated with the five generic caustics.

3.3.1 Chekanov’s formula for the singular set †. Chekanov’s formula is a relationbetween the Euler characteristic�.†/ of the singular set† and the number ]D4.�1=2/of umbilics of index �1=2. More precisely, one has [27]:

�.†/C 2]D4.�1=2/ D 0 (3.1)

In order to understand the definition of the index, let us recall at first that the eikonalequation expresses the fact that the Lagrangian submanifold L lies on a hypersurfaceE � T �R3. The rays correspond the (skew) orthocomplements of E, and are calledcharacteristic l . Moreover, an umbilic point T 2 L is a singular point where thesurface † � L is locally a cone (in fact a double cone) with vertex at T . Then, sincethe corank of the projection p at T is equal to 2, a 2D plane … D kerp is defined.Finally, cusp lines A3 � † pass through T . Now, the index is defined according tothe relative positions of these elements. If l and A3 are separated by …, the index isequal toC1=2, and �1=2 in the other case (see Figure 7).

One shows that the index of an elliptic umbilic is always equal to�1=2. The indexof a hyperbolic umbilic may be equal either to �1=2, and it is denoted by DCt

4 , or toC1=2, and it is denoted by DCd

4 . Thus, formula (3.1) writes:

�.†/C 2.]D�4 C ]DCt

4 / D 0 (3.2)


characteristic d characteristic d characteristic d

†††

A3

A3

A3

(a) (b) (c)

DCd4

DCt4

………

Figure 7. In the neighborhood of a hyperbolic umbilic DC4

, the critical set † is a cone (DC4

is at its vertex). The kernel … of the Lagrangian projection at the point DC4

cuts the cone. If… separates the characteristic l from the cusp line A3, the index is C1=2 and the umbilic isdenoted by DCd

4(a). In the other case, the index is �1=2 and umbilic is denoted by DCt

4

(b). Simulation of the singular set† in the neighborhood of a hyperbolic umbilic of the causticrepresented in Figure 8(a). By comparison with (a) and (b), one sees that, in this case, the indexis C1=2.

Chekanov’s theorem requires some assumptions. In particular, the hypersurfaceE is supposed to be convex with respect to the wave vector Ep. This special conditionis always satisfied in geometrical optics, because of the general form of the eikonalequation (2.2). For that reason, in this framework, the Lagrangian singularities arecalled optical singularities. It is also assumed that † is a compact surface.

Since it contains elements defined in the abstract space T �R3, Chekanov’s formulacannot be directly checked by experiment. Nevertheless, it may be possible, in thebest cases, to obtain experimental informations about the ray congruence sufficient tocalculate numerically these elements. This reconstruction has been successfully madein the case of a biperiodic caustic produced by the deflection of a light beam througha nematic liquid crystal layer [28]. The biperiodicity in the plane of the layer makesthe emerging wave front topologically equivalent to a torus T 2. Now, through eachpoint of this torus passes one straight ray bearing two caustic points. These two pointscoincide only at the umbilics. In other words, † is a topological surface obtainedby gluing together two torus at the umbilics points. One deduces immediately that�.†/ is related to the number of umbilics ]D4 through the relation �.†/ D �]D4.In the experiment one counts eight umbilics per cell: �.†/ D �8. The remainingwork is a careful simulation of the deflection of the rays inside the liquid crystal, thenumerical calculation of the projection, and the determination of its Thom–Boardmansets †1 (giving the double cones at the umbilic points) and †1;1 (giving the A3 lineswhich pass through the vertices of the double cones). In the case under consideration,one finds that all hyperbolic umbilics (four per cell) have a positive index. Since onecounts per cell four elliptic umbilics (index �1=2) and four umbilics DCd

4 , one has�.†/C 2D.�1=2/ D �8C 2.4C 0/ D 0. The Chekanov relation is verified.

It is interesting to recall that M. Kazarian [29] gave an alternative characterizationof the indices of the umbilics in the configuration space. This characterization is based

14 Alain Joets

(a) (b)

Figure 8. Two particular sections of the biperiodic caustic produced in a certain experimentusing a liquid crystal as a light deflector. The caustic contains hyperbolic umbilics (a) andelliptic umbilics (b).

on the behavior of the ray direction along a cusp line A3 passing through the umbilicpoint. At each point of a cusp line a tangent plane to the caustic surface is defined(even if the surface is a non-regular surface there) and the ray lies inside this plane. Atthe umbilic point, the ray becomes parallel to the cusp line. Along A3, there are twopossibilities for the ray direction. If it points inside the cuspidal edge, the line is saidto be AC

3 , and A�3 when it points outside the cuspidal edge. Now, the direction of the

ray at the umbilic point T defines an orientation of the cusp lines passing through T.Following this orientation, a cusp-line AC

3 (resp. A�3 ) becomes A�

3 (resp. AC3 ) at the

umbilic point and the index is equal to C1=2 (resp. �1=2). To our knowledge thisnew characterization has not yet been exploited experimentally.

Chekanov’s relation has an important consequence on the caustic bifurcations.Among the eleven possible caustic bifurcations, considered as bifurcations of gen-eral Lagrangian singularities, four of them cannot be realized as bifurcations of op-tical Lagrangian singularities: they are incompatible with the Chekanov relation. SoChekanov’s relation reduces the number of optical metamorphoses to seven (see Fig-ure 9).

3.3.2 Topological formula for caustics. Chekanov’s formula describes the topologyof to the singular set † but gives no information about its image C D �.†/ in theconfiguration space. We want to give here new elements for this issue.

It is known that the Euler characteristic of a regular surfaceAwith boundaryB andcorners Ci is determined by the total curvature associated with the surface (gaussiancurvature �), with the boundary (geodesic curvature �g ) and with the corners (external


1 2 3 4

5 6 7

Figure 9. Chekanov’s relation implies that only seven caustic bifurcations can be realizedoptically. Each drawing shows the caustic before, at and after the bifurcation (after [21]).

angles ˛i ). More precisely, one has:

2��.A/ DZA

� ds CZB

�g dl CXi

˛i : (3.3)

A natural issue is then to generalize this formula to the case of caustics C. We havefound that such a generalization may be made. For a caustic without boundary, thenew formula writes [30]:

2��.C / DZA2

� ds CZA3

2�g dl C �.2]A2A2A2 C ]A4 C 2]D�4 /: (3.4)

The first contribution is the gaussian contribution of the fold surface A2. The secondcontribution is the geodesic contribution. The factor 2 means that the cusp lines

16 Alain Joets

A3 may be considered as a kind of double boundary, along which 2 sheets A2 jointogether. In the third contribution, that each type of Lagrangian point singularitygives a different contribution proportional to � , the factor being an integer: 0 forthe hyperbolic umbilics, 1 for the swallowtails, and 2 for the elliptic umbilics. Thisnumber may be interpreted as the number of Whitney umbrellas (contribution �)“contained” in the singularity [30]. There is also a contribution coming from the triplepoints A2A2A2.

In conclusion, optical caustics are complex physical objects, structured in differ-ent types. Because of this complexity, they were analyzed as the physical realizationof various mathematical notions: envelopes, evolutes, focals, centers of curvature,asymptotics, etc. Their local properties are now satisfactorily understood when theyare modelled as singularities, obtained by projecting the Lagrangian manifold repre-senting the set of rays in the phase space into the physical space where the causticsare observed. However, the understanding of their global properties has necessitated arefinement of the model, taking into account the particular form of the light propaga-tion, expressed by the eikonal equation. At present, a consequence of the new model,namely the existence of a topological invariant, has been experimentally checked.However new experiments are needed to verify the other theoretical consequences ofthe model.

References

[1] M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, inter-ference and diffraction of light, Pergamon Press, Oxford etc. 1965. 3, 7, 11

[2] J. F. Nye, Natural focusing and fine structure of light. Caustics and wave dislocations,Institute of Physics Publishing, Bristol 1999. 3, 11

[3] A. Gullstrand, Einiges über optische Bilder, Naturwissenschaften 28 (1926), 653–664. 3

[4] W. Glaser and H. Grumm, Die Kaustikfläche der Elektronenlinsen, Optik 7 (1950), 96–120.3

[5] S. Leisegang, Zum Astigmatismus von Elektronenlinsen, Optik 10 (1953), 5–14. 3

[6] H. Levine, A. O. Petters and J. Wambsganss, Singularity theory and gravitational lensing,Progress in Mathematical Physics 21, Birkhäuser, Basel 2001. 3

[7] W. Merzkirch, Flow visualization, Academic Press, Orlando 1987. 3

[8] A. Joets, Caustics and visualization techniques, in Singularity theory – Dedicated toJean-Paul Brasselet on his 60th birthday. Proceedings of the 2005 Marseille Singular-ity School and Conference, CIRM, Marseille, France, 24 January – 25 February 2005,ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman, World Scrientific,Singapore 2007, 277–284. 3

[9] Apollonius, Conics, Books V to VII: the Arabic translation of the lost Greek original inthe version of the Banu Musa, ed. and transl. by G. J. Toomer, Springer, New York 2006,37–41. 4

[10] A. Joets, Apollonios, premier géomètre des singularités, Quadrature 66 (2006), 37–41. 4


[11] W. Tschirnhausen, Nouvelles découvertes proposées à Messieurs de l’Académie Royaledes Sciences, Journal des Sçavans (1682), 176–179. 5, 6

[12] W. Tschirnhausen, Curva Geometrica quae se ipsam sui evolutione describit, Acta Erudi-torum IX (1690), 169–172. 5

[13] C. Huygens, Traité de la lumière, Pieter van der Aa, Leiden 1690. 5, 6, 8

[14] G. F. A. Marquis de l’Hospital, Analyse des infiniment petits pour l’intelligence des lignescourbes, Imprimerie Royale, Paris 1696, reprint ACL-éditions, Paris 1988. 5

[15] G. Monge, Feuilles d’analyse appliquée à la géométrie à l’usage de l’Ecole Polytechnique,Baudolin, Paris 1795, reprint Editions Jacques Gabay, Sceaux 2008. 5

[16] A. Cayley, On the centro-surface of an ellipsoid, Transactions of the Cambridge Philo-sophical Society XII (1873), 319–365. 5, 6

[17] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriquesdu calcul infinitésimal, tome IV, Gauthier-Villars, Paris 1896, reprint Editions JacquesGabay, Sceaux 1993, 448–465. 5

[18] I. R. Porteous, Geometric differentiation for the intelligence of curves and surfaces, Cam-bridge University Press, Cambridge 1994. 5

[19] H. Whitney, On singularities of mappings of Euclidean spaces I. Mappings of the planeinto the plane, Ann. of Math. (2) 62 (1955), 374–410. 5

[20] R. Thom, Les singularités d’applications différentiables, Ann. Inst. Fourier, Grenoble 6(1956), 43–87. 5, 7

[21] V. I.Arnold, S. M. Gusein-Zade, andA. N.Varchenko, Singularities of differentiable maps.The classification of critical points, caustics and wave fronts, Vol. I, translated from theRussian by I. Porteous and M. Reynolds, Monographs in Mathematics 82, Birkhäuser,Boston 1985. 8, 9, 10, 11, 15

[22] R. Thom, Topological models in biology, Topology 8 (1969), 313–335. 8, 11

[23] R. Thom, Stabilité structurelle et morphogenèse, Interéditions, Paris 1977. 8

[24] M. V. Berry and C. Upstill, Catastrophe optics: morphologies of caustics and their diffrac-tion patterns, Progress in Optics XVIII (1980), 257–346. 11

[25] M. V. Berry, J. F. Nye, and F. J. Wright, The elliptic umbilic diffraction catastrophe, Trans.R. Soc. Lond. A 291 (1079), 453–484. 11

[26] V. A. Vassilyev, Lagrange and Legendre characteristic classes, Advanced Studies in Con-temporary Mathematics 3, Gordon and Breach Science Publishers, New York 1988. 11

[27] Yu. V. Chekanov, Caustics in geometrical optics, Funct. Anal. Appl. 20 (1986), 223–226.12

[28] A. Joets and R. Ribotta, Experimental determination of a topological invariant in a patternof optical singularities, Physical Review Letters 77 (1996), 1755–1758. 13

[29] M. E. Kazarian, Umbilical characteristic number of Lagrangian mappings of 3-dimen-sional pseudooptical manifolds, in Singularities and differential equations, Singularitiesand differential equations. Proceedings of a symposium, ed. by S. Janeczko, W. M. Za-jaczkowski, and B. Ziemian, Bogdan, Banach Center Publications 33, Polish Academy ofSciences, Inst. of Mathematics, Warsaw 1996, 161–170. 13

[30] A. Joets, Gauss–Bonnet formula for caustics, in preparation. 15, 16

On local equisingularity

Helmut A. Hamm�

Mathematisches Institut, Westf. Wilhelms-UniversitätEinsteinstr. 62, 48149 Münster, Germany


Abstract. We will prove some generalization of the theorems of Lê and Ramanujam resp. ofTimourian for the case where the ambient space is no longer Cm. Furthermore we will derivesome weaker result in the case of a family of non-isolated singularities.

1 Introduction

Essentially, a family of singularities is called “equisingular” if the topological typeof the singularities is constant. In order to be precise we will stick to the notions oftopological type and local topological triviality.

LetX andX 0 be topological spaces. ThenX andX 0 are said to have the same topo-logical type if they are homeomorphic. Suppose that we have a family of topologicalspaces Ft ; t 2 T , given by a continuous mapping

g W X �! T

between topological spaces such that

Ft D g�1.ftg/:Then we know that allFt have the same topological type as soon as T is connected andg is a locally trivial fibration, which means that for each t 2 T there is a neighborhoodT 0 of t in T , a topological space F and a homeomorphism

h W g�1.T 0/ �! F � T 0

such that the following diagram is commutative:

g�1.T 0/

g��

��

��

h �� F � T 0

pr2��

T 0

;

�Work partially supported by Deutsche Forschungsgemeinschaft.


20 Helmut A. Hamm

where pr2 is the projection onto the second factor. This is obvious because Ft ishomeomorphic to F for t 2 T .

In fact we are only interested in the complex-analytic context. Let

g W X �! T

be a holomorphic mapping between complex spaces. We call g a topologically locallytrivial fibration if the underlying continuous map is a locally trivial fibration.

Lemma 1.1 (Thom’s first isotopy lemma). Suppose thatg is proper, T smooth and thatthere is a Whitney regular stratification ofX such that g is a stratified submersion, i.e.the restriction ofg to each stratum ofX defines a submersion. Theng is a topologicallylocally trivial fibration.

This is proved using integration of suitable stratified vector fields, see [3] I 1.5.

Now let us pass to the local situation. Let .X; x/ be a germ of a complex space.We can assume that it is a subgerm of .Cm; x/; after translation we can assume x D 0.Let k:::k be the Euclidean norm in Cm, X a representative of .X; 0/ in Cm and

B� D fz 2 Cm j kzk � �g:If � > 0 is sufficiently small the topological type of X \ B� is independent of �, wecall it the topological type of .X; 0/. (In fact, the embedding plays no role too.)

If two germs .X; 0/ and .X 0; 0/ of complex spaces have the same topological typethey are homeomorphic but it is doubtful whether the inverse implication holds.

We may extend the notion of topological type to pairs of spaces or to mappings.If .X; 0/ and .X 0; 0/ are embedded into .Cm; 0/ it is not clear whether a given

homeomorphismX \ B� �! X 0 \ B�

extends to a homeomorphism

.B�; X \ B�/ �! .B�; X0 \ B�/:

For this question it is useful to look at germs of pairs of complex spaces.Now .X; 0/ can be viewed as the zero locus of a holomorphic map germ

f W .Cm; 0/ �! .Ck; 0/:

This motivates the transition from spaces to functions. We restrict to the case k D 1

(and change the notation, taking .X; 0/ to be the domain of f ).Let

f W .X; 0/ �! .C; 0/

and

f 0 W .X 0; 0/ �! .C; 0/

On local equisingularity 21

be holomorphic map germs. Then f and f 0 have the same topological type if for

0 < ˛ � � � 1

and

D˛ D ft 2 C j jt j � ˛gthe mappings

f W X \ B� \ f �1.D˛/ �! D˛

and

f 0 W X 0 \ B� \ .f 0/�1.D˛/ �! D˛

have the same topological type, i.e. if there is a homeomorphism h such that thefollowing diagram is commutative:

X \ B� \ f �1.D˛/

f��

��

��

h �� X 0 \ B� \ .f 0/�1.D˛/

f 0

��

��

��

D˛

:

Note that the exact choice of � and ˛ plays no role. (In the case k > 1 useDk˛ instead

of D˛ .)

Now let

f W .X; 0/ �! .C; 0/

be as above, .X; 0/ being a subgerm of .CmC1; 0/, and

g W CmC1 �! C

the projection onto the last coordinate. Put

X�;˛ D fz 2 X j k.z1; : : : ; zn; 0/k � �; jf .z/j � ˛g:Then we may ask whether for 0 < max.˛; ˇ/� � � 1 the family of mappings

f W g�1.ftg/ \X�;˛ �! D˛; jt j � ˇ;is locally trivial in the following sense. There is a homeomorphism

h W X�;˛ \ g�1.Dˇ / �! F �Dˇand a continuous mapping

f0 W F �! D˛

22 Helmut A. Hamm


X�;˛ \ g�1.Dˇ /h ��

.f;g/ ��

��

�F �Dˇ

.f0;id/��

��

D˛ �Dˇ

:

(The numbers ˛ and ˇ will always be supposed to be positive.)If this is the case, then the mappings

f W g�1.ftg/ \X�;˛ �! D˛; t 2 Dˇ ;have the same topological type. However this does not imply automatically thatthe corresponding mapping germs at .0; t/, provided that .0; t/ 2 X , have the sametopological type too: for fixed t ¤ 0, we get the topological type for the correspondinggerm if 0 < ˛ � � � jt j, so it is not clear whether we can use the same ˛ and �simultaneously for all t which are small enough.

So we are led to two type of questions. In the context of isolated hypersurfacesingularities, the second question has been dealt with by Lê and Ramanujam, the firstby Timourian, as we will see in the next section.

2 Results

Strong results may be achieved in the case of a family of isolated singularities. Inorder to avoid heavy conditions it is reasonable to start from assumptions which arenatural in view of the aim to be achieved.

Since we want to study the case of isolated singularities we make the followingassumption. Let U be an open neighborhood of 0 in CmC1, X a complex analyticsubset of U which is purely n-dimensional,

Z D U \ .f0g �C/ � X;X nZ smooth, g W CmC1 ! C the projection onto the last coordinate. Let

f W X �! C

be holomorphic, f jZ D 0, and

.f; g/jX nZ W X nZ �! C2

be submersive.For t 2 C put

Ft D X \ g�1.ftg/:The hypothesis implies that the spaces Ft is smooth except at .0; t/ and that f jFt hasat most an isolated singularity at .0; t/.


We want to take the Milnor number�t off jFt at .0; t/ into account. It is reasonableto suppose that X is a complete intersection or at least that rhd.X/ D n, whererhd denotes the rectified homotopical depth, see e.g. [6]. Fix t and suppose that0 < jsj � � � 1. Then the space fz 2 X jg.z/ D t; f .z/ D s; kzk � �g has thehomotopy type of a bouquet of .n�2/-spheres, let �t be the number of these spheres.

We will see in Lemma 5.1 below that�t is constant, cf. also [7] p. 2 (our hypothesisabove excludes the “coalescing” of critical points).

Then we have the following theorem which constitutes essentially a generalizationof the theorem of Timourian in the case of one complex parameter.

Theorem 2.1. For 0 < max.˛; ˇ/� � � 1 the family of mappings

f W g�1.t/ \X�;˛ �! D˛; jt j � ˇ;is locally trivial, i.e. there is a homeomorphism

h W X�;˛ \ g�1.Dˇ / �! F �Dˇand a continuous mapping

f0 W F �! D˛


X�;˛ \ g�1.Dˇ /h ��

.f;g/ ��

��

�F �Dˇ

.f0;id/��

��

D˛ �DˇCf. Timourian [14] in the case X D Cn.

In order to get equisingularity it is important to have the following too.

Theorem 2.2. For 0 < ˛ � ı � jt j � � � 1, the mappings

f W X�;˛ \ g�1.f0g/ �! D˛

andf W Xı;˛ \ g�1.ftg/ �! D˛

have the same topological type.

Cf. Lê and Ramanujam [8] in the case X D Cn.

Now the mappingf W X�;˛ \ g�1.f0g/ �! D˛

represents the topological type of f jF0 at 0. By Theorem 2.1, the mapping

f W X�;˛ \ g�1.ftg/ �! D˛; t ¤ 0;

24 Helmut A. Hamm

has the same topological type, by Theorem 2.2 we may replace � by ı, so we can passto the topological type of f jFt at .0; t/. Altogether, the topological type of f jFt at.0; t/ does not depend on t .

As we will see, we can weaken the hypothesis considerably if we content ourselveswith the following question. Suppose that 0 < max.˛; ˇ/� � � 1. Is the family ofmappings

f W .X�;˛ n Y / \ g�1.ftg/ �! PD˛; jt j � ˇ;locally trivial? Here

Y D f �1.f0g/and

PD˛ D D˛ n f0g:A seemingly weaker question is the following. Is the family of spaces

.X�;˛ n Y / \ g�1.t/; jt j � ˇ;locally trivial?

A natural condition which ensures a positive answer is the following: there is aWhitney regular stratification of .X; Y / such that g�1.f0g/ is transversal to Y at 0,i.e. to the stratum of Y which contains 0 at 0. Note that g�1.f0g/ is then transversalto Y in a neighborhood of 0 too.

We want to have a cohomological condition instead. Suppose that the family islocally trivial: then, for 0 < jt j � ˇ, we have, for all k,

H k..X�;˛ n Y / \ g�1.Dˇ /IC/ ' H k..X�;˛ n Y / \ g�1.ftg/I C/;

i.e.H k..X�;˛ n Y / \ g�1.Dˇ /; .X�;˛ n Y / \ g�1.ftg/I C/ D 0:

For 0 < jt j � ˇ � min.˛; �/ the latter coincides with the stalk of sheaves ofvanishing cycles .ˆkgj�CXnY /0, where j W XnY ! X is the inclusion and j� (orRj�)is the direct image in the derived category. So it is natural to replace the transversalitycondition by the assumption that ˆkgj�CXnY D 0 for all k.

Theorem 2.3. Suppose thatˆkgj�CXnY D 0

for all k. Then the mapping

.f; g/ W X�;˛ \ g�1.Dˇ / �! PD˛ �Dˇdefines a locally trivial fibration for 0 < max.˛; ˇ/ � � � 1. In particular, thefamily of mappings

f W .X�;˛ n Y / \ g�1.t/ �! PD˛; jt j � ˇ;is locally trivial.


Note that the use of the sheaves ˆkgj�CXnY is motivated by the study of absenceof vanishing cycles in the global case: Let g W Cn ! C be a polynomial mapping.More generally, we can look at a surjective morphism g W Z ! C, Z being a smoothaffine variety of dimension n. Let Ng W xZ ! C be a compactification such that Z1 DxZ n Z is locally defined by one equation and j W Z ! xZ be the inclusion. If thesheaves ˆkNgj�CZ vanish we have that g defines a locally trivial fibration over someneighborhood of 0. This follows from Theorem 3.5 of [5], in the special caseZ D Cn

see also [10].

3 The use of vanishing cycles

Here we will give a proof of Theorem 2.3. However we change our point of viewslightly: we will assume already that the sheaves ˆkgj�CXnY vanish in a puncturedneighborhood of 0, a hypothesis which is fulfilled if g�1.f0g/ intersects the strata ofY transversally in some punctured neighborhood of 0.

Let U be an open neighborhood of 0 in CmC1, X a complex analytic subset ofU which is purely n-dimensional, g W CmC1 ! C holomorphic, g.0/ D 0. Afterpassing to the graph of g if necessary we may and do assume that g.z/ D zmC1. Letf W X ! C be holomorphic, f .0/ D 0,

Y D f �1.f0g/; dim Y D n � 1:We assume thatX nY is smooth. As in Section 2, let j W X nY ! X be the inclusion.

PutB� D fz 2 C j k.z1; : : : ; zn; 0/k � �g

andS� D @B�:

(In this and the next section we could also take the usual ball, resp. sphere, instead.)First, using a suitable transversality condition, we obtain the following result; see [4],Theorem 1.1.

Theorem 3.1. Let us fix a Whitney regular stratification of .X; Y /. Assume that thehyperplane fg D 0g intersects all strata of X transversally within some puncturedneighborhood of 0. Then the following conditions are equivalent:

a) for 0 < jt j � � � 1;

�.B� \ .X n Y / \ fg D tg/ D 0Ib) for 0 < max.˛; ˇ/� � � 1,

.f; g/ W B� \X \ f �1. PD˛/ \ g�1.Dˇ / �! PD˛ �Dˇdefines a C1 fibre bundle.

26 Helmut A. Hamm

We want to weaken the hypothesis of this theorem by assuming thatˆgj�CXnY isacyclic in some punctured neighborhood of 0. Note that the condition thatˆgj�CXnYis acyclic outside Y just means that gjX n Y has no critical points which are mappedonto 0. First we have the following lemma.

Lemma 3.2. Suppose that ˆgj�CXnY is acyclic outside 0. Then we have that themapping .f; g/jS� \ .X n Y / has no critical points with jf j � ˛; jgj � ˇ, 0 <max.˛; ˇ/� � � 1.

Proof. Let us fix a Whitney regular stratification ofX such thatY andY \fg D 0g/ areunions of strata. It satisfies automatically Thom’s af -condition, see [1]. If 0 < � � 1,thenS� intersects each stratum ofY \fg D 0g transversally. Letp 2 Y \S�\fg D 0gand let S be the stratum which contains p. If � > 0 is small enough we know that S�intersects S transversally at p. According to [5] Lemma 3.1 we have that fg D 0g istransversal toL ifL D limLnwhereLn is the tangent space to .XnY /\ff D f .pn/gat pn and pn ! p. Because of Thom’s af -condition we have TpS � L. On theother hand, we have TpS � fg D 0g, of course. So TpS � L \ fg D 0g. Now S�is transverse to S , so S� intersects L \ fg D 0g transversally at p. Altogether S� , Land fg D 0g are transversal. If s and t are sufficiently small compared with �, s ¤ 0,we obtain that S� , X \ ff D sg and fg D tg are transversal too. This implies thelemma.

Now we can prove the following which is to a large extent a generalization of [4]Theorem 1.1.

Theorem 3.3. Suppose that ˆgj�CXnY is acyclic outside 0. Then the followingconditions are equivalent:

a) .ˆgj�CXnY /0 is acyclic;

b) for 0 < max.˛; ˇ/� � � 1,

.f; g/ W B� \X \ f �1. PD˛/ \ g�1.Dˇ / �! PD˛ �Dˇdefines a C1 locally trivial fibration;

c) for 0 < jt j � � � 1,

�.B� \ .X n Y / \ fg D tg/ D 0Id) for 0 < jt j � � � 1,

�.B� \X \ fg D tg/ D �.B� \ Y \ fg D tg/Ie) for 0 < jsj � jt j � � � 1,

�.B� \X \ ff D sg/ D �.B� \X \ fg D t; f D sg/I


f) .f; g/jX n Y has no critical points with jf j � ˛; jgj � ˇ, and 0 < max.˛; ˇ/�� 1.

Proof. a)() c) Let us fix a Whitney regular stratification of .X; Y /. By a result ofSullivan [11] we know that �.B� \ .X n Y // D �.S� \ .X n Y // D 0 because thestrata of S� \X are odd-dimensional. Therefore c) is equivalent to

�.B� \ .X n Y /; B� \ .X n Y / \ fg D tg/ D 0;i.e.

�..ˆgj�CXnY /0/ D 0;but CXnY Œn� is perverse (with respect to the middle perversity, see [12]), becauseX nYis smooth of dimension n, so j�CXnY Œn� and henceˆgj�CXnY Œn� too, see [12]. Sinceˆgj�CXnY is acyclic outside 0 by assumption we have that .ˆkgj�CXnY /0 D 0 fork ¤ n, so �..ˆgj�CXnY /0/ D 0 () .ˆgj�CXnY /0 is acyclic. This implies ourassertion.

b) H) a) Obvious.

a) H) b) By Lemma 3.2 we have that .f; g/jS� \ .X n Y / is submersive abovePD˛\Dˇ . On the other hand, Lemma 3.1 of [5] implies that .f; g/jX nY is submersive

in B� \f �1. PD˛/\g�1.Dˇ /. So we can construct vector fields which lead to a localtrivialization.

b) H) f) Obvious.

f) H) b) This follows from Lemma 3.2 and the assumption f), see proof of a) H)b).

c)() d) We look at jf j W B� \ .X n Y / \ fg D tg. We know that

�.B� \X \ fjf j D ˛; g D tg/ D �.B� \X \ fjf j D ˛; g D 0g/D �.B� \X \ f0 < jf j � ˛; g D 0g/D �.B� \ .X n Y / \ fg D 0g/D �.S� \ .X n Y / \ fg D 0g/D 0

by [11] (see above), 0 < jt j � ˛ � � � 1. Because of Lemma 3.2, we obtain:

c) () �.B� \X \ f �1. PD˛/ \ fg D tg/ D 0() �.B� \X \ f �1. PD˛/ \ fg D tg; B� \X \ f �1.@D˛/ \ fg D tg/ D 0() f j.X n Y / \ fg D tg has no critical point in B� \ f �1.D˛/() �.B� \X \ f �1.D˛/ \ fg D tg; B� \ Y \ fg D tg/ D 0() d),

since �.B� \X \ f �1.D˛/ \ fg D tg/ D �.B� \X \ fg D tg/.

28 Helmut A. Hamm

f) H) e) We look at gjB�\X \ff D sg. Because of f) we have no critical pointsin the interior above Dˇ , because of Lemma 3.2 there are no critical points for therestriction to the boundary (above Dˇ ). This implies our assertion, because

�.B� \X \ ff D sg/ D �.B� \X \ ff D sg \ g�1.Dˇ //:

e) H) f) Assume that f) is wrong. Let us then look at the critical locus C of.f; g/j.X n Y / \ VB� . We have dimC � 1; because of Lemma 3.2,

.f; g/jC \ f �1. PD˛/ \ g�1.Dˇ / �! PD˛ �Dˇis proper, so the image is analytic of dimension � 1, because of Sard’s theorem: D 1.The image curve must intersect s D const, where 0 < jsj � ˛. So we have thatgjB� \X \ ff D sg has critical points. These cause a difference between the Eulercharacteristics considered in e), contradiction.

Proof of Theorem 2.3. This follows from the preceding theorem: a) H) b).

Now the question arises how to verify the hypothesis of Theorem 3.3 or condition a).Here the following proposition is useful.

Proposition 3.4. Assume that ˆgj�CXnY is acyclic outside some analytic set ofdimension � k. Then the following conditions are equivalent:

a) ˆgj�CXnY is acyclic outside some analytic subset of dimension � k � 1;

b) for all 1 � j1 < < jk � m there is a subset V of Ck whose complement hasLebesgue measure 0 such that for all z� 2 g�1.f0g/ \ Y with .z�

j1; : : : ; z�

jk/ 2 V

the following holds:

�.B�.z�/\.XnY /\fzj1

D z�j1; : : : ; zjk

D z�jk; g D tg/ D 0; 0 < jt j � � � 1:

Here B�.z�/ D fz 2 CmC1 j kz � z�k � �g.Proof. We take aWhitney regular stratification ofg�1.f0g/\X adapted toˆgj�CXnY .If S is a stratum the set of points where S ! Ck: z 7! .zj1

; : : : ; zjk/ has rank < k

is mapped onto a set of measure 0.

a) H) b) Let 1 � j1 < < jk � m. Take z� 2 g�1.f0g/ \ X such that.z�j1; : : : ; z�

jk/ lies outside a suitable set of measure 0. Then z� is contained in some

stratum S of dimension � k and S ! Ck W z 7! .zj1; : : : ; zjk

/ has rank k at z�, sowe have transversality of fzj1

D z�j1; : : : ; zjk

D z�jkg to S at z�. Therefore

�.B�.z�/\.XnY /\fzj1

D z�j1; : : : ; zjk

D z�jk; g D tg/D��..ˆgj�CXnY /z�/D 0:

b) ) a): Look at a stratum S of dimension k. Choose j1; : : : ; jk such thatS ! Ck: z 7! .zj1

; : : : ; zjk/ is of rank k somewhere. If z� 2 S is such that


.z�j1; : : : ; z�

jk/ lies outside a suitable set of measure 0 we have that

�..ˆgj�CXnY /z�/ D ��.B�.z�/ \ .X n Y / \ fzj1D z�

j1; : : : ; zjk

D z�jk; g D tg/

D 0;so ˆgj�CXnY jS D 0.

4 An auxiliary result

The situation of Section 3 is not sufficient in order to discuss equisingularity: weshould look at a one-parameter family of space or map germs. The family shouldbe given by the function g, the germ should be taken at .0; t/. So it is reasonable tosuppose that Z D .f0g �C/ \ U is contained in Y , see Section 2.

First we will give equivalent conditions which are necessary in order to have thestatement of Theorem 2.1. Recall the notion of rectified homological depth (withcomplex coefficients): rHd.X;C/ D n means that CX Œn� is perverse, cf. [6] Corol-lary 1.10. This condition holds in particular if X is locally a complete intersection ofdimension n or, more generally, if rhd.X/ D n. Let B� , S� be defined as in the lastsection.

Theorem 4.1. Suppose thatˆgj�CXnY is acyclic outside 0,Z � Y , rHd.X;C/ D n,

f CX\fgD0g is acyclic outside 0, and gjsuppˆkf

CX is finite for every k. Then thefollowing conditions are equivalent:

a) .ˆgCX /0 is acyclic, and �t D dim.ˆn�1f

CX\fgDtg/.0;t/ is independent of t ,

b) f CX is acyclic outside Z, and .ˆgj�CXnY /0 is acyclic.

Note that the condition that .ˆgCX /0 is acyclic means that

�.B� \X \ fg D tg/ D 1for 0 < jt j � � � 1.

Proof. Let t ¤ 0. We have . f CX\fgDtg/z D . f CX Œ�1�/z , see [12]. NowCX\fgDtg is perverse, because CX is perverse, so f CX\fgDtg too. By hypothesis,

f CX\fgDtg is acyclic outside a finite set, so ˆkf

CX\fgDtg D 0 for k ¤ n � 1.Similarly for f �sCX\fgDtg, where s ¤ 0, because here we are looking at the

vanishing cycles at critical points of f j.X n Y / \ fg D tg. These are isolated, seeproof of Theorem 3.3 e) H) f).

30 Helmut A. Hamm

Choose s with jsj < ˛ general enough so that f �sCX\fgDtg is acyclic. We obtain

�.B� \X \ fg D tg/ D �.B� \X \ fjf j � ˛; g D tg/D �.B� \X \ ff D s; g D tg/C .�1/n�1 X

z W jf .z/j�˛dim.ˆn�1

f �f .z/CX\fgDtg/z:

Note that

�.B� \X \ ff D s; g D tg/ D �.B� \X \ ff D s; g D 0g/D 1C .�1/n�0;

and

�.B� \X \ fg D tg/ D 1C .�1/n�1 dim.ˆngCX /0:

Altogether, we obtain the following equation:

dim.ˆngCX /0 D ��0 C �t CX

z¤.0;t/dim.ˆn�1

f �f .z/CX\fgDtg/z: (*)

a) H) b) Equation (*) yields

0 DX

z¤.0;t/dim .ˆn�1

f �f .z/CX\fgDtg/z :

So ˆn�1f

CX\fgDtg is acyclic outside .0; t/, hence f CX jfg D tg too. Furthermore,by Theorem 3.3 f) H) a) we get that .ˆgj�CXnY /0 is acyclic.

b) H) a) By the second assumption we obtain from Theorem 2.3, a) H) d):

�.B� \X \ fg D tg/ D �.B� \ Y \ fg D tg/; 0 < jt j � � � 1;

which means

1C .�1/n�1 dimHn�1.B� \X \ fg D tgIC/D 1C .�1/n�2 dimHn�2.B� \ Y \ fg D tgIC/:

Here we use the fact that rHdCY D n�1. ThereforeHn�1.B�\X\fg D tgIC/ D 0,which implies .ˆgCX /0 D 0.

So (*) yields

0 D ��0 C �t CX

z¤.0;t/dim .ˆn�1

f �f .z/CX\fgDtg/z;

and the sum on the right vanishes by the assumption.Indeed note that f CX is acyclic outsideZ, so f .z/ ¤ 0 if .ˆn�1

f �f .z/CX\fgDtg/z¤ 0, so z is a critical point of .f; g/jX nY , which contradicts Theorem 3.3, a) H) f).

So �0 D �t for t ¤ 0.


5 Proof of Theorem 2.1 and Theorem 2.2

Now we want to follow the arguments of Lê and Ramanujam [8]. Here, we needstronger hypotheses.

In particular we will use the notion of rectified homotopical depth rhd introducedby A. Grothendieck, see [6]. For example, rhd.X/ D n as soon as X is locally acomplete intersection of dimension n.

Assume thatX nZ and Y nZ are smooth and that g�1.f0g/ intersects these spacestransversally. Then ˆgj�CXnY is concentrated upon Z \ g�1.f0g/ D f0g. Assumefurthermore that rhd.X/ D n, n ¤ 4. Let �t be defined as in Theorem 4.1.

First, we have the following lemma (where we could use rHd.X;C/ instead ofrhd.X/).

Lemma 5.1. The following conditions are equivalent:

a) .ˆgCX /0 is acyclic, and �t is constant;

b) .f; g/ W X nZ ! C2 is submersive above D˛ �Dˇ .

Proof. By hypothesis, ˆgj�CXnY and f CX\fgD0g are acyclic outside 0, and thesheaves f CX are acyclic outside Z, in particular gjsuppˆk

fCX is finite.

Because of Theorem 4.1, a)() .ˆkgj�CXnY /0 D 0 for all k. The equivalenceof Theorem 3.3 a) () f) implies that the last condition is equivalent to the conditionthat .f; g/ W X n Y has no critical points above D˛ �Dˇ .

The last condition can be rewritten as follows: .f; g/ W X n Z has no criticalpoints above D˛ � Dˇ , because gjY n Z is submersive. Altogether we obtain theassertion.

Now we have the following result which is related to [8]; here

Dˇ .t0/ D ft 2 C j jt � t0j � ˇg:Theorem 5.2. Assume that �t is constant. Then there is a homeomorphism h suchthat the following diagram is commutative:

X�;˛ \ g�1.Dˇ .t0//h ��

.f;g/ ��

��

��Xı;˛ \ g�1.Dˇ .t0//

.f;g/

D˛ �Dˇ .t0/

;

where 0 < max.˛; ˇ/� ı � jt0j � � � 1.

Proof. We use the definition of B� and S� introduced before Theorem 3.1!Let t 2 Dˇ .t0/, s ¤ 0, n � 5. First of all, we observe that S� \ Y \ fg D 0g is

.n � 4/-connected, hence simply connected, because of the local Lefschetz theoremfor X , see [6] Theorem 2.9: We have rhd.X/ D n. Therefore we have that S� \X is

32 Helmut A. Hamm

.n�2/-connected and has the homotopy type of a space obtained fromS�\Y \fg D 0gby attaching cells of dimension � n � 2.

The same holds for S� \ Y \ fg D tg because this space is homeomorphic to thespace S� \Y \fg D 0g before. Similarly, Sı \Y \fg D tg is simply connected too.By Morse theory (use z 7! �kzk2), .B� n VBı/\ Y \ fg D tg has the homotopy typeof a space obtained from S� \ Y \ fg D tg by attaching cells of dimension � n� 2,so it is simply connected too. (**)

Furthermore, B� \ X \ fg D 0g is contractible, and by stratified Morse theory,we have that B� \ X \ fg D 0g has the homotopy type of a space obtained fromB�\X\fg D 0; f D sg by attaching�0 .n�1/-spheres, soB�\X\fg D 0; f D sghas the homotopy type of a bouquet of �0 .n � 2/-spheres. The same holds for thehomeomorphic spaceB�\X \fg D t; f D sg. Similarly, Bı \X \fg D t; f D sghas the homotopy type of a bouquet of �t .n � 2/-spheres. Using z 7! kzk2 as aMorse function we see thatB�\X\fg D t; f D sg has the homotopy type of a spaceobtained from Bı \X \ fg D t; f D sg by attaching cells of dimension � n� 2. So

H k.B� \X \ fg D t; f D sg; Bı \X \ fg D t; f D sgIZ/ D 0for k > n � 2. Since �0 D �t by assumption we get that

H k.B� \X \ fg D t; f D sgIZ/ ' H k.Bı \X \ fg D t; f D sgIZ/;i.e.

Hk..B� n VBı/ \X \ fg D t; f D sg; Sı \X \ fg D t; f D sgIZ/' H k.B� \X \ fg D t; f D sg; Bı \X \ fg D t; f D sgIZ/D 0

for all k. The same holds for homology instead of cohomology. Also, by duality, wecan deduce that

Hk..B� n VBı/ \X \ fg D t; f D sg; S� \X \ fg D t; f D sgIZ/ D 0for all k. (***)

Now .f; g/j.B� n VBı/\X defines a C1 fibre bundle over D˛ �Dˇ .t0/ because

gj.Y n f0g/ �C is a submersion. The fibre is .B� n VBı/ \X \ fg D t; f D sg, it issimply connected, the same holds for its boundary components: this follows from (**)because we may pass to the case s D 0. The inclusion of each boundary componentinto .B� n VBı/\X\fg D t; f D sg defines a homotopy equivalence, by Whitehead’stheorem, see [13], and (***). By the h-cobordism theorem (cf. [9]) we conclude that.B�n VBı/\X\fg D t; f D sg is diffeomorphic to .S�\X\fg D t; f D sg/�Œ0; 1�.So there is a diffeomorphism of .B� n VBı/ \ X \ f �1.D˛/ \ g�1.Dˇ .t0// onto.S� \ X \ f �1.D˛/ \ g�1.Dˇ .t0/// � Œ0; 1� which is compatible with .f; g/. Thisimplies our assertion.


The case n � 3 is easier: Assume n D 3. Put

A D .B� n VBı/ \X \ fg D t; f D sg;B D Sı \X \ fg D t; f D sg;B 0 D S� \X \ fg D t; f D sg:

We have that H0.BIZ/ ' H0.AIZ/, so the irreducible components Aj of A and Bjof B correspond to each other. Now we haveHk.Bj IZ/ ' Hk.Aj IZ/. FurthermoreBj must be homeomorphic to S1. Similarly for B 0, B 0

j instead of B , Bj . So Ajhas the homotopy type the complement of two points in a compact Riemann surface.Let g be its genus, then �.Aj / D �2g. On the other hand, �.Aj / D �.Bj / D 0.Therefore g D 0, and Aj is diffeomorphic to the complement of two disjoint disks inthe Riemann sphere. So Aj ' S1 � Œ0; 1�, as expected.

In the case n � 2 we must have A D B D B 0 D ;.

Proof of Theorem 2.2. This follows from Lemma 5.1 and Theorems 5.2 and 5.3 below.In fact,Y nZ is smooth andg�1.f0g/ intersects the spacesXnZ andY nZ transversallybecause .f; g/ W X nZ ! C2 is a submersion.

Now we want to prove 2.1. Using Lemma 5.1 we reformulate it as follows.

Theorem 5.3. Under the equivalent hypotheses of Lemma 5.1, we have that, for0 < max.˛; ˇ/� � � 1 the family of mappings f W g�1.ftg/\X�;˛ ! D˛ , jt j � ˇ,is locally trivial, i.e. there is a homeomorphism h W X�;˛\g�1.Dˇ /! F �Dˇ and acontinuous mapping f0 W F ! D˛ such that the following diagram is commutative:

X�;˛ \ g�1.Dˇ /h ��

.f;g/ ��

��

�F �Dˇ

.f;id/��

��

D˛ �Dˇ

:

Example. We modify the example of Briançon and Speder [2]. Let

X D f.z1; z2; z3; t / 2 C4 j z53 C tz62z3 C z72z1 C z151 D 0g;f .z1; z2; z3; t / D z2;g.z1; z2; z3; t / D t:

Then we may apply Theorem 5.3: in particular,�t is constant because of the weightedhomogeneous situation; furthermore, .ˆgCX /0 is acyclic. By [2], the pair .X nZ;Z/does not satisfy Whitney’s regularity condition, so the local triviality is not merely aconsequence of stratification theory.

34 Helmut A. Hamm

Remark. The hypothesis that .ˆgCX /0 is acyclic is fulfilled in particular whenX D CmC1 (with n D m C 1, of course). Then it is sufficient to suppose that �tis constant, see Lemma 5.1.

It is possible that we may apply Theorem 5.2 but not Theorem 5.3.

Example. Let

X D f.x; y; t/ 2 C3 jy2 D x2.x � t/g;f .x; y; t/ D x:

Then Y D f0g�C, . f CX\fgDtg/.0;t/ has dimension 1, so �t is constant. However,the conclusion of 5.3 does not hold.

On the other hand, Theorem 5.2 is applicable.

Proof of Theorem 5.3. Put

�.z/ D k.z1; : : : ; zn; 0/k:Let † be the critical set of .�;Reg/jY \ fImg D 0g n Z. After shrinking U ifnecessary, each branch of the closure of † is parametrized by a real analytic curve 7! ./, with ./ D 0, and we may choose the parametrisation in such a way that�..// D ˙� . In this way we see that there is k > 0 such that along † we havethe inequality jRegjk < � < jRegj1=k .

We may assume ˇ < 1 and 2ˇ1=k < �.On .Y n Z/ \ fIm g D 0g, in a neighborhood of fjRegj � ˇ; � � 1=2jRegjkg

we can find a vector field v with d�.v/ 0; dg.v/ 1. Similarly in a neighborhoodof fjRegj � ˇ; 2jRegj1=k � � � �g.

Note that Regj.Y nZ/ \ fImg D 0; jRegjk � � � jRegj1=k ; 0 < jRegj � ˇgdefines a fibre bundle over Œ�ˇ; ˇ� n f0g which is trivial over Œˇ; 0Œ, resp. �0; ˇ�; bythe h-cobordism theorem (for n � 5) or a simple direct argument (for n � 3), seeproof of Theorem 5.2, the fibre is diffeomorphic to F� � Œ0; 1�, resp. FC � Œ0; 1�,where fjRegjk D �g corresponds to F˙�f0g and fjRegj1=k D �g to F˙�f1g. Theprojection onto Œ0; 1� induces a mapping

W .Y nZ/ \ fIm g D 0; jRegj � ˇ; jRegjk � � � jRegj1=kg �! R:

Along jRegjk D � and jRegj1=k D � we have no z with dz.�jfg D constg/ D�dz. jfg D constg/, with� � 0. Therefore there is a vector fieldv on a neighborhoodof fjRegj � ˇ; 1=2jRegjk � � � jRegjkg in .Y n Z/ \ fImg D 0g such that�g d�.v/ � 0, dg.v/ D 1, and d�.v/ D 0 along f1=2jRegjk D �g, d .v/ D �1=galong fjRegjk D �g. On a neighborhood of fjRegj � ˇ; jRegjk � � � jRegj1=kgin .Y nZ/ \ fImg D 0g we can find a vector field v such that dg.v/ D 1, d .v/ D�1=g, and �gd�.v/ � 0 along f� D jRegjkg or f� D jRegj1=kg. Finally, on aneighborhood of fjRegj � ˇ; jRegj1=k � � � 2jRegj1=kg in .Y nZ/\ fIm g D 0g


we can find a vector field v such that �g d�.v/ � 0, dg.v/ D 1, and d�.v/ D 0

along f2jRegj1=k D �g, d .v/ D �1=g along fjRegj1=k D �g.We may arrange that the vector fields match together. The resulting vector field can

be extended on .Y nZ/\fjgj � ˇ; � � �g such that it is controlled on a neighborhoodof Z \ fImg ¤ 0g, dg.v/ 1, and along f� D �g, d�.v/ 0. Finally we mayextend to .X nZ/\ fjf j � ˛; jgj � ˇ; � � �g such that dg.v/ 1; df .v/ 0 and,along � D �, d�.v/ 0. Note here that .f; g/jX n Y and .f; g/jX \ f� D �g arewithout critical points in this set.

Now fix ˇ > 0 sufficiently small. Then the flow ˆ corresponding to v is definedon f.z; t/ j jReg.z/j � ˇ; t 2 Œ�ˇ � Reg.z/; ˇ � Reg.z/�g. Assume that the lowerbound of the interval is not correct. Then there is a t0 > 0 and an integral curve c suchthat c.t/ is defined for �t0 < t � 0 and .0; t1/ is an accumulation point of c.t/ fort ! �t0. Then necessarily t1 D 0 andg.c.t// D tCt0. We must have that c is a curvein Y \ fIm g D 0g. If �.c.t// < .Reg.c.t///k for all these t we have that �.c.t//is monotonously decreasing, contradiction. So there is a t� with .Reg.c.t�///k ��.c.t�//. Assume that .Reg.c.t///k � �.c.t// � .Reg.c.t///1=k for�t0 < t � t�;then

d . Pc.t// D � 1

g.c.t//D � 1

t C t0for these t , so

.c.t�// � .c.t// Dt�Zt

d . Pc.t//dt

D �t�Zt

1

t C t0dt

D ln.t C t0/ � ln.t� C t0/ �! �1for t ! �t0, in contradiction to the fact that .c.t�// and .c.t// are contained inŒ0; 1�. So we must have a t�� with�.c.t��// � .Reg.c.t��///1=k; then for t � t�� wehave that �.c.t// is monotonously decreasing too, which gives again a contradiction.Similarly if the upper bound is not correct.

Now ˆ can be extended continuously to y defined by

y ..0; t/; / D .0:t C /on X \ fjf j � ˛; jgj � ˇ; � � �g. Let .pl/ be a sequence in the complementof Z which converges to p 2 Z. Then ˆ.pl ; t / cannot accumulate to a point inthe complement of Z. Otherwise we get a contradiction using the continuity of theopposite flow.

Similarly we can proceed interchanging the role of Reg and Im g: we find asuitable vector field w on B� \X nZ such that dg.w/ i; df .w/ 0.

In this way we obtain the desired trivialization.

36 Helmut A. Hamm

As a consequence of Theorem 2.2 and Theorem 2.1, we have that the topologicaltype of f jX \ g�1.ftg/ at .0; t/ does not depend on t , as we have seen in Section 2.In fact we can say more.

Corollary 5.4. Under the hypothesis of Theorem 5.3, the topological type of .f; g� t/at .0; t/ does not depend on t , where jt j < ˇ � 1.

Proof. Assume 0 < max.˛; ˇ0/� jt j < ˇ � � � 1.Then the topological type of .f; g/ at 0 is represented by the topological type of

the mapping

.f; g/ W X \ f� � �g \ f �1.D˛/ \ g�1.D0ˇ / �! D˛ �D0

ˇ :

This mapping has, by Theorem 5.3, the same topological type as

.f; g � t/ W X \ f� � �g \ f �1.D˛/ \ g�1.D0ˇ .t// �! D˛ �D0

ˇ ;

which represents, by Theorem 5.2, the topological type of .f; g � t/ at .0; t/.

Of course, the condition that �t is constant is essential in Lemma 5.1 and Theo-rem 5.2 and Theorem 5.3:

Example. Let

X D C2;

f .x; t/ D x.x � t/;g.x; t/ D t:

Then .ˆkgCX /0 D 0 for all k, �0 D 1; �t D 0, for t ¤ 0.

References

[1] J. Briançon, P. Maisonobe, and M. Merle, Localisation des systèmes différentiels, strati-fications de Whitney et condtion de Thom, Invent. Math. 117 (1994), 531–550. 26

[2] J. Briançon and J.-P. Speder, La trivialité topologique n’implique pas les conditions deWhitney, C. R. Acad. Sci. Paris Sér. A 280 (1975), 365–367. 33

[3] M. Goresky and R. MacPherson, Stratified Morse theory, Ergebnisse der Mathematik undihrer Grenzgebiete 3, Springer, Berlin 1988. 20

[4] H. A. Hamm, Complements of hypersurfaces and equisingularity, in Singularity theory –Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedings of the 2005 MarseilleSingularity School and Conference, CIRM, Marseille, France, 24 January – 25 Febru-ary 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman, WorldScrientific, Singapore 2007, 625–649. 25, 26


[5] H. A. Hamm, Euler characteristics and atypical values, in Real and complex singularities.Selected papers from the 10 th Workshop held at São Paulo University, São Carlos, July 27–August 2, 2008, ed. by M. Manoel, M. C. Romero Fuster, and C. T. C. Wall, London Math.Soc. Lecture Notes 380, Cambridge University Press, Cambridge 2010, 167–184. 25, 26,27

[6] H. A. Hamm and D. T. Lê, Rectified homotopical depth and Grothendieck conjectures, inThe Grothendieck Festschrift. A collection of articles written in honor of the 60 th birthdayof Alexander Grothendieck, vol. II, ed. by P. Cartier, L. Illusie, N. M. Katz, G. Laumon,Y. Manin, and K. A. Ribet, Birkhäuser, Boston 1990, 311–351. 23, 29, 31

[7] H. C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), 1–13. 23

[8] D. T. Lê and C. P. Ramanujam, The invariance of Milnor’s number implies the invarianceof the topological type, Amer. J. of Math. 98 (1970), 67–78. 23, 31

[9] J. Milnor, Lectures on the h-cobordism theorem. Princeton University Press, Princeton1965. 32

[10] A. Parusinski, A note on singularities at infinity of complex polynomials, in Symplecticsingularities and geometry of gauge fields. Papers from the Banach Center Symposium heldinWarsaw, 1995, ed. by R. Budzynski, S. Janeczko, W. Kondracki, andA. F. Künzl, BanachCenter Publications 39, Polish Academy of Sciences, Inst. of Mathematics, Warsaw 1997,131–141. 25

[11] D. Sullivan, Combinatorial invariants of analytic spaces, in Proceedings Liverpool Singu-larities Symposium I (1969/70), ed. by C. T. C. Wall, Lecture Notes in Mathematics 192,Springer, Berlin 1971, 165–168. 27

[12] J. Schürmann, Topology of singular spaces and constructible sheaves, Instytut Mate-matyczny Polskiej Akademii Nauk. Monografie Matematyczne 63, Birkhäuser, Basel2003. 27, 29

[13] E. H. Spanier, Algebraic topology, McGraw-Hill, New York etc. 1966. 32

[14] J. G. Timourian, The invariance of Milnor’s number implies topological triviality, Amer.J. Math. 99 (1977), 437–446. 23

Jet schemes of homogeneous hypersurfaces

Shihoko Ishii,�Akiyoshi Sannai,�and Kei-ichi Watanabe

Graduate School of Mathematical Science, University of TokyoKomaba, Meguro, Tokyo, 153-8914, Japan


Graduate School of Mathematics, Nagoya UniversityFurocho, Chikusaku, Nagoya, 464-8602, Japan


Department of Mathematics, College of Human and Science, Nihon UniversitySetagaya, Tokyo, 156-0045, Japan


Abstract. This paper studies the singularities of jet schemes of homogeneous hypersurfaces ofgeneral type. We obtain the condition of the degree and the dimension for the singularities of thejet schemes to be of dense F -regular type. This provides us with examples of singular varietieswhose m-jet schemes have rational singularities for every m.

1 Introduction

The concept jet schemes over an algebraic variety was introduced by Nash in hispreprint in 1968 which is later published as [10]. These spaces represent the nature ofthe singularities of the base space. In fact, papers [1], [2], [8], and [9] by Mustata, Ein,and Yasuda show that geometric properties of the jet schemes determine properties ofthe singularities of the base space. To summarize, their results among others are asfollows.

LetX be a variety of locally a complete intersection over an algebraically closedfield of characteristic zero. Then Xm is of pure dimension (resp. irreducible,normal) for all m � 1 if and only if X has log-canonical (resp. canonical,terminal) singularities.

According to this form, it is natural to formulate the question.

Problem 1.1. Does the following hold? X is non-singular if and only if Xm has atworst certain “mild” singularities for every m � 1.

�Partially supported by JSPS grant in aid (B) 22340004.�Partially supported by JSPS research fellow 08J08285.�Partially supported by JSPS grant in aid (C) 20540050.




40 Shihoko Ishii, Akiyoshi Sannai, and Kei-ichi Watanabe

Does the bound of a certain invariant of the singularities on Xm characterize thesmoothness of X?

The easiest candidate for “certain mild singularities” is a rational singularity. Inthis paper, we show that a rationality is not appropriate for the required statementin the problem. This is proved by providing with counter examples. We study thesingularities of the jet schemes of homogeneous hypersurface of general type andobtain the condition of the degree and the dimension for the singularities of the jetschemes to be dense F -regular type.

Theorem 1.2. LetX be a hypersurface in ANk

over a field k of characteristic 0 definedby a polynomial of general type of degree d , i.e.,

f .x1; : : : ; xN / DX

i1�i2;��id�i1;i2;:::;idxi1xi2 xid ;

where f�i1;i2;:::;id g are algebraically independent over Q. If d2 � N , then the jetscheme Xm has at worst rational singularities for every m 2 N.

A rational singularity is defined by using a resolution of the singularities. Since it isalmost impossible to construct a resolution of the singularities of the jet scheme evenfor the simplest singularities on the base variety because of too many variables on thejet scheme, we use the positive characteristic method. The theorem shows examplesof singular X whose jet schemesXm for allm have at worst rational singularities. Wealso show that X is non-singular if and only if the F -pure threshold does not changebetween Xm’s for different m.

Theorem 1.3. Let X be a variety of locally a complete intersection at 0 over a fieldof characteristic p > 0. For m < m0, assume also Xm; XmC1; : : : ; Xm0 are completeintersections at the trivial jets 0m; : : : ; 0m0 . Then, the following are equivalent:

(i) .X; 0/ is non-singular;

(ii) fpt.Xm; 0m; 0m/ D fpt.Xm0 ; m0m�1.0m/; 0m0/, where m0m W Xm0 ! Xm is

the truncation morphism.

Throughout this paper the base field k is an algebraically closed field.

2 Preliminaries on jet schemesand positive characteristic methods

2.1. For a schemeX of finite type over an algebraically closed field k, we can associatethe space ofm-jet (or them-jet scheme) Xm for everym 2 N. The exact definition of

Jet schemes of homogeneous hypersurfaces 41

the m-jet scheme and the basic properties can be seen in [6]. We use the notation andthe terminologies in [6]. The canonical projection Xm ! X is denoted by �m.

If X is a closed subscheme of ANk

defined by an equation f D 0, then the m-jetscheme Xm is defined in

A.mC1/Nk

D Spec kŒx.j /i j 1 � i � N; j D 0; 1; : : : m�by the equations fF .j / D 0gjD0;:::;m. Here, the F .j / 2 kŒx.j /i j 1 � i � N; j D0; 1; : : : m� is defined as follows:

f�X

jx.j /1 tj ; : : : ;

Xjx.j /N tj

�DX

jF .j /tj :

For the simplicity of the notation, we write x.j / D .x.j /1 ; : : : ; x

.j /N /. For a point

P 2 X , let Pm 2 Xm be the trivialm-jet at P . In particular if P is the origin 0 2 X �ANk

, then 0m is defined by the maximal ideal .x.0/; : : : ; x.m// � kŒx.0/; : : : ; x.m/� in

A.mC1/Nk

.

2.2. The Frobenius map of rings of positive characteristic has been important tool tostudy the singularities of positive characteristic. The concepts F -pure, strongly F -regular, weakly F -regular and F -rational appear in this stream. These notions haveclose relations with rationality and log-canonicity: A singularity is of denseF -rationaltype (i.e. it is F -rational by the reduction to characteristic p for infinitely many primenumber p) if and only if it is rational by Smith [11], Hara [4], and Mehta and Srinivas[7]. If a normal Q-Gorenstein singularity is of dense F -pure type (i.e., it is F -pure bythe reduction to characteristic p for infinitely many prime number p), then it is log-canonical by Hara and Watanabe [5]. In the Gorenstein case, the three notions stronglyF -regular, weakly F -regular and F -rational coincide. When we restrict ourselves inthe case of a complete intersection, we call it justF -regular. The definitions of F -pureand F -regular can be found in the papers above and we do not repeat them here.

Lemma 2.3. The m-jet scheme Xm is F -pure (resp. strongly F -regular, rational)along the fiber��1

m .P / if and only ifXm isF -pure (resp. stronglyF -regular, rational)at Pm.

Proof. Note that these conditions, F -pure, strongly F -regular, rational, are open con-ditions. Therefore, if Xm has one of these conditions at Pm, then Xm has that onan open neighborhood U � Xm of Pm. Remember that the multiplicative algebraicgroup A1

kn f0g acts on Xm and the closure of the orbit of every point y in ��1

m .P /

contains Pm (see, for example, [6]). This shows that on Xm there is an isomorphismwhich sends y into U . Hence, Xm has the condition at y.

Lemma 2.4 ([12, Lemma 3.9]). Let .R;m/ be a local ring at a closed point of anon-singular variety over an algebraically closed field of characteristic p and I � Ran ideal. Fix any ideal a � R and any real number t � 0. Write S D R=I .


(i) The pair .S; .aS/t / is F-pure if and only if for all large q D pe � 0, we haveabt.q�1/c.I Œq� W I / 6� mŒq�.

(ii) The pair .S; .aS/t / is strongly F-regular if and only if for every elementg 2 RnI ,there exists q D pe > 0 such that gadtqe.I Œq� W I / 6� mŒq�.

In the case of a complete intersection, we can regard the following criteria of Feddertype as the definition of F -pure and F -regular.

Corollary 2.5. If .S;m/ is a regular local ring of characteristic p > 0, f1; f2; : : : ; fris an S -sequence and f DQr

iD1 fi , then the following are equivalent:

(i) S=.f1; : : : ; fr/ is F -pure (resp. F -regular)

(ii) f p�1 62 mŒp� (resp. for any non-zero g 2 S , there is q D pe > 0 such thatgf q�1 62 mŒq�).

Proof. The statement on F -purity is in [3]. If I D .f1; : : : ; fr/ is generated byS -regular sequence, then I Œq� W I D I Œq� C f q�1S and our assertion on F -regularityfollows from Lemma 2.4.

To apply the criteria, we need to show that our jet schemes are complete intersec-tions. The following is a characteristic free statement and is a refinement of a specialcase of the statement obtained by Mustata [8] for characteristic zero.

Lemma 2.6. Let X be a hypersurface of ANk

defined by a homogeneous polynomialf of degree d . Assume X has an isolated singularity at the origin 0 2 X . Then, itfollows:

(i) if d � N , then Xm is not irreducible for every m � N � 1;

(ii) if d � N �1, thenXm is irreducible, therefore a complete intersection, for everym 2 N.

Proof. First of all, we note that for a hypersurface X with the isolated singularity at0, the jet scheme Xm is irreducible if and only if

dim ��1m .0/ < .mC 1/.N � 1/: (2.1)

Indeed, as dim ��1m .X n f0g/ D .m C 1/.N � 1/, “only if” part is trivial. For the

“if” part, note that Xm is defined by mC 1 equations in A.mC1/Nk

. Therefore, everyirreducible component ofXm has dimension greater than or equal to .mC 1/.N � 1/.If we assume the inequality (2.1), then ��1

m .0/ does not provide with an irreduciblecomponent of Xm.

For the proof of (i), assume d � N . The fiber ��1m .0/ is defined by

F .0/.0/; F .1/.0; x.1//; : : : ; F .m/.0; x.1/; : : : ; x.m//

on AmNkD Spec kŒx.1/; : : : ; x.m/� and the first d polynomials are trivial because

F .j / is homogeneous of degree d and of weight j, therefore every monomial in F .j /


(j < d ) has the factor x.0/i for some i , where the weight of a monomialQx.j /i is

defined asPj.

Therefore, for m � N � 1dim ��1

m .0/ � mN �maxf0; .mC 1/ � dg � .mC 1/.N � 1/:Hence, Xm is not irreducible for m � N � 1.

For the proof of (ii), assume d � N � 1. Form such thatm � d � 1, as we see inthe previous argument, ��1

m .0/ D AmNk

. As m � N � 2, the inequality (2.1) holds,therefore Xm is irreducible. For m such that m � d , we will show it by inductionon m. Assume that X0 D X; : : : ; Xm�1 are irreducible. We note that for j � d

F .j /.0; x.1/; : : : ; x.j // D F .j�d/.x.1/; : : : ; x.j�dC1//;

because f is homogeneous of degree d. Since ��1m .0/ is defined by F .j /.0; x.1/; : : : ;

x.j // (j D d; : : : ; m) in Spec kŒx.1/; : : : ; x.m/�, we obtain

��1m .0/ D Spec kŒx.1/; : : : ; x.m/�=.F .j�d/.x.1/; : : : ; x.j�dC1///j�dD0;::;m�d

' Xm�d �A.d�1/Nk

:

By this we have dim ��1m .0/ D .m � d C 1/.N � 1/C .d � 1/N and it follows the

inequality (2.1). Now we obtain the irreducibility of Xm and in this case we have thecodimension of Xm equal to the number of the defining equation in A.mC1/N

k.

3 Singularities of the jet schemes

Definition 3.1. Under the notation in 2.1, let k be a field of characteristic zero and pa prime number. Let m be the maximal ideal .x.0/; x.1/; : : : ; x.m// � kŒx.0/; x.1/; : : : ;x.m/�. Take a polynomial F in the ring. A monomial x 2 kŒx.0/; x.1/; : : : ; x.m/� iscalled a good monomial for .F; p/ if x 62 mŒp� and x 2 F p�1 by modulo p reduction.Here “x 2 F p�1” means x appears in F p�1 with non-zero coefficient.

Theorem 3.2. LetX be a hypersurface in ANk

over a field k of characteristic 0 definedby a polynomial of general type of degree d , i.e.

f .x1; : : : ; xN / DX

i1�i2��id�i1;i2;:::;idxi1xi2 xid ;

where f�i1;i2;:::;id g are algebraically independent over Q. If d2 � N , then the jetscheme Xm is dense F -regular type for every m 2 N.

Proof. Fix m 2 N. Let p be a prime number satisfying p > m.d � 1/ C d . ByLemma 2.6, we may assume that Xm is a complete intersection. For the polynomialf 2 kŒx1; : : : ; xN �, let F .j / 2 kŒx.0/; x.1/; : : : ; x.j /� be as in 2.1 and put F DQmjD0 F .j /. Let g be any polynomial in kŒx.0/; x.1/; : : : ; x.m/�. We will show that


there exist e > 0 and a monomial M 2 F pe�1 with gM 62 mŒpe� by modulo p. For

any e, we can decompose

pe � 1 D .pe � pe�1/C .pe�1 � pe�2/C .pe�2 � 1/:

Let a D pe � pe�1; b D pe�1 � pe�2; c D pe�2 � 1. Define three monomialsL1; L2; L3 2 F DQm

jD0 F .j /. First pick up the term L1.j / from F .j / as follows:

L1.0/ D �1;2;:::;dx.0/1 x.0/2 x.0/d ;

L1.1/ D �dC1;dC2;:::;2dx.0/dC1 x.0/dC2 x.0/2d�1x.1/

2d;

L1.2/ D �2dC1;2dC2;:::;3dx.0/2dC1 x.0/2dC2 x.0/3d�2x.1/

3d�1x.1/

3d;

:::

L1.d � 1/ D �d2�dC1;d2�dC2;:::;d2x.0/

d2�dC1 x.1/

d2�dC2 x.1/

d2�1x.1/

d2 ;

L1.d/ D �1;2;:::;dx.1/1 x.1/2 x.1/d�1x.1/

d;

L1.d C 1/ D �dC1;dC2;:::;2dx.1/dC1 x.1/dC2 x.1/2d�1x.2/

2d;

:::

L1.2d/ D �1;2;:::;dx.2/1 x.2/2 x.2/2d�1x.2/

2d;

:::

Define L1 DQmjD0L1.j /=(coefficients). Then, note that every variable x.j /i appears

in L1 at most once. We can see that La1 2 F a by modulo p, by noting that thecoefficients �i1;:::;id are algebraically independent over Q.

Next, pick up the term L2.j / from F .j / as follows

L2.0/ D �2d;:::;2d .x.0/2d /d ;

L2.1/ D d�2d;:::;2d .x.0/2d /d�1x.1/2d;

L2.2/ D d�2d;:::;2d .x.0/2d /d�1x.2/2d;

:::

L2.m/ D d�2d;:::;2d .x.0/2d /d�1x.m/2d:


Define L2 D QmjD0L2.j /=(coefficients). Then, note that a variable with positive

weight (i.e., x.j /2d

.j > 0/) appears in L2 at most once and a variable x.0/2d

appearsm.d � 1/C d times in L2. We can see that Lb2 2 F b by modulo p.

Finally, pick up the term L3.j / from F .j / as follows:

L3.0/ D �d;:::;d .x.0/d /d

L3.1/ D d�d;:::;d .x.0/d /d�1x.1/d

L3.2/ D d�d;:::;d .x.0/d /d�1x.2/d

:::

L3.m/ D d�d;:::;d .x.0/d /d�1x.m/d:

Define L3 D QmjD0L3.j /=(coefficients). Then, note that a variable with positive

weight (i.e., x.j /d

.j > 0/) appears in L3 at most once and a variable x.0/d

appearsm.d � 1/C d times in L3. We can see that Lc3 2 F c by modulo p.

DefineM D La1 Lb2 Lc3. Noting that .ps�1/Š has exactly .Ps�1iD1.pi�1//-powers

of p as a factor and .ps � ps�1/Š has exactly .ps�1 � 1/-powers of p as a factor forevery positive integer s, we obtain that .pe � 1/Š=.aŠbŠcŠ/ does not have p as a factor.Hence, it follows that M 2 F pe�1 by modulo p. Every variable of weight 0 appearsin M at most maxfb.md �mC d/; aC c.md �mC d/g times and

pe �maxfb.md �mC d/; aC c.md �mC d/g �! 1 .e !1/:On the other hand, every variable of positive weight appears inM at most aCb times(here, we used the fact that aC c � aC b). We can also see that

pe � .aC b/ �!1 .e !1/:Therefore, for any polynomial g 2 kŒx.0/; x.1/; : : : ; x.m/� we obtain gM 62 mŒp

e� forsufficiently large e.

Corollary 3.3 (Theorem 1.2). Let k be a field of characteristic zero. Let X be ahypersurface in AN

kdefined by a homogeneous polynomial of general type of degree

d . If d2 � N , then the jet scheme Xm has at worst rational singularities for everym 2 N.

Remark 3.4. It is expected that Theorem 3.2 and Corollary 3.3 also hold for thehypersurface X of Fermat type of degree d such that d2 � N.

Theorem 3.5. Assume char k D p > 0. Let X be a hypersurface in ANk

defined by ahomogeneous polynomial f 2 kŒx1; : : : ; xN � of degree d . If the jet scheme Xm is acomplete intersection and F -pure for every m 2 N, then d2 � N.


Proof. As Xm is a complete intersection and F -pure, there exists a good monomial xfor .F; p/. Fix an expression of x into a product of monomials of .F .j //p�1’s. Writex DQm

jD0 x.j /, where x.j / is the contribution from .F .j //p�1. Let aijk be the power

of x.i/k

in x.j / and let aij D .PNkD1 aijk/=.p�1/. Then the matrixA D .aij /0�i;j�m

satisfies the following conditions:

(i) A is an upper triangular matrix,

(ii)PmiD0 aij D d; .0 � j � m/,

(iii)PmiD0 iaij D j; .0 � j � m/ and

(iv)PmjD0 aij � N; .0 � i � m/.

Under these conditions we will prove that ifm is sufficiently large, for any real numbers < d2, there exists i such that ˛i D Pm

jD0 aij > s, which shows d2 � N by (4).Let C be the matrix as follows:

C D

0BBBBBBBBB@

d d � 1 d � 2 1

1 2 d � 1 d d � 1 1

1 d � 1 d d � 11

: : :

1CCCCCCCCCA:

In other words, C D .cij /0�i;j�m be defined as cij D d � u (if j D di ˙ u) forevery u D 0; 1; : : : ; d � 1 and cij D 0 (otherwise). Then, C also has the properties(1)�(3). Let i DPm

jD0 cij , and assume m D dl for an integer l > 0, then

i D

8ˆ<ˆ:

d.d C 1/2

; if i D 0,

d2; if 1 � i � l � 1,

d.d C 1/2

; if i D l ,0; if i � l C 1.

If we put ıi D ˛i � i for i D 0; : : : ; m, thenmXiD0

ıi D 0; andmXiD0

iıi D 0:

Now, assume max ˛i � s D d2 � e for some e > 0. Put D D Pi�l�1 ıi and

D0 D Pi�l ıi , so that D CD0 D 0. Since ıi � d.d�1/

2� e for i D 0 and ıi � �e

for 1 � i � l � 1, we have D � �le C d.d�1/2

and D0 � le � d.d�1/2

. If we put

ıi D �e � �i for 1 � i � l � 1 and D D �le C d.d�1/2� �, then � �Pl�1

iD1 �i and


D0 D le � d.d�1/2C �. By this we have

l�1XiD0

iıi D �el�1XiD0

i �l�1XiD0

i�i � �e l.l � 1/2

� .l � 1/�:

On the other hand, noting that ıi � 0 for i � l C 1, we havemXiDl

iıi � lD0:

Thus, we conclude that

mXiD0

iıi � �el�1XiD1

i C l�le � d.d � 1/

2

�D 1

2.el2 C .e � d2 C d//l/:

But if l is sufficiently large, then the latter will be positive and this contradicts to thefact that

PmiD0 iıi D 0.

3.6. Takagi and Watanabe [13] introduced the invariant F -pure threshold (denotedby fpt.X;Z; P /) for a scheme X over a field of positive characteristic and the closedsubschemeZ � X at a pointP 2 X . It is closely related to the log-canonical thresholdfor characteristic zero. Here, we refer the formula for a complete intersection case.

Let k be a field of characteristic p > 0. Let X be a subscheme of ANk

definedby polynomials f1; : : : ; fr where dimX D N � r . Let f D Qr

iD1 fi . Let a closedsubscheme Z be defined by an ideal I � kŒx1; : : : ; xN � and m be the maximal idealof a point P 2 X . Let q D pe . Then, by Lemma 2.4

fpt.X;Z; P / D limq!1

maxfr j I rf q�1 6� mŒq�gq

:

As we think of only local a complete intersection case, we can regard this formula asthe definition of F -pure threshold.

Theorem 3.7. Let X be a variety of locally a complete intersection at 0 over a fieldof characteristic p > 0. For m < m0, assume also Xm; XmC1; : : : ; Xm0 are completeintersections at the trivial jets 0m; : : : ; 0m0 . Then, the following are equivalent:

(i) .X; 0/ is non-singular;

(ii) fpt.Xm; 0m; 0m/ D fpt.Xm0 ; m0m�1.0m/; 0m0/, where m0m W Xm0 ! Xm is

the truncation morphism.

Proof. Assume (i), then Xi is non-singular for every i 2 N and the truncation mor-phism m0m W Xm0 ! Xm is smooth. In this case, Xm0 ; Xm; m0m

�1.0m/ and f0mgare all non-singular. Therefore by the formula in 3.6, we have fpt.Xm; 0m; 0m/ Dcodim.f0mg; Xm/ D codim. m0m

�1.0m/; Xm0/ D fpt.Xm0 ; m0m�1.0m/; 0m0/.

For the proof of (ii) H) (i), we first show

fpt.Xm; 0m; 0m/ > fpt.XmC1; mC1;m�1.0m/; 0mC1/; (3.1)


if .X; 0/ is singular and Xm and XmC1 are complete intersections at the trivial jets.Let m and m0 be the maximal ideals of OXm;0m

and of OXmC1;0mC1, respectively.

Let f1; : : : ; fr define X in AN , where r D codim.X;AN /. Then, under the notationin 2.1, Xi is defined by F .j /

l.l D 1; : : : ; r; j � i/ in A.iC1/N

kD .AN

k/i . Let

G.j / DQrlD1 F

.j /

land G DQm

jD0G.j /, G0 DQmC1jD0 G.j /. For q D pe , let

rq D maxfs j msGq�1 6� mŒq�g (3.2)

r 0q D maxfs j msG0q�1 6� m0Œq�g: (3.3)

Let x be a monomial in G0q�1 and c be an element of mr0q such that cx 62 m0Œq�.

Then x is factored as x D x0x00, where x0 and x00 are contributions fromGq�1 and from.G.mC1//q�1, respectively. As F .mC1/

lis of weightmC 1, each monomial of F .mC1/

l

has at most one variable x.mC1/i of weightmC 1. Then, if we factorize x00 D zz0 with

z 2 kŒx.0/; : : : ; x.m/� and z0 2 kŒx.mC1/�, we have

deg z � .q � 1/rX

jD1.dj � 1/;

where dj D ord fj . Here, we note that dj � 1 for all j D 1; : : : ; r and dj � 2 forsome j , since .X; 0/ is singular. The condition cx 62 m0Œq� gives .cz/x0 62 mŒq�. Notingthat x0 2 Gq�1 and cz 2 mr

0qC.q�1/Pr

j D1.dj �1/, we obtain

r 0q C .q � 1/

rXjD1

.dj � 1/ � rq;

which yields

fpt.XmC1; mC1;m�1.0m/; 0mC1/CrX

jD1.dj � 1/ � fpt.Xm; 0m; 0m/

as required in (3.1).Now we can see the following in a similar and easier way as in the above discus-

sions:

fpt.Xm; I; 0m/ � fpt.XmC1; IOXmC1; 0mC1/ (3.4)

for an ideal I � OXm;0m. (This follows by just replacing ms by I s in (3.2) and (3.3).)

By (3.1) and (3.4), we obtain that if .X; 0/ is singular, then fpt.Xm; 0m; 0m/ > fpt.Xm0 ;

m0m�1.0m/; 0m0/. Therefore we conclude (ii) H) (i).


References

[1] L. Ein, M. Mustata, and T. Yasuda, Jet schemes, log discrepancies and inversion of ad-junction, Invent. Math. 153 (2003), 519–535. 39

[2] L. Ein and M. Mustata, Inversion of Adjunction for local complete intersection varieties,Amer. J. Math. 126 (2004), 1355–1365. 39

[3] R. Fedder,F -purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461–480.42

[4] N. Hara, A characterization of rational singularities in terms of injectivity of Frobeniusmaps, Amer. J. Math. 120 (1998), 981–996. 41

[5] N. Hara and K.-i. Watanabe,F -regular and F-pure rings vs. log terminal and log canonicalsingularities, J. Algebraic Geom. 11 (2002), 363–392. 41

[6] S. Ishii, Jet schemes, arc spaces and the Nash map, C. R. Math. Rep. Acad. Sci. Canada 29(2007), 1–21. 41

[7] V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math. 1(1997), 249–271. 41

[8] M. Mustata, Jet schemes of locally complete intersection canonical singularities, with anappendix by D. Eisenbud and E. Frenkel, Invent. Math. 145 (2001), 397–424. 39, 42

[9] M. Mustata, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599–615(electronic). 39

[10] J. F. Nash, Arc structure of singularities, Duke Math. J. 81 (1995), 31–38. 39

[11] K. Smith,F -rational rings have rational singularities, Amer. J. Math. 119 (1997), 159–180.41

[12] S. Takagi, F -singularities of pairs and Inversion of Adjunction of arbitrary codimension,Invent. Math. 157 (2004), 123–146. 41

[13] S. Takagi and K.-i. Watanabe, On F -pure thresholds, J. Algebra 282 (2004), 278–297. 47

Singularities in relativity

Tatsuhiko Koike

Department of Physics, Keio UniversityYokohama, 223-8522 Japan


Abstract. Many phenomena of importance in general relativity theory are related to singularitiesin mathematics. For a simple example, the spacetime regions of extreme gravitational field suchas the beginning of the Universe and the fate of a massive star are described by singularities inthe differential-geometric sense, i.e., curvature singularities of pseudo-Riemannian manifolds.This type of singularity is one of the main objects of interest in general relativity. A less trivialexample is that the formation of a black hole horizon can be described as a blow-up solutionof some partial differential equations in a certain coordinate system, which is a singularity inthe analytic sense. Another is that the “shape” of the black hole horizon is fully characterisedby the set of its nondifferential points which are singularities in the sense of singularity theory.I will explain these connections between singularity and relativity with some comments on myrelated works.

1 Introduction: A very brief review of general relativity

Space and time are not absolute contents of the nature but constitute a spacetime whichcan bend and can be deformed, ant that is what we feel as gravitational phenomena.The spacetime is described by a Lorentzian 4-manifold .M; g/, i.e., a 4-manifold M

endowed with a pseudo-Riemannian metric g of signature .� C C C / (Figure 1).Table 1 shows the brief correspondence of physical and mathematical objects,

where a timelike or null curve refers to a curve whose tangent vector has negative orzero squared norm, i.e., g.V; V / < 0 or g.V; V / D 0, respectively.

The matter field bends the spacetime, and the matter’s equation of motion must beconsistent with how the spacetime bends. This fact is described by Einstein’s equation

Rab � 12Rgab D 8�G

c4Tab;

where Rab D Racbc is the Ricci tensor and R D Rabg

ab is the scalar curvatureconstructed by the Riemann curvature tensor Rabcd of the metric g, and Tab is theenergy-momemtum tensor of the matter field which depends on the type of matter inconsideration. The constant c is the speed of light, andG is Newton’s constant.We shalltake the units c D 1 and G D 1 below. The Bianchi identity rŒaRbc�d e D 0 (where


52 Tatsuhiko Koike

.M; g/

Figure 1. The physical world is represented by a curved Lorentzian manifold. The verticaldirection corresponds to time and the horizontal directions to space.

Œ� denotes the anti-symmetrisation) implies raTab D 0. This must be consistent withthe matter field’s own equation of motion.

Table 1. Correspondence of physical and mathematical objects in general relativity.

Physics Mathematics

Spacetime Lorentzian manifold .M; g/

Gravitational field strength Riemann curvature Rabcd

Spacetime with no gravitational field Minkowski space .R4;�dt2 C jdxj2/Motion of massive particles Timelike geodesics

Motion of photons Null geodesics

In this article, I present singularities in a wider sense appearing in general relativity.I briefly discuss geodesic incompleteness in Section 2, and curvature singularity inSection 3. These are the most widely discussed singularities in general relativity. Theyare the singularities of the spacetime itself and physically represent extreme gravita-tional situations. One can consider them as “singularities in the geometric sense.” Inthe later sections, I discuss other classes of objects that can be considered as “singular-ities”. They are in general not treated as singularities in physics and they have nothingto do with the above-mentioned geometric singularities. One such object is the end-point set of the event horizon (Section 4) which characterise the qualitative physicalfeature of a black hole. We see that this set is closely related to the singularities inmathematical singularity theory. Another is the apparent horizon (Section 5) whichphysically characterises formation of a black hole. We see that this is closely related tosingular behaviour of the solutions of a partial differential equation. One can considerthese two as “singularities in the analytic sense.” To demonstrate the usefulness ofthis viewpoint that the apparent horizon is a singularity, I discuss critical behaviour

Singularities in relativity 53

.M; g/

singularity

Figure 2. Existence of singularity implies that there is some observer whose history suddenlyends in finite time.

in gravitational collapse in Section 6. I explain what one can call “singularities in thetopological sense” in Section 7. Section 8 is a summary.

2 Geodesic incompleteness

Geodesic incompleteness is what is usually called singularity in general relativity. Itis one of the most important object because of its intimate relation with black holesand with the initiation of the universe.

Geodesic incompleteness is defined as follows. An affinely parametrised geodesic W � 7!M is a curve satisfying

r P� P D 0;where the dot denotes the differentiation by� andr is the Riemannian connection withrespect to g. An affinely-parametrised geodesic W D !M is complete ifD D R. Apoint p is a future (respectively, past) endpoint a of a causal (i.e., timelike or null atall points) curve � W D ! M if for each neighbourhood U of p, there is t such that�.fs 2 D j s > tg/ � U (respectively, �.fs 2 D j s 6 tg/ � U ). An incompletegeodesic without endpoints is said to have a singularity. Physically, existence ofa singularity along a causal geodesic represents that there is some observer whosehistory suddenly ends in a finite time.

Since the gravitational force is attractive, positive mass attracts matter and the light.Using this fact, Penrose established that occurrence of the singularity is inevitable incommon situations. The following version states that existence of the cosmologicalinitial singularity in usual situations.

Theorem 2.1 (Penrose [1]). Spacetime .M; g/ is null geodesically incomplete if

(i) RabKaKb > 0 for all null vectors Ka;

(ii) there is a noncompact Cauchy surface † in M; and

(iii) there is a closed trapped surface T in the future of †.

54 Tatsuhiko Koike

The first condition is essentially the requirement of positivity of mass. In the secondcondition, a Cauchy surface is defined as a 3-hypersurface† such that all causal curveswithout endpoints intersect †. In the third, a closed trapped surface is a compact 2-surface T without boundary such that the area form is converging to both null directionsorthogonal to T (Figure 3).

T

†

Figure 3. The conditions in Theorem 2.1. A trapped surface lies in the future of a Cauchy surface.

Existence of the singularity is neither necessary nor sufficient for the existence ofblack holes, but they are closely related. The domain of future outer communicationDOCC.M/ of M is the set of points p such that there is a future-directed causalcurve from p which can escape to infinity. To be precise, the curve above must reachto future null infinity I C. The definition of the future and past null infinity I ˙requires a notion of conformal completion of M (see e.g. [2] for details). The blackhole region B of M is the complement of DOC.M/ in M. The event horizon H is theboundary of B.

The Schwarzschild spacetime is the simplest spacetime containing a black holeregion. In the most common coordinates, the metric has the form

g D� f .r/dt2 C dr2

f .r/C r2hS2 ; f .r/ D 1 � 2MBH

r; (2.1)

where hS2 is the metric of a unit 2-sphere and MBH is called the black hole mass.Figure 4 shows the causal structure of the Schwarzschild spacetime, where two angulardirections are suppressed. In fact, at r D 2MBH, called the Schwarzschild radius, themetric components in (2.1) become singular, and the coordinate system covers any oneof the four regions in Figure 4 (two diamonds and two triangles). There is a coordinatesystem which covers the whole spacetime. The event horizon H is a null surface whosespatial section is a two-sphere with the radius being the Schwarzschild radius. Thereare singularities at r D 0; geodesics toward them are incomplete.


r D 0

r D 1

H

B

JC

J�

Figure 4. Causal structure of the Schwarzschild black hole. The white part is the domain offuture outer communication DOCC.M/. whose complement B and boundary H are the blackhole region and the event horizon, respectively.

3 Curvature singularities

Curvature singularity is most commonly encountered “singularity” in general relativ-ity. A scalar singularity is said to exist when some scalar combination of the Riemanntensor such as Ricci scalar R, Ricci tensor squared RabRab , Riemann tensor squaredRabcdR

abcd , etc., diverge along a curve in the spacetime manifold .M; g/.Physically, it represents the existence of infinitely strong tidal force where every-

thing approaching there would be destroyed. In most common situations in relativity,a singularity (geodesic incompleteness) is a curvature singularity. A simple example isthe one at r D 0 in the Schwarzschild spacetime represented by (2.1). If a singularityis not a curvature singularity, the observer’s life ends in a finite time (geodesic incom-pleteness) without a catastrophe (curvature singularity). Physically, this is a somewhatpuzzling and unwanted situation. In those cases, the notions of singularities weakerthan curvature singularity are defined and discussed.

Since curvature singularities are most common and discussed in enormous numberof works, which the reader should be able to access easily, we do not go into themhere (see e.g. [2] for further reading on basic concepts).

4 The endpoint set of the event horizon

Mathematical singularity theory gives a good characterisation of an event horizon inrelativity.

An event horizon H is generated by null geodesics. A future event horizon cannothave future endpoints but can have past endpoints. The endpoint set E of a horizon H

is an arc-wise connected acausal set, where an acausal set is a set such that no twopoints thereof can be connected by a causal curve in the spacetime M (Figure 5).

56 Tatsuhiko Koike

E

H

JC

Figure 5. The endpoints set E of the event horizon H .

In relativity, there is no preferred time coordinate. For example, if E is a two-dimensional surface, the spacetime allows the following interpretations, among manyothers, according to different choices of time slices (Figure 6); see [3]:

(i) an S2 black hole forms, and grows;

(ii) two black holes form, and collide;

(iii) a torus black hole forms, and the handle pinches to make an S2 black hole.Note that if E consists of a point, e.g. in the case a spherically symmetric black hole,the first interpretation above applies for any choice of time slices. Thus, the endpointset determines the qualitative feature of the black hole.

EEt D const

JCJC

Figure 6. Physical interpretation depends on the choice of time slices: formation of a singleblack hole, collision of two formed black holes, …, etc.

Points u 2 E are classified by the multiplicity m.u/ of u, the number of the nullgeodesic generators emanating from u:

E D C tD ; C D fu 2 E j m.u/ > 1g; D D fu 2 E j m.u/ D 1g: (4.1)

The set C is called the crease set of the horizon. The crease set contains the interiorof the endpoint set, i.e., the closure of C contains E . The crease set C coincides withthe set of points of E on which the horizon is not differentiable, i.e., the horizon isdifferentiable at u 2 E if and only if u 2 D [4].

The sets C and D can be naturally understood in the context of singularity theory inmathematics. Assume M is globally hyperbolic with a Cauchy surface †. Then thereis a global time function t , and M is a direct product of the time R and the space †:

� W R �† 3 .t; q/ 7�! �.t; q/ 2M:


We assume that in the sufficiently late times a t=constant section H of the eventhorizon H is stable. We fix a compact surface S in† so that �.t; S/ D H with that t.

The generic structure of the endpoint set E can be classified by the singularitytheory [5]. Let us define the Fermat potential F for .x; q/ 2 S �† by

F.x; q/ D � supft 2 R j there is a future-directed causal curve from �.t; q/ to xg:(4.2)

Let BMaxwell.F / be the Maxwell set of F , i.e.,

BMaxwell.F / D fq 2 † j F. ; q/ has two or more global minimum pointsg:Let B.F / be the bifurcation set of F, i.e., the set of points where the minimum bifur-cates. Then we have

C Š BMaxwell.F /; D Š .BMaxwell.F / \ B.F //:This correspondence can be used for classifying the stable structure of E . The Maxwellset of a generic l 6 6-parameter family of functions is locally stably diffeomorphic toone of the types Am1

Amk, where themi are odd and

Pmi 6 l C 1. The concrete

types other than Ak1 are shown in Table 2 (see e.g. [6]). For example, one has

A21 D .x21 C y21 ; x22 C y22 C q1/;A3 D x4 C q2x2 C q1x C y2:

In particular, the Maxwell set Ak1 is merely the intersection of walls separating kdomains.

In the case of four-dimensional spacetime M, we have l 6 dim† D 3 and thepossible locally stable structure of C , hence of E , can be summarised by the followingdiagram:

A31��

A3A1

��

��

��

A21

A41

��

A3 ,��

where an arrow means that the structure at the origin of the arrow has the structuresat the target of the arrow in the vicinity and the box means that the structure appearsat the boundary of the Maxwell set and is not contained therein. See Figure 7 for anexample.

By making use of Table 2, one can easily construct such diagrams for the casesdim M D 5; 6; 7.

58 Tatsuhiko Koike

A3A1

A3A1

A1A1

A1A1

A1A1A1

Figure 7. An example of a generic endpoint set.

Table 2. Locally stable Maxwell sets.

l � 2 3 4 5 6

Type A3 A1A3 A21A3; A5 A23; A31A3; A1A5 A1A

23; A

41A3; A

21A5; A7

5 Apparent horizons

The event horizon is an important object in relativity but it may not be determinedeasily because it depends on the causal structure of the whole spacetime. i.e., it dependsheavily on the behaviour at infinitely late times. For the same reason, it is hard to relatethe event horizon with the dynamics of the gravitational field. Thus one often discussesapparent horizons which can be defined (quasi-)locally. An apparent horizon usuallyemerges near the event horizon, and, if the cosmic censorship holds, inside thereof(see e.g. [2]). In particular, if the event horizon exists and if the spacetime is static(i.e. if it has a one-parameter family of isometries with image curve of each pointbeing timelike outside the black hole), the event horizon coincides with the apparent


horizon. The apparent horizon is not considered as a singularity in general relativity,but the viewpoint that it is so is possible and useful.

An apparent horizon is a closed 2-surface (or 3-hypersurface foliated thereby) suchthat the surface area form is converging in one null direction and is stationary in theother null direction (see Figure 8). In terms of coordinate systems consisting of a“time” coordinate and “space” coordinates, the apparent horizon is characterised byits singularity.

ordinary surface apparent horizon

��

Figure 8. An ordinary surface and an apparent horizon.

O rAH

a.r/

1

r

Figure 9. An apparent horizon formation can be described by a blow-up of a certain componentof the metric.

A simplest example is a spherically symmetric spacetime. In a commonly usedcoordinate system, the metric reads

g D �˛.t; r/2dt2 C a.t; r/2dr2 C r2hS2 :

The product of expansion rates in null directions is

�C�� D lC.r2/r2

l�.r2/r2

D 2

r2a.t; r/2;

60 Tatsuhiko Koike

where �˙ are the expansion rates of the area form in null directions

l˙ D 1p2

�@t˛˙ @r

a

�:

Thus, from the viewpoint of partial differential equation, the blow-up a.t; r/ % 1of the solution corresponds to the emergence of an apparent horizon. Thus, when andwhere a.t; r/ blows up have the information on the apparent horizon (see Figure 9).

The massMBH of the black hole, i.e. of the event horizon is usually well estimatedby the massMAH of the apparent horizon. In spherically symmetric spacetimes, thereis a natural definition of the mass (or energy) M.t; r/ contained in a given spherespecified by .t; r/. It is simply M.t; r/ D r=2. Therefore, if we find an apparenthorizon appearing at r D rAH, i.e., if limt!tAH a.t; rAH/ D 1with some tAH, then themass contained in the apparent horizon is given by MAH DM.tAH; rAH/ D rAH=2.

An application of this fact is presented in the following section.

6 Critical behaviour in gravitational collapse

In this section, we shall present an application of the occurrence of the apparent horizonas a “singularity” discussed in the previous section.

Formation of black holes is important in astrophysics as well as in general rela-tivity. By the quasistationary analysis of realistic stars, it is suggested that relativelyheavy stars (those more than several times heavier than the sun) become a black hole,while the minimum mass of resulting black hole is around twice that of the sun. Ingeneral relativity, one is also interested in more dynamical situations of black holeformation called gravitational collapse, and in fundamental theoretical problems suchas the conditions for formation of singularities, event horizons and Cauchy horizons.1

However, it is difficult in general since Einstein’s equation is highly nonlinear.Critical behaviour in gravitational collapse presented here is a universal and char-

acteristic phenomenon appearing in the limiting situation that the initial matter distri-bution evolves into a black hole with infinitesimal mass. It was found by numericalsimulation [7] and allowed an interpretation similar to critical phenomena in statisticalphysics. The mechanism of this peculiar phenomenon was not known for a while, butwas later revealed that a certain structure of the phase space as a dynamical system [8]is responsible. The understanding has natural connection to that of critical phenomenain statistical physics and also gives a method for quantitative analysis. Though thestudy of the subject had been driven by purely theoretical interests, it now has directcosmological applications, related to the nature of black holes in an early stage of theUniverse [9].

1A Cauchy horizon is a hypersurface in the spacetime beyond which the physics is not predictable, i.e. it isthe boundary of globally hyperbolic region of the spacetime.


6.1 The phenomena

Let us consider some matter field and its gravitational collapse, i.e., the formationof a black hole. If the initial configuration of the matter is sufficiently “weak”, thenthe matter will disperse and the black hole will not form. For a sufficiently “strong”one, the black hole will form (for examples of rigorous analysis, see Christodoulou’sworks [10]). Then the questions arise. What happens if one gradually changes theinitial configuration? What happens at the threshold of the formation of the black hole(Figure 10)?

matter

gone!

threshold?

black hole

Figure 10. What is the behaviour of the initial data near the threshold between the ones whichwill evolve into a black hole and those which will not?

By numerical simulations, Choptuik [7] found critical behaviour in gravitationalcollapse which resembles critical behaviour in statistical physics, characterised byscaling and universality, in a spherically symmetric system of gravitational and realmassless scalar fields. The energy-momentum tensor of the matter is given by Tab D12ra�rb�, where � W M ! R is the real massless scalar field, and the spacetime

metric is given by g D �˛.t; r/2dt2 C a.t; r/2dr2 C r2hS2 .The behaviour is as follows (the description here relies on [11]) Let I.x/ be a

generic 1-parameter family of initial data such that a black hole will form in the futurefor sufficiently large x and it will not form for sufficiently small x. Then the followingproperties hold (Figure 11).

(E) There is xc such that for x > xc a black hole forms and for x < xc no black holeforms.

(S1) The x � xc solutions once approach a discretely self-similar solution and theneither forms a black hole or approaches the flat spacetime.

(S2) For x >� xc , the mass MBH of the formed black hole satisfies a scaling lawMBH / .x � xc/ˇBH .

62 Tatsuhiko Koike

(U1) The self-similar spacetime in (S1) is unique for all x � xc .

(U2) The critical exponent ˇBH, which was approximately 0:37, is universal in thesense that it does not depend on the choice of the one-parameter family I.x/ inthe space of initial data.

�

r

0

t

MBH

xc x

MBH / .x � xc/ˇBH

Figure 11. Critical behaviour in gravitational collapse. Up: any near-critical spacetime ap-proaches a self-similar one before it evolves into a black hole or diverges. Down: the massof the formed black hole in the super-critical spacetimes satisfies a universal power law.

Similar behaviour was found in the axial symmetric system of pure gravity (i.e. withno matter field) and in the spherically symmetric system of gravity and a radiationfluid. The former system showed discrete self-similarity and the critical exponentwas ˇBH � 0:37; see [12]. In the latter, the energy-momentum tensor was Tab D�uaub C p.uaub C gab/, where p D �=3 is the pressure, � > 0 is the density, andua is the velocity vector of the fluid. The system showed continuous self-similarityinstead of discrete self-similarity in (S1) and the value of ˇBH was approximately 0:36;see [13]. The mechanism of this interesting phenomenon was not understood.


6.2 Renormalisation group method

The mechanism of the critical behaviour in gravitational collapse is revealed by Hara,Adachi and the present author [8] by using the method of renormalisation group (RG).The method of RG for partial differential equation was used for the analysis of inter-mediate asymptoticswhere self-simlar solutions play an important role. In the gravi-tational system, as is discussed in the previous section, the formation of an apparenthorizon can be considered as blow-up of the solution. We apply the RG method to theanalysis of blow-up solutions.

Consider a partial differential equation (PDE)

L�@u@t;@u

@r; u; t; r

�D 0; u.�1; r/ D U.r/; (6.1)

where L is a function from R5 to Rn. We use the notation as if n D 1 for simplicity;the recovery to general n is simple. We assume that the system is invariant under thescaling transformation �s ,

�su.t; r/ D e˛su.e�st; r�ˇs/; ˛, ˇ, fixed reals:

Namely, if u is a solution to the PDE, so is �su.Let us call� D fU g (the space of initial data) the phase space. The renormalisation

group transformation (RGT) Rs on � is defined by

RsU.r/ D �su.�1; r/ D e˛su.�e�s; r�ˇs/;

where u is a solution to the PDE. The RGT depends on the real parameters ˛ andˇ but we omit them in the notation Rs . The RGT Rs can be described as “the timeevolution from t D �1 to t D �es , followed by a spatial scaling transformation”.2

The family of RGTs has semi-group property Rs1Cs2 D Rs2 ıRs1 .For simplicity, we explain the case of continuous self-similarity in the follow-

ing. The generalisation to the case of discrete self-similarity is straightforward. Thegenerator of the RGT Rs is defined by

PR D lims!0

R � 1s

;

and thereby the RGT is expressed as Rs D exp.s PR/. The generator PR defines a vectorfield on � , which we call the RG flow. A fixed point U � of the flow PR satisfyingPRU � D U � corresponds to a self-similar solution uss D �suss of (6.1) by

uss.t; r/ D .�t/˛U �� r

.�t/ˇ�:

2We use negative t because we consider a “shrinking” spatial scaling (and we want t to increase towardfuture).

64 Tatsuhiko Koike

The tangent map of Rs around the fixed point U � is defined by

TsF D lim"!0

Rs.U� C "F / � U �

":

The tangent map sends the tangent space at U � to that at RsU�. Its generator is given

by

PT D PTU� D lims!0

Ts � 1s

;

so that

Ts D exp.s PT /;An eigenmode of PT is defined by

PT F D �F:The eigenmode F is called relevant if Re � > 0, irrelevant if Re � < 0, and marginalif Re � D 0.

6.3 The mechanism

The mechanism and all aspects of the critical behaviour discussed in Section 6.1 canbe understood as the structure of the RG flow. We describe them as assumptions.

Assumption 6.1 (Spectrum of T at U �). The spectrum of T is

�.T / D f�g [ � 0;

where Re� > 0 and � 0 � f�0 2 C j Re�0 6 �g with some � < 0.

Assumption 6.2 (Global information). The two directions ofW u.U �/ are the flat andthe black hole spacetimes. If U is sufficiently far from U �, it becomes a black hole ina finite “time” s.

The stable manifold of U � is defined by

W s.U �/ D fU 2 � j lims!1 Rs.U / D U �g:

The unstable manifold of U � is defined by

W u.U �/ D fU 2 � j lims!�1 Rs.U / D U �g:

The properties (E), (S1), and (U1) are direct consequences of the assumptions.

Proposition 6.3 (E, S1, U1: global structure of the flow). We have

dimW u.U �/ D 1


andcodimW s.U �/ D 1:

The two directions of W u.U �/ are the flat and the black hole spacetimes (Figure 12).

W s

W uU �

Uinit

Q

I

Uc.x D xc/

Flatspacetime

Blackhole

x > xcx < xc

Figure 12. Schematic diagram of the structure of the RG flow on the phase space.

We can also show (S2) and (U2) from the same assumptions.

Proposition 6.4 (S2, U2: mass of the black hole). The mass of the black hole formsis given by

MBH ' K.x � xc/ˇBH ; ˇBH D ˇ

�;

when x ! xc .

We have also assumed that the mass MBH of the black hole is well estimated bythe mass MAH of the apparent horizon, as discussed in the previous section.

Let U be a small neighbourhood of U �. Let U D UcC "F be a near-critical initialdata. Because U � is the attractor of the critical surface W u.U �/, RsU enters in U atsome s D s0 (Figure 13):

Rs0U D Rs0.Uc C "F / � Rs0Uc C "F 0:

Near U �, we can use the linearised equation:

Rs0Cs1U D Rs1.Rs0Uc C "F 0/ ' Rs0Cs1Uc C "Ts1F 0 ' U � C "e�s1F 0rel:

66 Tatsuhiko Koike

Choose s1 as the time spent in U. Then we have

"e�s1 � 1 (i.e. independent from "):

From Assumption 6.2, RsU will blow up at s D s0 C s1 C s2 with finite s2. Theblow-up occurs at position � 1 (independent from "). However, its physical lengthscale is given by rAH � expf�ˇ.s0 C s1 C s2/g/ � "ˇ=� because of the scalingtransformation contained in the RGT Rs . Thus rAH � "ˇ=� . Since MBH D rAH=2,

MBH ' rAH � e�ˇs1 � "ˇ=� :We have shown the scaling relation and have obtained the formula for the criticalexponent ˇBH D ˇ

�. This is obviously independent of the choice of the 1-parameter

family I.x/ of initial data.

Initial data

Black hole

Minkowskispacetime

U �

s0s0 C s1

s0 C s1 C s2

Figure 13. Illustration of how the structure of the RG flow explains all characteristics of thecritical behaviour. Note that the flow spends essentially the whole “time” near the fixed pointwhen x ! xc .

For a radiation fluid, we found, with ˛ D 1 andˇ D 1, a unique relevant mode with� D 2:81055255 by numerically solving the eigenmode equation of T, which is a two-point boundary problem of an ordinary differential equation [8]. We further establishedthe uniqueness of the relevant mode by Liapunov analysis (numerical scheme thatselectively finds large Re �0 modes) [11]. These show that our interpretation of thecritical behaviour is correct and give the critical exponent ˇBH D 0:35580192 (recallthat the simulation value was ˇBH � 0:36).

7 Topological singularities due to quotienting

Making the quotient of a manifold by a group action may yield a space with “topo-logical” singularities, and this may have some physical implications.


Many cosmological observations suggest that the universe underwent an inflation.This means that the spacetime was close to the de Sitter spacetime, i.e., the spacetimeof positive constant curvature. Observations also suggest that the present universehas negative spatial curvature so that the spatial sections can be approximated byhyperbolic 3-manifolds. The de Sitter spacetime does not have a curvature singularityand can be extended smoothly to the past.

On the other hand, spatial compactness of the universe is an appealing notion,especially in the context of the canonical treatments of the universe or quantum grav-ity. Compactness provides a finite value of the action integral and gives the naturalboundary conditions for the matter and gravitational fields in the universe.

It has been shown [15] that these three conditions, inflation, spatial hyperbolicityand spatial compactness cannot hold simultaneously. Namely, though the universalcovering space of the universe can be extended analytically, beyond the so-called pastCauchy horizon, the extended region has densely many points which correspond tosingularities of the compact universe. This is done by carefully analysing the groupaction for quotienting around the Cauchy horizon, and the proof relies on the ergodicityof the geodesic flow on a compact negatively curved manifold (Figure 14).

X0

X1

X2

H2

A1

B1

C1

De Sitter2

O

Figure 14. Topological singularities appear in a spatially compact, spatially hyperbolic, de Sitter-like spacetime.

68 Tatsuhiko Koike

8 Summary

I have reviewed singularities in a wider sense which appear in relativity. Geodesicincompleteness, curvature singularity, the endpoint set of the event horizon, apparenthorizons with an application of the critical behaviour in gravitational collapse, andtopological singularity. Since relativity is formulated as differential geometry, theideas and the techniques of the latter have been applied to the former. I hope thatsingularity theory and other field of mathematics will have more and more interactionswith relativity and lead to new discoveries.

References

[1] R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965),57–59. 53

[2] S. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Mono-graphs on Mathematical Physics 1, Cambridge University Press, London and New York1973. 54, 55, 58

[3] M. Siino, Topology of event horizons, Phs. Rev. D 58 (1998), 104016. 56

[4] J. K. Beem and A. Królak, Cauchy horizon end points and differentiability, J. Math.Phys. 39 (1998), 6001–6010 56

[5] M. Siino and T. Koike, Topological classification of black hole: Generic Maxwell set andcrease set of horizon, Internat. J. Mod. Phys. D 20 (2011), 1095–1122. 57

[6] V. I. Arnold (ed.), Dynamical systems VIII: Singularity Theory II, Classification and Apli-cations, Encyclopedia of Mathematical Science 39, Springer-Verlag,Berlin etc., 1991.57

[7] M. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field,Phys. Rev. Lett. 70 (1993), 9–12. 60, 61

[8] T. Koike, T. Hara, and S. Adachi, Critical behavior in gravitational collapse of radiationfluid: A renormalization group (linear perturbation) analysis, Phys. Rev. Lett. 74 (1995),5170–5173. 60, 63, 66

[9] J. C. Niemeyer and K. Jedamzik, Near-critical gravitational collapse and the initial massfunction of primordial black holes Phys. Rev. Lett. 80 (1998), 5481. 60

[10] D. Christodoulou, A mathematical theory of gravitational collapse, Comm. Math.Phys. 109 (1987), 613–647, and references therein. 61

[11] T. Koike, T. Hara, and S. Adachi, Critical behavior in gravitational collapse of a perfectfluid, Phys. Rev. D 59, 104008; see T. Hara and T. Koike, in Blowup and aggregation, ed.by M. Mimura, University of Tokyo Press, Tokyo 2006. 61, 66

[12] A. M. Abrahams and C. R. Evans, Critical behavior and scaling in vacuum axisymmetricgravitational collapse, Phys. Rev. Lett. 70 (1993), 2980–2983. 62

[13] C. R. Evans and J. S. Coleman, Critical phenomena and self-similarity in the gravitationalcollapse of radiation fluid, Phys. Rev. Lett. 72 (1994), 1782–1785. 62


[14] J. Bricmont, A. Kupiainen, and G. Lin, Renormalization group and asymptotics of solu-tions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47 (1994), 893–922.

[15] A. Ishibashi, T. Koike, M. Siino, and S. Kojima, Compact hyperbolic universe and singu-larities, Phys. Rev. D 54 (1996), 7303–7310. 67

On the universal degenerating familyof Riemann surfaces

Yukio Matsumoto�

Department of Mathematics, Gakushuin UniversityMejiro, Toshima-ku, Tokyo 171-8588, Japan


Abstract. Let †g be a closed oriented (topological) surface of genus g .>D 2/. Over the Teich-müller space T .†g/ of †g , Bers constructed a universal family V.†g/ of curves of genus g,which would be well called “the tautological family of Riemann surfaces”. The mapping classgroup �g of †g acts on V.†g/! T .†g/ in a fibration preserving manner. Dividing the fiberspace by this action, we obtain an “orbifold fiber space” Y.†g/!M.†g/, where Y.†g/ andM.†g/ denote V.†g/=�g and T .†g/=�g , respectively. The latter quotient M.†g/ is calledthe moduli space of †g . The fiber space Y.†g/ ! M.†g/ can be naturally compactified toanother orbifold fiber space Y.†g/! M.†g/. The base space M.†g/ is called the Deligne–Mumford compactification. Since this compactification is constructed by adding “stable curves”at infinity, it is usually accepted that the compactified moduli spaceM.†g/ is the coarse modulispace of stable curves of genus g. In this paper, we will sketch our argument which leads to aconclusion, somewhat contradictory to the above general acceptance, that the compactified fam-ily Y.†g/!M.†g/ is the universal degenerating family of Riemann surfaces, i.e. it virtuallyparametrizes not only stable curves but also all types of degenerate and non-degenerate curves.

Our argument is a combination of the Bers–Kra theory and Ashikaga’s precise stable reduc-tion theorem.

Contents

1. Main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

2. Complex orbifolds and fiber spaces over orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3. Types of mapping classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4. Fenchel–Nielsen coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5. Compactification process of M.†g/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6. Bers’ deformation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7. Subdeformation spaces D".C/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8. The universal degenerating family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100�Supported by the Grant-in-Aid for Scientific Research (B) 20340014.


72 Yukio Matsumoto

1 Main theorem

First we will recall the notation and some basic results. Let †g be a closed orientedtopological surface of genus g .>D 2/. Let �g D �.†g/ be the mapping class groupof †g , which is defined as follows:

�g D ff W †g �! †g j orientation preserving homeomorphismg=isotopy:

Let Tg D T .†g/ denote the Teichmüller space of†g . The definition of T .†g/ beginswith pairs .S; w/ consisting of a Riemann surface S and an orientation preservinghomeomorphism

w W S ! †g :

Two such pairs .S1; w1/ and .S2; w2/ are said to be equivalent and denoted by.S1; w1/ � .S2; w2/, if there exists a biholomorphic map

t W S1 �! S2

such that the following diagram is homotopically commutative1:

S1w1 ��

t

��

†g

id

��s2 w2

�� †g :

The Teichmüller space of †g is defined2 by

T .†g/ D f.S; w/g= � :The equivalence class to which .S; w/ belongs will be denoted by ŒS; w�. By theworks of Ahlfors [3], and Weil [40], [41], the Teichmüller space T .†g/ is a complexmanifold of complex dimension 3g � 3. Bers [6] embedded T .†g/ in C3g�3 as abounded domain. The mapping class group �g acts on T .†g/ through

Œf ��.ŒS; w�/ D ŒS; f ı w�:Here Œf � is an element of �g and ŒS; w� is a point of T .†g/. It can be proved that thisaction is properly discontinuous and holomorphic [7]. See also [30], Chapter 2, and[18], Chapter 6. Consequently, the moduli space M.†g/ D T .†g/=�g is a normalcomplex analytic space. See [12].

Bers [7] constructed a family of Riemann surfaces V.†g/! T .†g/ and showedthat �g acts on the total space V.†g/ and on the base space T .†g/ simultaneously in

1In our situation, this condition is equivalent to saying that the diagram is isotopically commutative.2The usual definition of Teichmüller spaces starts with pairs .S;w/ in whichw is an orientation preserving

homeomorphism of†g ontoS, being in the opposite direction to our homeomorphism which is fromS onto†g .Cf. [18]. The author thinks that our definition is more natural so far as the action of �g on T.†g/ is concerned

On the universal degenerating family of Riemann surfaces 73

such a way that the projection is equivariant with respect to these actions3 (in short, theaction of �g preserves the fibered structure of V.†g/ ! T .†g/). His constructionuses quasiconformal machinery and is quite involved, but as a final product, his fiberspace is “tautological” in the sense that over the point ŒS; w� 2 T .†g/ the identicalsurface S is situated. The action of �g on the fiber space is also tautological: ifŒf ��.ŒS; w�/ D ŒS 0; w0�, we have a homotopically commutative diagram in which t isa biholomorphic mapping

S

t

��

w �� †g

f

��S 0

w0�� †g

(1.1)

and the action of Œf � on the total space V.†g/ takes the fiber S onto the fiber S 0 bythe mapping t .

Let us recall Bers’ theorem.

Theorem 1.1 ([7]). There exists an orbifold fiber space

Y.†g/ �!M.†g/

such that over each point p D ŒS� 2 M.†g/, one has a fiber which is isomorphic tothe quotient S=Aut.S/.

Note that Aut.S/ ¤ f1g if and only if p D ŒS; w�.2 T .†g// is fixed under theaction of a finite subgroup of �g , isomorphic to Aut.S/. Any finite subgroup of �ghas a non-empty fixed point set in T .†g/; see [21]. The fiber space of Theorem 1.1 isthe quotient of V.†g/! T .†g/ by the action of �g , i.e. Y.†g/ D V.†g/=�g andM.†g/ D T .†g/=�g .

The moduli space M.†g/ is compactified by adding “stable curves” at infinity.This compactification is called the Deligne–Mumford compactification (see [14]) andthe compactified moduli space will be denoted by M.†g/. Because of this process

of compactification, it is usually accepted that M.†g/ is the coarse moduli space ofstable curves. Somewhat contradictory (at first sight) to this consensus, our main resultstates the following.

3As a matter of fact, Bers did not mention this fiber space explicitly in his paper [7]. He constructed insteada fiber space F.G/ ! T.G/, where G is a Fuchsian group isomorphic to the fundamental group �1.†g/,and T.G/ is canonically identified with T.†g/. Each fiber of F.G/ is a (Jordan) domain of discontinuity ofa quasi-Fuchsian group isomorphic toG. The groupG acts on F.G/ so that it preserves the fibering structure.To get the fiber space V.†g/ ! T.†g/, we have only to take the quotient F.G/=G ! T.G/ and putV.†g/ D F.G/=G. Cf. [22] §4.6.

74 Yukio Matsumoto

Theorem 1.2. (1) The orbifold fiber space Y.†g/ ! M.†g/ can be compactifiedto an orbifold fiber space

Y.†g/!M.†g/:

The total and the base spaces of this fiber space are compact complex normal analyticspaces (hence Hausdorff ).

(2) The compactified fiber space is the universal degenerating family, or moreprecisely, the universal orbifold fiber space in the following sense: given a degenerat-ing family of Riemann surfaces ' W M ! B of genus g( >D 3) over a (either compactor non-compact) Riemann surfaceB, one can canonically associate with it an orbifoldfiber space

'# W M# �! B;

and if ' W M ! B is almost asymmetric, then there exists an orbifold pull-backdiagram

M#

'#

��

orbifold pull-back �� Y.†g/

��B

b#

�� M.†g/ :

Note that the statement of the main theorem belongs to “biregular geometry”, notto “birational geometry”. In birational sense, the main theorem (especially as statedin the first version of this paper4) is almost straightforward. See Remark 8.3.

A degenerating family of Riemann surfaces of genus g over a base Riemann sur-face B is, by definition, a proper surjective holomorphic map ' W M ! B of a 2-dimensional complex manifold M to B , whose general fiber is a Riemann surfacehomeomorphic to †g . It may admit isolated degenerate fibers. A Riemann surface S

4In the first version of this paper, the main theorem was stated somewhat vaguely as follows.

Theorem (1) The orbifold fiber space Y.†g/ ! M.†g/ can be compactified, in a fiber preserving manner,to an orbifold fiber space

Y.†g/ �! M.†g/:

The total and the base spaces of this fiber space are compact complex normal analytic spaces.

(2) The compactified fiber space is the universal degenerating family, in the sense that for any degeneratingfamily of Riemann surfaces of genus g. >D 3/ over a (either compact or noncompact) Riemann surface B ,' W M ! B , which is almost asymmetric, there exists a pull-back/blowing up diagram

M

'

��

blowing up �� M#

'#

��

pull-back �� Y.†g/

��B

id�� B �� M.†g/ :


is said to be asymmetric, ifAut.S/ D f1g:

We will say that a degenerating family of Riemann surfaces overB is almost asymmet-ric, if there exists a set of isolated points Symm inB such that for eachp 2 B�Symm,the fiber over p is asymmetric.

Our asymmetry condition5 rules out several exceptional cases. For example, thecase whereg <D 2 is excluded, because in this case every Riemann surface is symmetricwith respect to the hyperelliptic involution.

It is well known that by blowing down .�1/-spheres inM , we get relatively minimaldegenerating family containing no .�1/-spheres in the fibers. In what follows, we willassume that ' W M ! B is a relatively minimal degenerating family. Also by blowingup M at non-normal crossing singular points of fibers, one can make a degeneratingfamily ' W M ! B a normally minimal family ' 0 W M 0 ! B , in which the reducedscheme F red of each fiber F is a nodal surface and every .�1/-sphere in F red has atleast three nodes. The normally minimal family '0 W M 0 ! B is uniquely determinedby ' W M ! B , and is called the normally minimal model of ' W M ! B .

By Ashikaga’s theorem (Theorem 8.1), we obtain an orbifold fiber space

'# W M# �! B

from the normally minimal model '0 W M 0 ! B by contracting certain (explicitlyknown) linear and/or Y-shaped configurations of rational curves in fibers of M 0. Fortheir exact shapes, see Figure 1, [32], and [33]. The length r and the self-intersectionnumbers (�a1;�a2; : : : ;�ar or�b3; : : : ;�br ) are read off from the topological mon-odromy around the singular fiber on which the singular point in question is situated;see [4], [27], and [28]. The total spaceM# may have two types of singularities: cyclicquotient singularities which are the contraction images of linear configurations, anddihedral quotient singularities which are the contraction images of Y-shaped configu-rations. The orbifold fiber space '# W M# ! B is uniquely and explicitly determinedfrom ' W M ! B . We will call it the orbifold model of ' W M ! B .

�a1 �a2 �a3 �ar

�b3 �b4 �br

�2

�2

: : :

: : :

Figure 1. A linear configuration and a Y-shaped configuration.

5This condition was inspired by Ashikaga and Yoshikawa’s A-generality condition imposed on fibered sur-faces, [5].

76 Yukio Matsumoto

Since the normally minimal model '0 W M 0 ! B is obtained from '# W M# ! B

by minimal resolution (just converse to the contraction) of cyclic and/or dihedralsingularities in M# ([32] and [33]), the following theorem is an immediate corollaryof Theorem 1.2.

Theorem 1.3. Let ' W M ! B be a degenerating family of Riemann surfaces whichis relatively minimal and almost asymmetric. Then there exists the following diagram(the arrows are holomorphic maps):

M

'

��

M 0blow down�� minimal resolution ��

'0

��

M#orbifold pull-back ��

'#

��

Y.†g/

��B B

id��

id�� B �� M.†g/:

2 Complex orbifolds and fiber spaces over orbifolds

Orbifolds were introduced by Satake [34] and [35] (under the name of V-manifolds)and independently by Thurston [38] (who created the terminology of orbifolds). Inthis section, we will recall orbifolds, especially in the category of complex analyticspaces, and will introduce “fiber spaces over orbifolds”.

2.1 Complex orbifolds

There are several ways of defining orbifolds; see [34], [35], [38], [19], [11], and [26].We will follow Satake’s definition. A complexm-dimensional orbifold (briefly a com-plex m-orbifold) is a � -compact Hausdorff space X which is covered by an atlas offolding charts A D f. zUi ; Gi ; �i ; Ui /gi2I , each chart consisting of a connected openset zUi of Cm, a finite group Gi acting on zUi holomorphically and effectively, an openset Ui ofX and a folding map �i W zUi ! Ui which induces a natural homeomorphism�i=Gi W Gin zUi ! Ui . The atlas A must satisfy the following conditions.

(1) If a point p of X is contained in the intersection Ui \ Uj , where Ui D �i . zUi /and Uj D �j . zUj /, then there exists a folding chart . zUk; Gk; �k; Uk/ such thatp 2 Uk � Ui \ Uj .

(2) If the open set Ui D �i . zUi / is contained in the open set Uj D �j . zUj /, then thereexists a holomorphic embedding �j i W zUi ! zUj (called an injection) of zUi ontoan open subset of zUj such that �i D �j ı �j i .

It is proved that if �j i and �0j i W zUi ! zUj are two injections, there exists a uniquely

determined elementg0 ofGj such that�0j i D g0�j i (see Lemma 1 in [35]). In particular,


if g is an element of Gi , then there exists a uniquely determined element g0 of Gjsuch that �j ig D g0�j i . The correspondence g 7! g0 is an isomorphism of Gi onto asubgroup of Gj (see [35], p. 466), which we will denote by

�[j i W Gi �! Gj :

The injection �j i W zUi ! zUj becomes an equivariant map with respect to �[j i .Two atlases A and B of folding charts are said to be equivalent, if the union A[B

is an atlas of folding charts satisfying the above two conditions. An orbifold structureon the space X is an equivalence class of atlases of folding charts. By Cartan [12], acomplex orbifold X is a normal complex analytic space.

A typical example of a complexm-orbifold is given by anm-dimensional complexmanifold M on which a discrete group G is acting holomorphically and properlydiscontinuously: the quotient space GnM has a structure of a complex m-orbifold.

2.2 Orbifold maps

LetX andY be two (complex) orbifolds possibly of different dimensions.A continuousmap h W X ! Y is said to be a (holomorphic) orbifold map if it satisfies the followingconditions (cf. [35], p. 469).

For each pointp 2 X , there exist a folding chart . zUi ; Gi ; �i ; Ui / ofX containingp(which means thatUi containsp) and a folding chart . zVk;Hk; �k; Vk/ of Y containingh.p/ with the following properties.

(I) h.Ui / � Vk , and there exists a lifted holomorphic map hki W zUi ! zVk such thatthe diagram

zUi hki ��

�i

��

zVkk

��Ui

hjUi

�� Vk

commutes.

(II) Suppose that folding charts . zUi ; Gi ; �i ; Ui /; . zUj ; Gj ; �j ; Uj / ofX and foldingcharts . zVk ;Hk; �k; Vk/; . zVl ;Hl ; �l ; Vl/ of Y satisfy Ui � Uj , Vk � Vl ,h.Ui / � Vk , and h.Uj / � Vl , and that there exist lifted holomorphic mapshki W zUi ! zVk and hlj W zUj ! zVl . Then for any injection �j i W zUi ! zUj , there

78 Yukio Matsumoto

exists an injection �0lkW zVk ! zVl such that �0

lkı hki D hlj ı �j i :

zUjhlj �� zVl

zUihki

��

ji

��

zVk :

0lk

��

Let p be a point of a complex m-orbifold X , . zUi ; Gi ; �i ; Ui / a folding chart con-taining p. Let Qp be a point (2 zUi ) of ��1

i .p/. The isomorphism class of the isotropysubgroup .Gi / Qp is proved to depend only on p and to be independent of the choiceof the folding chart containing p and of the choice of Qp (cf. [26], Appendix A). If.Gi / Qp ¤ f1g, we call p an isotropic point, and the totality†.X/ of isotropic points iscalled the isotropic set. From the definition of orbifolds, it follows that the isotropicset †.X/ is an analytic subset of complex dimension <D m � 1.

Definition 2.1. A (holomorphic) orbifold map h W X ! Y is said to be generic, if

h.Ui / \ .Vk �†.Y // ¤ ;:for each pair of folding charts . zUi ; Gi ; �i ; Ui / of X and . zVk ;Hk; �k; Vk/ of Y suchthat h.Ui / � Vk .

Lemma 2.2. A generic orbifold map h W X ! Y satisfies the following equivariancecondition (III).

(III) (Equivariance condition). For each pair of open sets, Ui D �i . zUi /; Vk D�k. zVk/; such that h.Ui / � Vk , a (not necessarily injective) group homomor-phism h[

kiW Gi ! Hk is associated with each lifted holomorphic map hki ,

with respect to which hki W zUi ! zVk is an equivariant map, that is, for all.g; u/ 2 Gi � zUi we have

hki .gu/ D h[ki .g/hki .u/:Proof. The argument is similar to the proof of Lemma 1 in [35]. By the assumption,we have ��1

i .Ui � h�1.†.Y /// ¤ ;. Choose a point Qp in it. Let g be any element ofGi . Since �k.hki .g Qp// D h�i .g Qp/ D h�i . Qp/ D �k.hki . Qp//, there exists an elementg0 2 Hk such that

hki .g Qp/ D g0hki . Qp/: (2.1)

The element g0 is uniquely determined by g and (2.1) holds independently of thechoice of Qp, which follows from the connectedness of ��1

i .Ui � h�1.†.Y ///. Thecorrespondence g! g0 is our homomorphism h[

kiW Gi ! Hk .


2.3 Fiber spaces over orbifolds

We will introduce the notion of fiber spaces over orbifolds.In this subsection, the spaces denoted by E will not in general be manifolds, but

will always be (not necessarily normal) complex analytic spaces. Let E be a complexanalytic space. Suppose X is a complex m-manifold. Then a map ' W E ! X is saidto be a fiber space if ' is a surjective, proper and flat holomorphic map. We wouldlike to extend this notion of a fiber space over a manifold X to the case where X is anorbifold.

Let X be a complex m-orbifold. A map ' W E ! X is said to be a fiber spaceover the orbifold X , if it is a surjective, proper holomorphic map and additionally itsatisfies the following two conditions (i) and (ii).

(i) For each p 2 X , there exist a folding chart . zUi ; Gi ; �i ; Ui / ofX containing p, afiber space Q'i W zEi ! zUi over anm-manifold zUi , a holomorphic action ofGi onzEi , and a fibration preserving holomorphic map Q�i W zEi ! '�1.Ui / (called the

fibered folding map) which induces an isomorphism Q�i=Gi W Gin zEi ! '�1.Ui /.We assume that the actions of Gi on zEi and on zUi preserve the fibered structureof Q'i W zEi ! zUi and that the following diagram commutes

Gin zEi

induced projection

��

Q�i=Gi .Š/ �� '�1.Ui /

'j'�1.Ui /

��Gin zUi

�i=Gi .Š/�� Ui :

We will call . Q'i W zEi ! zUi ; Gi ; Q�i ; �i ; Ui / a fibered folding chart of ' W E ! X.

(ii) Let . Q'i W zEi ! zUi ; Gi ; Q�i ; �i ; Ui / and . Q'j W zEj ! zUj ; Gj ; Q�j ; �j ; Uj / be twofibered folding charts of ' W E ! X with Ui � Uj . Then there exits a fibrationpreserving holomorphic embedding Q�j i W zEi ! zEj (called a fibered injection)which covers �j i W zUi ! zUj , and satisfies Q�i D Q�j ı Q�j i .

A fiber space over an orbifold ' W E ! X is called an orbifold fiber space if E is anorbifold and ' is an orbifold map.6

2.4 Orbifold pull-back diagram

Let ' W E ! X be a fiber space over a complexm-orbifoldX. LetX 0 be another com-plex orbifold whose dimension may be different fromm. Let h W X 0 ! X be a genericorbifold map. Then we can canonically construct a new fiber space '0 W E 0 ! X 0, and

6The reason that we do not confine ourselves to the orbifold fiber spaces is that a pull-back of an orbifoldfiber space is not necessarily an orbifold fiber space.

80 Yukio Matsumoto

a commutative diagram

E 0 Qh ��

'0

��

E

'

��X 0

h�� X;

which we will call an orbifold pull-back diagram.The construction is as follows. Let p 2 X 0 be a point and let . zWk; Jk; k; Wk/ a

folding chart ofX 0 containingp. If we takeWk small enough, then there exists a fiberedfolding chart . Q'i W zEi ! zUi ; Gi ; Q�i ; �i ; Ui / of ' W E ! X such that h.Wk/ � Ui . Lethik W zWk ! zUi be a lifted holomorphic map. Since h W X 0 ! X is generic, we mayassume by Lemma 2.2 that the following equivariance condition is satisfied:

(*) there exists a group homomorphism h[ikW Jk ! Gi with respect to which

hik W zWk ! zUi is an equivariant map.

Step 1 (Construction of a fiber space over zWk). Let Q'0kW zE 0

k! zWk be the fiber space

over zWk pulled back from the fiber space Q'i W zEi ! zUi by the lifted holomorphic maphik W zWk ! zUi , that is, zE 0

kis defined by

zE 0k D f.w; e/ 2 zWk � zEi j hik.w/ D Q'i .e/g;

and Q'0kW zE 0

k! zWk is the projection: Q'0

k.w; e/ D w. Also Qhik W zE 0

k! zEi is the

projection: Qhik.w; e/ D e. The map Q'0k

is proper, surjective and flat holomorphicmap. See the diagram below:

zE 0k

Q'0k

��

Qhik �� zEi

Q'k

��zWk

hik

�� zUi :

Step 2 (Group action of Jk on the fiber space Q'0kW zE 0

k! zWk). By the definition of

orbifolds, the group Jk acts on zWk holomorphically and effectively. We let Jk act onzE 0k

as follows:

g.w; e/ D .gw; h[ik.g/e/; g 2 Jk; .w; e/ 2 zE 0k � zWk � zEi : (2.2)

If .w; e/ 2 zE 0k

, then g.w; e/ 2 zE 0k

. In fact, assuming hik.w/ D Q'i .e/, we have

hik.gw/ D h[ik.g/hik.w/ D h[ik.g/ Q'i .e/ D Q'i .h[ik.g/e/:The action of Jk is holomorphic and preserves the fibered structure of Q'0

kW zE 0

k! zWk ,

(i.e. Q'0k.g.w; e// D gw).


Step 3 (Constructions of the pulled back fiber space '0 W E 0 ! X 0). Let E 0k

bethe quotient space Jkn zE 0

k, and let Qk W zE 0

k! E 0

kbe the quotient map. We define

'0kW E 0

k! Wk to be the map

E 0k

. Q�k=Jk/D Jkn zE 0k �! Jkn zWk .�k=Jk /D Wk;

which is naturally induced from the projection Q'0kW zE 0

k! zWk . The fiber space

'0 W E 0 ! X 0 is constructed by pasting together all the fiber spaces '0kW E 0

k! Wk

over all folding charts . zWk ; Jk; k; Wk/. The quotient map Qk W zE 0k! E 0

kcan be con-

sidered the fibered folding map Qk W zE 0k! .'0/�1.Wk/.

Note that we have constructed fibered folding charts

. Q'0k W zE 0

k �! zWk; Jk; Qk; k; Wk/

of '0 W E 0 ! X 0. We have to check the compatibility condition (ii) for them.Let . Q'0

kW zE 0

k! zWk; Jk; Qk; k; Wk/ and . Q'0

lW zE 0

l! zWl ; Jl ; Ql ; l ; Wl/ be the

two fibered folding charts constructed as above. We assume Wk � Wl , and let�0lkW zWk ! zWl be an injection. Choose fibered folding charts . Q'i W zEi ! zUi ; Gi ; Q�i ;

�i ; Ui / and . Q'j W zEj ! zUj ; Gj ; Q�j ; �j ; Uj /of' W E ! X such thatUi � Uj ; h.Wk/ �Ui and h.Wl/ � Uj . Let �j i W Ui ! Uj (or Q�j i W zEi ! zEj ) be a corresponding injec-tion (or a fibered injection). Then, since

zE 0k D f.w; e/ 2 zWk � zEi j hik.w/ D Q'i .e/g;zE 0l D f.w0; e0/ 2 zWl � zEj j hjl.w0/ D Q'j .e0/g;

the map Q�0lkW zE 0

k! zE 0

ldefined by

Q�0lk.w; e/ D .�0

lk.w/;Q�j i .e//

is a fibration preserving holomorphic embedding onto an open subset, which covers�0lkW zWk ! zWl .We have the actions of Jk on zE 0

kand of Jl on zE 0

l, defined by (2.2). The embedding

Q�0lkW zE 0

k! zE 0

lis equivariant with respect to these actions together with the (injective)

group homomorphism .�0lk/[ W Jk ! Jl .

The embedding Q�0lkW zE 0

k! zE 0

ldescends to the quotient fiber spaces to identify

the fiber space '0kW E 0

k! Wk with a fibered subspace of '0

lW E 0

l! Wl , and the

compatibility condition (ii) is verified.The construction of the pulled back fiber space '0 W E 0 ! X 0 is now completed.

82 Yukio Matsumoto

3 Types of mapping classes

Let us recall the Thurston-Bers classification of mapping classes of†g . We will followBers’ terminology:

Theorem 3.1 ([39], [10], and [16]). Let f W †g ! †g be an orientation preservinghomeomorphism. Then the mapping class Œf � is one of the four types.

(1) periodic (elliptic): there exists a positive integer n such that Œf �n D 1 2 �g .

(2) parabolic: Œf � is reduced by a system of disjoint simple closed curves C D C1 [C2 [ [ Cr on †g and its component maps are periodic, in other words, fpreserves C and its restriction to the complement of C is isotopic to a periodicself-homeomorphism of †g � C .

(3) hyperbolic: f is a pseudo-Anosov homeomorphism in Thurston’s sense.

(4) pseudo-hyperbolic: “reducible” by a system of disjoint simple closed curves, butnot parabolic.

The dynamical aspects of the actions of �g on T .†g/ and of the Fuchsian groupon the upper half plane are much alike: a periodic mapping class Œf � has fixed pointson T .†g/. The other types of mapping classes have fixed points on the Thurstonboundary of T .†g/; see [39]. But for our purpose, Bers’ “extremum” formulation ismore suitable; see [10].

Let p1 D ŒS1; w1� and p2 D ŒS2; w2� be two points of the Teichmüller spaceT .†g/. Then the distance (Teichmüller distance) between the two points is defined as

d.p1; p2/ D 1

2log inf K.g/

where g W S1 ! S2 is a homeomorphism isotopic to w�12 ı w1, and K.g/ is the

dilatation7 of g.For Œf � 2 �g , let a.Œf �/ denote the infimum of d.Œf ��.p/; p/ for p 2 T .†g/.

Bers’ classification of mapping classes is the following; see [10], p. 80: Œf � is ellipticif it has a fixed point in T .†g/ (cf. [21]), parabolic if there is no fixed point buta.Œf �/ D 0, hyperbolic if a.Œf �/ > 0 and there is a p 2 T .†g/ with d.Œf ��p; p/ Da.Œf �/, pseudo-hyperbolic if a.Œf �/ > 0 and d.Œf ��.p/; p/ > a.Œf �/ for all p 2T .†g/. The property of being elliptic, parabolic, hyperbolic, or pseudo-hyperbolic ispreserved by inner automorphisms of the mapping class group �g .

Definition 3.2. An orientation preserving homeomorphism f W †g ! †g or its map-ping class Œf � is called pseudo-periodic, if it is periodic or parabolic.

7For the definition of the dilatation K.g/, see [10]. K.g/ takes real values between 1 and C1, withK.g/ D 1 meaning that g is conformal, andK.g/ D C1 meaning that g is not quasi-conformal.


Given a degenerating family ' W M ! � of Riemann surfaces of genus g over aunit disk � D fz j jzj < 1g, the topological monodromy f W †g ! †g is knownto be pseudo-periodic of negative twist.8 (We are assuming that there is a possiblesingular fiber only over the origin z D 0.) For a pictorial explanation of the topologicalmonodromy, see Figure 2.

M

�

'

f

†g

Figure 2. Degenerating family and its monodromy f.

The pseudo-periodic nature of the homological and topological monodromy andits negativity have been clarified by several mathematicians: in Milnor fiberings, thesefacts were discovered by Lê [23], A’Campo [2], Lê, Michel, and Weber [24], andMichel and Weber [29], and in families of Riemann surfaces, by Imayoshi [17], Shigaand Tanigawa [36], and Earle and Sipe [15].

The converse is also true. Two degenerating families of Riemann surfaces of genusg, ' W M ! � and '0 W M 0 ! �, are said to be topologically equivalent and will be

denoted by .M; ';�/TopŠ .M 0; '0; �/, if there exist orientation preserving homeomor-

8 A pseudo-periodic homeomorphism f W †g ! †g is of negative twist, if it is periodic or, in the paraboliccase, if all of its screw numbers (see [31] and [28]) about the reducing curves Ci are negative: s.Ci / < 0.

84 Yukio Matsumoto

phisms H W M !M 0 and h W �! � (with h.0/ D 0) such that the diagram

MH ��

'

��

M 0

'0

��

h��

commutes.

Theorem 3.3 ([27], [28], and [25]). The set �g of all topological equivalence classesof (relatively minimal) degenerating families of Riemann surfaces of genus g ( >D 2)over � is in a bijective correspondence with the set P �

g of all conjugacy classes (in�g ) of pseudo-periodic mapping classes of negative twists. The correspondence isgiven by the topological monodromy

�g 3 ŒM; ';�� 7�! Œf � 2 P �g :

A right-handed full Dehn twist about an essential simple closed curveC on†g is thesimplest example of a pseudo-periodic homeomorphism of negative twist (we considerthat such a Dehn twist gives a .�1/-twist about C.) To this Dehn twist corresponds,under Theorem 3.3, a degenerating family over�whose central fiber is a stable curvewith one node, obtained by pinching the curve C to a point.

4 Fenchel–Nielsen coordinates

The closed surface †g has a system of disjoint simple closed curves

L D fL1; L2; : : : ; L3g�3gsuch that the closure of each connected component of†g �L is a pair of pants (i.e. acompact surface homeomorphism to a 2-sphere with three disjoint open disks deleted);see [1] and [13]. Let p D ŒS; w� be a point of T .†g/. Since the Riemann surface Shas a natural hyperbolic metric (descending from the Poincaré metric of the upper halfplane H D QS ), each simple closed curve w�1.Li / on S is isotopic to a simple closedgeodesic yLi . Moreover, these simple closed geodesics yL D fyL1; yL2; : : : ; yL3g�3g aredisjoint; see [13]. The hyperbolic length li D l.yLi / depends on p 2 T .†g/ real

analytically. Decomposing S along the simple closed geodesics yL, we obtain 2g � 2pairs of pants with geodesic boundaries. This process is called the pants decompositionof S by the system L of simple closed curves.

To recover the Riemann surface S, one has to glue these pants together along thegeodesics yLi using certain twisting parameters �i ; i D 1; 2; : : : ; 3g�3. The mapping

‰.p/ D .l1; : : : ; l3g�3; �1; : : : ; �3g�3/


gives a diffeomorphism of T .†g/ onto R3g�3C �R3g�3. These coordinates‰ are called

the Fenchel–Nielsen coordinates based on the system L of simple closed curves. Cf. [1]and [18], Chapter 3. For the proofs of the following two basic lemmas, see [20] and[1], p. 95.

Lemma 4.1. There exists a universal positive constant M0 such that distinct simpleclosed geodesicsC1 andC2 on a compact Riemann surface of genus g do not intersectif their hyperbolic lengths are shorter than M0, i.e.

l.C1/; l.C2/ < M0 H) C1 \ C2 D ;:

Figure 3 explains the intuitive meaning of Lemma 4.1.

longerlonger

shortershorter

Figure 3. When closed geodesics become shorter, transverse curves become longer.

Lemma 4.2. There exists a universal positive constant M1 such that every compactRiemann surface of genus g has a pants decomposition by a system of simple closedgeodesics yL D fyL1; : : : ; yL3g�3g each member of which has the length shorter thanM1, i.e.

l.yLi / < M1; i D 1; : : : ; 3g � 3:

5 Compactification process of M.†g/

First note that, up to the action of �g , there are at most a finite number of topologicalways to decompose a surface of genus g into 2g � 2 pairs of pants. Given an infinitesequence of points fpng1nD1 � T .†g/, Lemma 4.2 tells us that we may assume, underthe action of �g and in particular under the action of finite products of certain Dehntwists to pn, that there is a certain infinite subsequence, denoted by the same notationfpng1nD1 again, which is contained in

.0;M1� � � .0;M1�„ ƒ‚ …3g�3

� Œ0; 2�� Œ0; 2��„ ƒ‚ …3g�3

with respect to the Fenchel–Nielsen coordinates based on a certain system L of simpleclosed curves.

86 Yukio Matsumoto

Thus either

(1) there exists a subsequence fpn.j /g1jD1 converging to a point of T .†g/ or

(2) there exist a subsequence fpn.j /g1jD1 and a subset fLi.1/; : : : ; Li.k/g of L such

that the hyperbolic lengths l.yLi.1//; : : : ; l.yLi.k// of the corresponding geodesics(for pn.j / ) converge to 0 as j !1.

Therefore, considering .0;M1�� Œ0; 2�� to be the polar coordinates of a punctureddisk D.M1/ � f0g of radius M1, the disk D.M1/ compactifies .0;M1� � Œ0; 2�� byadding the origin, and the product

D.M1/ � �D.M1/

of 3g�3 copies ofD.M1/ compactifies the part ofM.†g/ corresponding to the directproduct .0;M1�� .0;M1�� Œ�2�; 2�� Œ�2�; 2��. The above product ofthe disksD.M1/ is real analytically isomorphic to a part near the “boundary” of Bers’deformation space D.L/, [9], [8]; see [1], p. 104.

6 Bers’ deformation spaces

Bers [9] defines Riemann surfaces with nodes. Let us quote relevant passages from [9](with notation slightly changed).

“A compact Riemann surface with nodes of (arithmetic) genus g.>D 2/ is aconnected complex space S, on which there are k.>D 0/ points P1; : : : ; Pk callednodes, such that (i) every node Pj has a neighborhood homeomorphic to theanalytic set fz1z2 D 0; jz1j < 1; jz2j < 1g, with Pj corresponding to .0; 0/;(ii) the set S � fP1; : : : ; Pkg has r.>D 1/ components †.1/; : : : ; †.r/, calledparts, each †.i/ is a Riemann surface of some genus gi compact except for nipunctures, with 3gi � 3C ni >D 0, and n1 C C nr D 2k; and (iii) we haveg D .g1 � 1/C C .gr � 1/C k C 1.”

Let S and S 0 be Riemann surfaces with nodes.

“A continuous surjection u W S 0 ! S is called a deformation if for every nodeP 2 S , u�1.P / is either a node or a Jordan curve avoiding all nodes and, forevery part †.i/ of S , u�1j†.i/ is an orientation preserving homeomorphism.”

Bers then proceeds to define an equivalence relation of deformations.

“Two deformations, u W S 0 ! S and v W S 00 ! S are called equivalent if there arehomeomorphisms ' W S 0 ! S 00 and W S ! S , homotopic to an isomorphismand to the identity, respectively, such that v ı ' D ı u.”

“The deformation space D.S/ consists of all equivalence classes ŒS 0; u� of de-formations onto S .”


Bers associates to each node P 2 S a subspace of D.S/ called a distinguishedsubset consisting of all ŒS 0; u� 2 D.S/ for which the preimage u�1.P / of P is a nodeof S 0.

Theorem 6.1 ([9]). D.S/ is a cell. There is a (canonical) homeomorphism of D.S/onto C3g�3 which takes each distinguished subset onto a coordinate hyperplane.

Bers realizedD.S/ as a bounded domain in C3g�3; see [9]. The complex structurecoming from the bounded domain is taken to be the natural complex structure ofD.S/.

A system of disjoint simple closed curves C D fC1; C2; : : : ; Ckg on†g is said to beadmissible, if each connected component of†g �C has negative Euler characteristic.Obviously, we have k <D 3g � 3, and if k D 3g � 3, we call C a terminal system. LetC D fC1; C2; : : : ; Ckg be an admissible system of simple closed curves. Let †g.C/denote a Riemann surface with nodes which is obtained by pinching each curve in C

to a point (i.e. a node). †g.C/ has arithmetic genus g.>D 2/. For simplicity, we willdenote the deformation space D.†g.C// by D.C/.

Let C 0 be a sub-system of simple closed curves of C , i.e. C 0 � C . Then thedeformation h W †g.C 0/ ! †g.C/ which pinches each curve in C � C 0 to a nodeinduces a mapping h� W D.C 0/! D.C/ called allowable mapping, which takes eachŒS; u� 2 D.C 0/ into ŒS; h ı u�. In particular, if C 0 D ;, then D.;/ is nothing but theTeichmüller space T .†g/, and the allowable mapping

h� W T .†g/ �! D.C/

is understood as follows (cf. Kra [22]).By adding certain simple closed curves CkC1; : : : ; C3g�3 to C D fC1; : : : ; Ckg,

we obtain a terminal system L which gives a pants decomposition of †g . We canspeak of the Fenchel–Nielsen coordinates based on L. Let �.C/ denote the subgroupof �g generated by .�1/-Dehn twists .Ci / about the simple closed curves Ci 2 C ,i D 1; : : : ; k. Then

�.C/ Š Z˚ ˚Z (with k summands).

Two pointsp1 D ŒS1; w1� andp2 D ŒS2; w2� have the same image under the allowablemapping h� W T .†g/ ! D.C/ if and only if p1 D Œf ��.p2/ for some Œf � 2 �.C/,i.e., in terms of the Fenchel–Nielsen coordinates, if and only if

‰.p1/ D .l1; : : : ; l3g�3; �1 C 2m1�; ; : : : ; �k C 2mk�; �kC1; : : : ; �3g�3/;

where we assume

‰.p2/ D .l1; : : : ; l3g�3; �1; : : : ; �k ; �kC1; : : : ; �3g�3/;

and m1; : : : ; mk are integers. We can adopt

.l1ei�1 ; : : : ; lke

i�k ; lkC1 C i�kC1; : : : ; l3g�3 C i�3g�3/

88 Yukio Matsumoto

as the coordinates for the image h�.T .†g// inD.C/. Thus h�.T .†g// is the quotientT .†g/=�.C/ and is real analytically isomorphic to

.C � f0g/ � � .C � f0g/„ ƒ‚ …k

�C � �C„ ƒ‚ …3g�3�k

:

Let …i denote the i-th coordinate plane in C3g�3:

…i D Ci�1 � f0g �C3g�3�i :

The deformation spaceD.C/ is the completion of the quotientT .†g/=�.C/ by adding

k[iD1

…i :

(See [22], Theorem 9.4.)The coordinate plane …i corresponds to the distinguished subset parametrizing

conformal structures on the Riemann surface with one node†g.fCig/. More generally,for a subset C 0 � C , the intersection \

Ci 2C 0

…i

corresponds to the subset ofD.C/ that parametrizes conformal structures on the Rie-mann surface with nodes†g.C 0/. Thus we can identify this subset with the Teichmüllerspace T .C 0/ of the Riemann surface with nodes †g.C 0/ 9. It is easy to see that

dimC T .C0/ D 3g � 3 � #C 0:

To summarize, D.C/ is the completion of the quotient T .†g/=�.C/ by the union ofthe Teichmüller spaces [

;¤C 0�C

T .C 0/:

Similarly to the above, the general allowable mapping h� W D.C 0/ ! D.C/ isthe projection map onto the quotient D.C 0/=�.C � C 0/, and this quotient space isidentified with an open subset

D.C/ �[

Ci 2C�C 0

T .fCig/:

Kra [22] announces the following result.

9We think that more precisely the notation T.C 0/ should be TC .C0/.


Theorem 6.2 ([22], §3.10). There exists a family of Riemann surfaces (with nodes)

�C W V.C/ �! D.C/:

The total space V.C/ is a .3g � 2/-dimensional complex manifold. Over a pointp 2 T .C 0/, C 0 being a subset of C , we have a Riemann surface with nodes canonicallyhomeomorphic to †g.C 0/.

Figure 4 illustrates this “tautological” fibration.

D.C/ D

i

i

j

j

Figure 4. The deformation space D.C/ parametrizes Riemann surfaces with nodes.

Let N�.C/ denote the normalizer of �.C/ in the mapping class group �g :

N�.C/ D fŒf � 2 �g j Œf ��.C/Œf ��1 D �.C/g:Let W.C/ denote the quotient group N�.C/=�.C/. This group is canonically iden-tified with the mapping class group of the surface with nodes †g.C/:

W.C/ D ff W †g.C/! †g.C/ j orientation preserving homeomorphismg=isotopy:

W.C/ acts on D.C/ as automorphisms. The action is defined as follows:

Œf ��.ŒS; u�/ D ŒS; f ı u�; for Œf � 2 W.C/; ŒS; u� 2 D.C/:The following theorem is an extension of Theorem 1.1. Cf. [22], §4.5, §4.6, and §7.

90 Yukio Matsumoto

Theorem 6.3. The group W.C/ acts on the fiber space �C W V.C/ ! D.C/ in afibration preserving manner. This action is properly discontinuous and holomorphic,and by taking quotients, we obtain an orbifold fiber space

� 0C W Y.C/ �!M.C/;

where Y.C/ and M.C/ denote V.C/=W.C/ and D.C/=W.C/, respectively.

Unfortunately, the relation between the spaces M.†g/ and M.C/ is very difficultto see. To remedy this difficulty, we will introduce “subdeformation spaces”.

7 Subdeformation spaces D".C /

Let C D fC1; : : : ; Ckg be an admissible system of disjoint simple closed curves on†g ,and let h� W T .†g/! D.C/ be the allowable mapping corresponding to the inclusion; � C . Choose positive real numbers "1; �1; "2; �2; : : : ; "3g�3; �3g�3 satisfying

0 < "1 < �1 < "2 < �2 < < "3g�3 < �3g�3 < M0; (7.1)

and fix them (M0 being the number appearing in Lemma 4.1). We define the subde-formation space D".C/ as follows:

D".C/ D fŒS; u� 2 D.C/ j 0 <D l.3u�1.ŒCi �// < "k; i D 1; : : : ; k;and other simple closedgeodesics on S are longer than �kg;

where ŒCi � denotes the node of †g.C/ which is obtained by pinching the simpleclosed curve Ci 2 C to a point, and the suffix k of "k and of �k is the number ofsimple closed curves in C . Thus u�1.ŒCi �/ is the simple closed curve or the node onS which is mapped to the point ŒCi � under the deformation u W S ! †g.C/. The

notation 3u�1.ŒCi �/ denotes the simple closed geodesic which is isotopic to u�1.ŒCi �/,and l.3u�1.ŒCi �// is its length. The equality l.3u�1.ŒCi �// D 0 is understood to mean

that u�1.ŒCi �/ is a node in S . In particular, l.3u�1.ŒCi �// D 0 for all i D 1; : : : ; k, ifand only if ŒS; u� is in the Teichmüller space T .C/. The subdeformation spaceD".C/is an open “handle” whose “core” is T .C/ \ D".C/. (See Figure 5.) The action ofW.C/ on D.C/ preserves the subdeformation space D".C/.

Lemma 7.1. Suppose that a mapping class Œf � 2 �g takes a point ŒS; w� 2 T .†g/into another point ŒS 0; w0� 2 T .†g/, and that h�.ŒS; w�/ and h�.ŒS 0; w0�/ are inD".C/, where h� W T .†g/ ! D.C/ is the allowable mapping. Then Œf � belongs toN�.C/.

Proof. Note thatS andS 0 are nonsingular Riemann surfaces. Since we have ŒS 0; w0� DŒf ��.ŒS; w�/, there exists a biholomorphic map t W S ! S 0 such that the following


Figure 5. Subdeformation space D".C/.

diagram homotopically commutes:

Sw ��

t

��

†g

f

��S 0

w0�� †g :

The map t W S ! S 0 preserves the hyperbolic (Poincaré) metric. Since h�.ŒS; w�/and h�.ŒS 0; w0�/ belong to the subdeformation space D".C/, the closed geodesics3w�1.Ci /; i D 1; : : : ; k; (or 3w0�1.Ci /; i D 1; : : : ; k) are only simple closed geodesicson S (or on S 0) that are shorter than "k . Thus for each i D 1; : : : ; k, the map t takes

the geodesic 3w�1.Ci / to a geodesic 3

w0�1.Cj / with some j 2 f1; : : : ; kg. Hencef W †g ! †g induces a permutation (up to isotopy) of the simple closed curves ofC . This implies that Œf � belongs to the normalizer N�.C/.

Lemma 7.1 is an analogy of Shimizu’s lemma on Fuchsian groups; see [18].

As we said above, the relation betweenM.†g/ andM.C/ is complicated. However,if we define the submoduli spaceM".C/ to be the quotientD".C/=W.C/, its relationto M.†g/ is clear: By Lemma 7.1, M".C/ projects homeomorphically onto an opensubset of M.†g/. We will identify M".C/ with the projected image. Note that thereare only finitely many subdeformation spaces D".C/ and submoduli spaces M".C/,because, if C and C 0 are equivalent under an orientation preserving homeomorphismf W †g ! †g (i.e. C 0 D f .C/), then .D".C/;W.C// and .D".C 0/;W.C 0// are“identical”.

Lemma 7.2. The “boundary” M.†g/�M.†g/ is covered by the submoduli spaces:

M.†g/ �M.†g/ �[

C¤;M".C/:

Proof. Let "1; �1; "2; �2; : : : ; "3g�3; �3g�3 be positive real numbers satisfying in-equality (7.1). Let S be a Riemann surface or a Riemann surface with nodes. Let

92 Yukio Matsumoto

C1; C2; : : : ; Cn be the totality of simple closed geodesics onS whose lengths l.Ci /; i D1; 2; : : : ; n; are shorter thanM0. By Lemma!4.1 we have n <D 3g� 3. (Here we adopta convention that a node is a special case of a simple closed geodesic whose lengthis equal to 0.) Put ci D l.Ci /, for simplicity. Permuting Ci ’s, if necessary, we mayassume

0 <D c1 <D c2 <D <D cn:

Claim. If c1 < "1, then there exists a suffix k with 1 <D k <D n such that

0 <D c1 <D c2 <D <D ck < "k < �k < ckC1 <D <D cn: (7.2)

In order to prove Lemma 7.2, we have only to prove Claim. In fact, for a nodal sur-face S , we have c1 D 0, and by Claim, the length sequence l.Ci /; i D 1; : : : ; n;

satisfies (7.2) for a certain suffix k, then ŒS; u� belongs to D".C/, where C DfC1; C2; : : : ; Ckg, and u W S ! †g.C/ is the map that pinches Ci to a node, i D1; : : : ; k.

Let us prove Claim. Our proof is a variant of the pigeonhole argument, but withfewer pigeons than pigeonholes. Let us denote a closed interval fx 2 R j ˛ <D x <D ˇgby Œ˛; ˇ� as usual. Among the n intervals

Œ"1; �1�; Œ"2; �2�; : : : ; Œ"n; �n�;

there exists at least one interval Œ"h; �h� that does not contain any ci from the n � 1numbers fc2; : : : ; cng. (We will call such an interval a gap.) Suppose h is the smallestamong the suffixes of such gap intervals. Then each Œ"j ; �j � with 1 <D j <D h � 1contains at least one ci from fc2; : : : ; cng. Thus taking account of c1, there are at leasth ci ’s satisfying ci < "h.

If the number of such ci ’s is precisely h, then we are done. (The suffix h turns outto be k of (7.2) in Claim.)

If not, there are at least hC 1 ci ’s satisfying ci < "h.Now there are two cases to be considered.

Case (i). There do not exist any gap intervals Œ"l ; �l � with h < l <D n.

In this case, there are at least n�h ci ’s satisfying �h < ci , because there are n�hintervals Œ"j ; �j � with h < j <D n, each containing at least one ci from fc2; : : : ; cng.This together with the assumption that there are at least hC 1 ci ’s satisfying ci < "hwould imply that the total number of ci ’s is at least .hC 1/C .n� h/ D nC 1. Thisis a contradiction, and Case (i) is impossible.

Case (ii). There exist gap intervals Œ"l ; �l � with h < l <D n.

Suppose l is the smallest suffix of a gap interval greater than h. Then we see thatthere are at least l ci ’s satisfying ci < "l , because there are at least hC1 ci ’s satisfyingci < "h, and each interval Œ"j ; �j � with h < j < l contains at least one ci . (Thus thenumber of ci ’s satisfying ci < "l is at least .hC 1/C .l � h� 1/ D l .) If the numberof such ci ’s is precisely l , then we are done. (The suffix l turns out to be k of (7.2) inClaim.)


If not, there are at least l C 1 ci ’s satisfying ci < "l .Now we can repeat the same argument as above and proceed inductively. This

completes the proof of Claim, and as we remarked above, the proof of Lemma 7.2 isfinished.

Now we know the “folding charts”10 of M.†g/ as an orbifold. They are

.T .†g/; �g ; proj;M.†g// and .D".C/;W.C/; proj;M".C//;

where C runs over all (isotopy classes of) non-empty admissible systems of simpleclosed curves on †g .

We remark that a pseudo-periodic (or pseudo-hyperbolic) mapping class Œf � 2�gwhich is reduced by an admissible system C acts onD".C/ periodically (or hyper-bolically).

8 The universal degenerating family

Recall from Section 1 that Bers’ fibration V.†g/ ! T .†g/ admits an action of �gwhich preserves the fibered structure. Taking the quotient, we obtain an orbifold fiberspace Y.†g/! M.†g/. This fiber space is compactified to Y.†g/! M.†g/. Thefibered compactification process is as follows.

We restrict the fibration of Theorem 6.2 to D".C/, then we obtain a fibration

�C ;" W V".C/ �! D".C/;

which admits a fibration preserving action of W.C/. Taking the quotient, we obtainan orbifold fiber space Y".C/!M".C/. The compactified fiber space

Y.†g/ �!M.†g/

is nothing but a fibered union

Y.†g/ [[

C¤;Y".C/ �!M.†g/ [

[C¤;

M".C/:

Bers’ fibration V.†g/ ! T .†g/ together with the action of �g is considered a“(generalized) fibered folding chart”of the compactified orbifold fibration Y.†g/!M.†g/. The fibration �C ;" W V".C/! D".C/ together with the action ofW.C/ givesanother type of a fibered folding chart. These two types of fibered folding charts givethe structure of an orbifold fiber space to Y.†g/!M.†g/.

The proof of the universality of Y.†g/ ! M.†g/ is as follows. (A prototypeof our argument is found in [17].) We are given a degenerating family of Riemannsurfaces over a Riemann surface B , ' W M ! B , which is almost asymmetric. Since

10Precisely speaking, these are “generalized folding charts”, because the groups �g andW.C/ are not finitegroups but infinite groups acting properly discontinuously.

94 Yukio Matsumoto

our argument is essentially local, we may assume that B is a unit disk � D fz 2C j jzj < 1g and that outside the origin 0 2 � every fiber is nonsingular.

Applying Theorem 3.3, Ashikaga [4] proved a precise stable reduction theorem,which plays a key role in our proof.

Let ' W M ! � be a degenerating family of Riemann surfaces of genus g over theunit disk�which is almost asymmetric. We assume as said above that all fibers outsidethe origin are nonsingular. Its topological monodromy f W †g ! †g is pseudo-periodic. Following Ashikaga [4], we call a positive integer N the pseudo-period off, if N is the smallest among the positive integers n having the property that f n isisotopic to a product of integral Dehn-twists about disjoint simple closed curves on†g . In what follows, N always denotes the pseudo-period of f.

Let '0 W M 0 ! � denote the normally minimal model of ' W M ! �: By con-tracting several (explicitly known) linear orY-shaped configurations of rational curvesin M 0, Ashikaga gets a complex analytic space M# with a fibration '# W M# ! �.The total space M# may have only isolated quotient singularities: cyclic quotient sin-gularities (corresponding to the contracted images of linear configurations of rationalcurves) or dihedral quotient singularities (corresponding to the contracted images ofY-shaped configurations of rational curves).

The following is the first half of Ashikaga’s precise stable reduction theorem.

Theorem 8.1 ([4], Theorem 2.2.1). (1) Let z D wN W �.N/ ! � be the cycliccovering of theN -th power. Let '.N/ W M .N/ ! �.N/ be the pureN -th root fibrationof '# W M# ! �, whereM .N/ is the normalization of the fiber product �.N/ ��M#.Then '.N/ W M .N/ ! �.N/ is a stable reduction of ' W M ! �.

(2) The cyclic group Z=N acts holomorphically on '.N/ W M .N/ ! �.N/ sothat the action preserves the fibered structure. The quotient fiber space coincides with'# W M# ! �.

We consider � as an orbifold with the folding chart .�.N/;Z=N;wN ; �/. ThenAshikaga’s stable reduction theorem gives to '# W M# ! � a structure of a fiber spaceover the orbifold�. To see this, let Q� W M .N/ !M#.D .Z=N/nM .N// be the quotientmap. Then the following diagram commutes:

M .N/

'.N /

��

Q� �� M#

'#

��.N/

wN

�� :

Thus .'.N/ W M .N/ ! �.N/;Z=N; Q�;wN ; �/ becomes a fibered folding chart for'# W M# ! � in the sense of Section 2.

Now we will complete the proof of Theorem 1.2.


Case I: f is periodic with period N. In this case, the fibration'.N/ W M .N/ ! �.N/

is topologically trivial, and M .N/ is a complex manifold. Thus there is a continuousmap H W M .N/ ! †g such that for each w 2 �.N/, the restriction to the fiber.'.N//�1.w/ is an orientation preserving homeomorphism

Hw W .'.N//�1.w/ �! †g :

The pair ..'.N//�1.w/;Hw/determines a point Œ.'.N//�1.w/;Hw �of the Teichmüllerspace T .†g/. Over this point, the identical Riemann surface .'.N//�1.w/ is situated.Define a (holomorphic) map b W �.N/ ! T .†g/ by

b.w/ D Œ.'.N//�1.w/;Hw �: (8.1)

Then we may identify

M .N/ D �.N/ �T.†g/ V.†g/; (8.2)

and we obtain a pull-back diagram (in the ordinary sense):

N .N/

'.n/

��

Qb �� †g

��.N/

b�� T .†g/:

(8.3)

The asymmetry assumption on ' W M ! � implies that the image b.�.N// inter-sects the isotropic points of the �g -action on T .†g/ only in b.f0g/. Thus passing tothe quotients, we have a (holomorphic) orbifold map

b# W � �!M.†g/

which is generic in the sense of Section 2.Let � W M .N/ ! M .N/ be the generator of the Z=N -action that descends to

the action of exp.2�i=N / on �.N/. Then the following diagram is homotopicallycommutative, because the pure N -th root fibration is constructed from the periodicmonodromy f of '# W M# ! �:

.'.N//�1.w/

��

Hw �� †g

f

��.'.N//�1.exp.2�i=N /w/

Hexp.2�i=N /w

�� †g :

(8.4)

This implies

Œ.'.N//�1.exp.2�i=N /w/;Hexp.2�i=N/w � D Œf ��Œ.'.N//�1.w/;Hw �: (8.5)

96 Yukio Matsumoto

As explained in Section 2, a homomorphism b[ W Z=N ! �g is associated withthe lifted holomorphic map b W �.N/ ! T .†g/ of the generic orbifold map b# W �!M.†g/. We have

b.�w/ D Œ.'.N//�1.�w/;H w � (by (8.1))

D Œ.'.N//�1.exp.2�i=N /w/;Hexp.2�i=N/w �

D Œf ��Œ.'.N//�1.w/;Hw � (by (8.5))

D Œf ��b.w/:In terms of b[, this equality is written as

b[.�/ D Œf ��: (8.6)

By the definition of the action of Z=N on the fiber product (see (2.2) in Section 2)

�.N/ �T.†g/ V.†g/;

we have

�.w; e/ D .�w; b[.�/e/D .exp.2�i=N /w; Œf ��e/;

(8.7)

for all .w; e/ 2 �.N/ �T.†g/ V.†g/. Recall that �g acts on T .†g/, and tauto-logically on V.†g/, see [7]. By this tautological action, Œf �� .2 �g/ maps theRiemann surface .'.N//�1.w/ .� V.†g// over the point Œ.'.N//�1.w/;Hw � .2T .†g// to the Riemann surface .'.N//�1.exp.2�i=N /w/ .� V.†g// over the pointŒ.'.N//�1.exp.2�i=N /w/;Hexp.2�i=N/w � .2 T .†g// by the biholomorphic map-ping �. (Compare (8.4) and (1.1) of Section 1.)

Thus from (8.7), we have

�.w; e/ D .exp.2�i=N /w; �e/;

and we can identify the action of � on �.N/ �T.†g/ V.†g/ with the action of �on M .N/. Thus the fibered folding chart .'.N/ W M .N/ ! �.N/;Z=N; Q�;wN ; �/of '# W M# ! � given by Ashikaga’s theorem (Theorem 8.1) coincides with thefibered folding chart constructed in Step 3 in the construction of the pulled back fiberspace (§2). Passing to the quotient of the pull-back diagram (8.3), we get the orbifoldpull-back diagram:

M#��

'#

��

Y.†g/

��

b#

�� M.†g/ :

(8.8)

This completes the proof of Theorem 1.2 in Case I.


Case II: f is a parabolic homeomorphism with pseudo-period N , and is reducedby an admissible system of simple closed curves C D fC1; : : : ; Ckg on †g . Inthis case, the central fiber of '.N/ W M .N/ ! �.N/ is a stable curve homeomorphic to†g.C/. The total space M .N/ admits only type Al cyclic quotient singularities (seeFigure 6) at the nodes of the central fiber; see [4], [32], p. 212.

�2�2�2�2�2: : :

Figure 6. Resolution diagram of a type Al cyclic quotient singularity (with l vertices).

Thus there is a continuous mapH W M .N/ ! †g.C/ such that for eachw 2 �.N/its restriction to the fiber .'.N//�1.w/ is a deformation in Bers’ sense:

Hw W .'.N//�1.w/ �! †g.C/:

The pair ..'.N//�1.w/;Hw/ determines a point Œ.'.N//�1.w/;Hw � of Bers’deforma-tion space D.C/, over which the identical Riemann surface with nodes .'.N//�1.w/is situated. Thus this construction gives a pull-back diagram:

M .N/Qb ��

'.N /

��

C

��.N/

b�� D.C/:

The image of the origin b.0/ is in the Teichmüller space T .C/ of the surface withnodes †g.C/, but outside the origin f0g, the fiber .'.N//�1.w/ is nonsingular. Thusshrinking the radius of �.N/, if necessary, we may assume that the image b.�.N// iscontained in the subdeformation space D".C/11. We get a pull-back diagram:

M .N/Qb ��

'.N /

��

V"

��.N/

b�� D".C/:

(8.9)

Analyzing the types of the singularities of M .N/ and monodromy homeomor-phisms around them ([37], [4] §3), and then comparing the results with the Fenchel–Nielsen coordinates, we know that b W �.N/ ! D".C/ has the information of the screw

11As the case may be, we might have to change the values of "j ’s and �j ’s in the sequence (7.1) of Section 8.Lemma 7.2 assures that any choice does not affect the fact that .T .†g/; �g/ and f.D".C/;W.C//gC givethe (generalized) folding charts of the orbifoldM.†g/.

98 Yukio Matsumoto

numbers s.Ci /, that is, if s.Ci / D mi ; i D 1; : : : ; k, then the induced homomorphism

b� W Z Š �1.�.N/ � f0g/ �! �1

�D".C/ �

k[iD1

T .fCig/�

Š Z˚ ˚Z .k summands/

maps the generator 1 to .m1; : : : ; mk/.The parabolic monodromy homeomorphism f gives a periodic map Œf � inW.C/.

Here W.C/ is identified with the mapping class group of †g.C/. Let

� W M .N/ �!M .N/

be the generator of the Z=N -action as in Case I. Then the following diagram ishomotopically commutative:

.'.N//�1.w/

��

Hw �� †g.C/

Œf �

��.'.N//�1.exp.2�i=N /w/

Hexp.2�i=N /w

�� †g.C/ :

This implies that

Œ.'.N//�1.exp.2�i=N /w/;Hexp.2�i=N/w � D Œf ��Œ.'.N//�1.w/;Hw �;

and that the maps .b; Qb/ in the diagram (8.9) are equivariant with respect to the actionof � on M .N/ ! �.N/ and that of Œf �� on V".C/! D".C/. By the same argumentas in Case I, we get the orbifold pull-back diagram:

M#

'#

��

�� Y".C/

��

b#

�� M".C/ :

This completes an outline of Case II.


Pasting together the local data of Cases I and II, we obtain the pull-back diagram12

M#

'#

��

�� Y.†g/

��B

b#

�� M.†g/:

This completes the proof of Theorem 1.2.

Remark 8.2. SinceM .N/ has only “mild” singularities, it is easy to see that the map'# W M# ! B is an orbifold fiber space.

Remark 8.3. Our main theorem (especially if it is vaguely stated as in footnote 4 ofSection 1) is almost straightforward in birational sense. The following argument isdue to the referee who cautioned the author about the possibility of misunderstanding.

Let � W xCg ! xMg be the “universal family”over the Deligne–Mumford compacti-fication xMg , i.e. xCg is the moduli of one-pointed stable curves. The orbifold structuregiven in §1 of [14] is as follows. For any point ŒC � 2 xMg , the orbifold chart .UŒC �; G/is given byUŒC � D Ext1.�1C ;OC /,G D Aut.C /where�1C is the Kähler differential.The duality theorem says that Ext1.�1C ;OC / ' H 0.C; !C ˝�1C / ' C3g�3 (!C isthe dualizing sheaf). The action of Aut.C / to the differential also induces the actionto UŒC �. Therefore xMg DS.UŒC �=Aut.C //.

The space UŒC � is the Kuranishi space of C and the versal family exists over UŒC �,which coincides with the restricted fibration of � over UŒC �. If C is automorphismfree, then UŒC � is non-singular and is effectively parametrized.

Let M.0/g be the open set of xMg which parametrizes the automorphism free non-

singular curves and let �.0/ W xC .0/g ! xM.0/g be the restriction. The family �.0/ behaves

as the fine moduli of the automorphism free non-singular curves. Namely a smoothfamily of smooth curves such that any fiber of it is automorphism free can be pulledback by this family.

Now let' W M ! B be an almost asymmetric degenerate family as in Theorem 1.2.Let '.0/ W M .0/ ! B be the restricted (open) family to the locus of automorphismfree non-singular fibers. Since B is complex one-dimensional, the induced map (themoduli map) B.0/ ! xM.0/

g has a holomorphic extension �' W B ! xMg (by thevaluative criterion or Imayoshi [17]).

Let � 0 W M 0 ! �'.B/ be the restricted family of � over �'.B/, and let '0 W zM !B be the fiber product of M 0 and B over �'.B/ (i.e. the natural pull back). Then therestriction of '0 over B.0/ coincides with '.0/ W M .0/ ! B.0/ Therefore '0 W zM ! B

is birationally equivalent to ' W M ! B . If we resolve the singularities of zM andcontract .�1/ curves if necessary, we reach the original family ' W M ! B by the

12Our asymmetry condition on ' W M ! B assures non-ambiguity of the pasting process.

100 Yukio Matsumoto

uniqueness of the relatively minimal model, i.e. ' is pulled back from � modulomodifications.

More details will appear elsewhere.

Acknowledgement

The author is grateful to the referees for their careful reading, valuable comments andsuggestions.

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102 Yukio Matsumoto

[27] Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degenerationof Riemann surfaces, Lecture Notes in Mathematics 2030, Springer, Berlin 2011. 75, 84

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http://library.msri.org/books/gt3m/

Algebraic local cohomologiesand local b-functions attached to

semiquasihomogeneous singularities with L.f / D 2

Yayoi Nakamura� and Shinichi Tajima

Department of Mathematics, School of Science and Engineering, Kinki University3-4-1, Kowakae, Higashi-Osaka, Osaka, 577-8502, Japan


Graduate School of Pure and Applied Sciences, University of Tsukuba1-1-1, Tenodai, Tsukuba, Ibaraki, 305-8571, Japan

email: [email protected]

Abstract. The role of the weighted degree of algebraic local cohomology classes in the com-putation of b-function is discussed. The result which is similar to quasihomogeneous cases isobserved for semiquasihomogeneous isolated singularities with L.f / D 2.

1 Introduction

For quasihomogeneous singularities, the Berenstein–Sato polynomial or the b-func-tion can be determined by a Poincaré polynomial. This result can be interpreted interms of the weighted degrees of algebraic local cohomology classes. For isolatedsingularity cases, since the b-function is defined as the characteristic polynomial ofthe action to a certain space of algebraic local cohomology classes, it is natural toexpect that there exist relations between b-function and weighted degrees of alge-braic local cohomology classes for semiquasihomogeneous singularity cases. For thispurpose, we adopt T. Yano’s method for computing b-function for semiquasihomoge-neous isolated singularities by taking account of weighted degrees of algebraic localcohomology classes in this paper.

In Section 2, we define the weighted degree of algebraic local cohomology classes.In Section 3, we recall the method introduced byYano in [9] for computing the b-func-tion. In Section 3.1, we give the definition of b-function in the context of DX -moduletheory. In Section 3.2, we define invariantL.f / and describe a method for computingthe b-function when L.f / D 2. In Section 4, we illustrate the method with exam-ples and show that it can be simplified by examining the structure of the action of s

�Work partially supported by KAKENHI(21740108).



104 Yayoi Nakamura and Shinichi Tajima

on the space of algebraic local cohomology classes. For the computations of exam-ples, we use computer algebra system Risa/Asir [3] in OpenXM, a project to integratemathematical software systems.

2 Algebraic local cohomology attached tosemiquasihomogeneous singularity

Let X be an open neighborhood of origin O of Cn (n 2 N). Denote by OX thesheaf of holomorphic functions and Bpt D Hn

ŒO�.OX / the sheaf of n-th algebraic local

cohomology groups supported at the origin.Any algebraic local cohomology class � in Bpt can be expressed in terms of a

relative Cech cohomology (cf. [1]):

� DX2ƒ�

c

�1

x

�D

X.`1;:::;`n/2ƒ�

c.`1;:::;`n/

"1

x`1

1 x`2

2 x`nn

#;

where c 2 C, x D x1

1 : : : xnn with � D .�1; : : : ; �n/ 2 ƒ�, ƒ� a finite subset of

NnC.Let us choose a weight vector w D .w1; : : : ; wn/ 2 QnC for a fixed coordinate

system x D .x1; : : : ; xn/ 2 X . Put

jwj D w1 C C wnand

hw; �i D �1w1 C C �nwnfor � D .�1; : : : ; �n/ 2 Nn. Then we define the weighted degree of algebraic localcohomology class �

1

x

�D �hw; �i:

Definition 2.1. The weighted degree degw.�/ of a cohomology class

� DX2ƒ�

c

�1

x

�(2.1)

for a weight vector w D .w1; : : : ; wn/ 2 QnC is the negative rational number definedby

dw.�/ D minf�hw; �i j � 2 ƒ�g;whereƒ� is a set of all exponents � D .�1; : : : ; �n/ 2 NnC with non-zero coefficientsc in expression (2.1) of cohomology class �.

Algebraic local cohomologies and local b-functions 105

Example 1. Let f D x3C xy4C y7. A function f, which is not weighted homoge-neous, defines a quasihomogeneous isolated singularity at the origin with weight vector�13; 16

�of weighted degree 1 (cf. [5]). The dual space of Milnor algebra OX=

�@f@x; @f@y

�as a vector space is spanned by the following ten algebraic local cohomology classes,and the number on the right hand side of each class is the corresponding weighteddegree:

�1

xy

� �� 12

�;

�1

xy2

� �� 23

�;�

1

x2y

� �� 56

�;

�1

xy3

� �� 56

�;�

1

x2y2

�.�1/;

�1

xy4

�.�1/;�

1

x2y3

� �� 76

�;

��13

1

x3yC 1

xy5

� �� 76

�;�

�13

1

x3y2C 1

xy6

� �� 43

�;

��74

1

x2y4� 13

1

x3y3C 1

xy7

� �� 32

�:

Note that in [1], by taking account of the notion of weighted degrees, we gavea method for constructing relative Cech cohomologies that constitute the dual spaceas a vector space of the Milnor algebra of the semiquasihomogeneous function withisolated singularity at the origin.

3 Yano’s method for computing b-function

In this section, we recall the method for computing the b-function introduced byT. Yano in [9] and the role of algebraic local cohomologies in b-function theory.

3.1 b-function

Denote by DX the sheaf of germs of the holomorphic linear partial differential oper-ators of finite order on X . Put

DX Œs� D DX ˝C CŒs�:

For holomorphic function f D f .x/ 2 OX that defines an isolated singularity atthe origin, there exist a differential operator P.s/ 2 DX Œs� and a polynomial b.s/ 2CŒs� such that

P.s/f sC1 D b.s/f s : (3.1)


The polynomial b.s/ is known to have negative rational roots and is called the Beren-stein–Sato polynomial or the b-function.

The notion of b-function can be interpreted in DX -module theory as follows. Set

N D DX Œs�fs :

We define the left DX Œs�-ideal J.s/ composed of the annihilators of f s:

J.s/ D fP.s/ 2 DX Œs� j P.s/f s D 0g:Then N is equivalent to DX Œs�=J.s/. N has a DX Œt; s�-module structure with t ands actions defined by

t W P.s/ 7�! P.s C 1/f;s W P.s/ 7�! P.s/s:

Set

M D N =tN :

Then

M D DX Œs�fs=DX Œs�f

sC1

D DX Œs�=.J.s/CDX Œs�f /

holds. Thus, the b-function is the minimal polynomial of action s to M:

s W M �!M;

P.s/f s 7�! sP.s/f s :

Definition 3.1. The minimal polynomial b.s/ of s 2 EndD.N =tN / is called theb-function of N .

SetzM D .s C 1/M:

Denote by A the Jacobi ideal of function f . Then, we have the isomorphism

zM D DX Œs�=.J.s/CDX Œs�.ACOXf //:

If f .x/ ¤ 0, then

P.�1/f �1C1 D b.�1/f �1

holds. Thus, the polynomial Qb.s/ exists such that

b.s/ D .s C 1/ Qb.s/:the polynomial Qb.s/ is the minimal polynomial of action s to zM. That is, the determi-nation of the b-function is reduced to the study of zM.


3.2 Computation of b-function

We describe the method for computing b-function for isolated singularity case intro-duced by T. Yano. He developed the method in [9] by introducing an invariant L.f /.Before giving the definition of L.f /, let us briefly recall the quasihomogeneous case.

Quasihomogeneous case. If the function f is quasihomogeneous with weight vectorw D .w1; : : : ; wn/ of weighted degree dw.f / D 1, then, after a suitable coordinatetransformation, f will satisfy

X0f D fwith

X0 DnX

jD1wjxj

@

@xj:

Then s � X0 2 J.s/ holds. Thus P.s/ D PsjPj .x;D/ 2 D Œs� and Pj .x;D/X

j0

are congruent modulo J.s/. Thus, for quasihomogeneous case,

zM Š DX=DXA

holds. This property means that L.f /, the precise definition is given later, is equal toone. Furthermore, it is known that function f is quasihomogeneous is equivalent tothe condition L.f / D 1 ([5]).

Applying functor HomDX.;Bpt/ to the representation

0 � zM � D.fi / �� Dn

of zM, we have0 �! F �! Bpt �! Bn

pt;

where F D HomD. zM;Bpt/ and .fi / D t .f1; : : : ; fn/ with fi D @f=@xi (i D1; : : : ; n). Since the action of s on F is X0 and the action of X0 to an algebraic localcohomology class computes the weighted degree of the algebraic local cohomologyclass, the sets fs j b.s/ D 0g and fdw.�/ j � 2 F g are clearly identical.

Example 2. Consider the b-function of f D x3Cxy4Cy7 in the previous example.The b-function is b.s/ D .sC 1/2.2sC 1/.2sC 3/.3sC 2/.3sC 4/.6sC 5/.6sC 7/.

Examples 1 and 2 show that this relation holds not only for weighted homogeneousfunctions but also for quasihomogeneous functions.

ForP.s/ D

XjsjPj .x;D/ 2 DX Œs�;

letordTP.s/ D maxj .j C ordPj /:


Definition 3.2. The total order L.f / for a function f 2 OX that defines an isolatedsingularity at the origin is given by

L.f / D minfordTP.s/ j P.s/ 2 J.s/g:

Let us consider the case L.f / D 2. There are non-constant functions in the idealquotient A W f . Let a� (� D 1; : : : ; r) be the generators of A W f . Let a�;i .x/ 2 OX.i D 1; : : : ; n/ be functions satisfying

a�.x/f CnXiD1

a�;i .x/fi D 0

for each a�.x/. Set

a0� D

nXiD1

a�;i .x/@

@xi:

A representation of zM is given by

0 � zM�1s

� �� D2

X

�fi 0f 0

a0� a�

�� DnCrC1

X ;

where 0@fi 0

f 0

a0� a�

1A D t

f1 : : : f1 f a0

1 : : : a0r

0 : : : 0 0 a1 : : : ar

!:

Applying the functor HomDX.;Bpt/, we have

0 �! F �! B2pt ! BnCrC1

pt

withF D HomDX

. zM;Bpt/:

Let

F1 D fu 2 Bpt j .ACOXf /u D 0gand

F2 D fv 2 Bpt j .A W f /v D 0g:Set

�1 D dimF1 and �2 D dimF2:

Then

�1 D dim OX=.ACOXf /;

�2 D dim OX=.A W f /


and thus

�1 C �2 D � D dim OX=A

holds. Since F2 � F1, for a basis .u1; : : : ; u�2/ of F2, we can take a basis of F1

as .u1; : : : ; u�2; u�2C1; : : : ; u�1

/. For each ui 2 F1, there exists algebraic localcohomology class vi such that

a�.x/v D �a0�.x;D/u mod F2:

Then�0ui

�; i D 1; : : : ; �2 and

�uivi

�; i D 1; : : : ; �1 form the basis of F.

There exist a first order differential operatorA 2 DX and a second order differentialoperator B 2 DX such that

s2 C As C B 2 J.s/:

The action of s on F is represented by

s W�u

v

7�!

�0 1

�B �A�

u

v

:

Then, the b-function b.s/ D .s C 1/ Qb.s/ is given as the minimal polynomial ofrepresentation matrix of s on the above basis of F.

By using the existing algorithms (see [2], [4], and [8]), this method can be realizedin computer algebra systems.

4 Semiquasihomogeneous singularities with L.f / D 2

For semiquasihomogeneous cases, there are annihilators of f s with similar propertiesto Euler operatorX0 of the quasihomogeneous case in §3.2. That is, some differentialoperators clarify the relations between the weighted degrees of algebraic local coho-mology classes and the roots of the b-function. Furthermore, analyzing annihilatorsfrom the viewpoint of weighted degrees, the computations of the b-function of casesof L.f / D 2 are reduced to small matrix calculations. Let us give an example first.

We use the notation h˛; ˇi to represent the algebraic local cohomology class 1

x˛yˇ

�.

Example 3. Let f D x4 C y5 C xy4. f is a semiquasihomogeneous function withweight vector .1

4; 15/ of weighted degree 1 and L.f / D 2.

The following twelve cohomology classes constitute a basis of the dual spacef� 2 Bpt j fj� D 0; j D 1; : : : ; ng of OX=A as a vector space:


h1; 1i�� 9

20

��10 Dh1; 2i

�� 1320

�; �9 Dh2; 1i

�� 7

10

�;

�8 Dh1; 3i�� 1720

�; �7 Dh2; 2i

�� 9

10

�; �6 Dh3; 1i

�� 1920

�;

�5 Dh1; 4i�� 2120

�; �4 Dh2; 3i

�� 1110

�; �3 Dh3; 2i

�� 2320

�;

�2 D h2; 4i � 45h1; 5i C 1

5h4; 1i

�� 1310

�;

�1 D h3; 3i�� 2720

�;

and

h3; 4i � 45h2; 5i C 16

25h1; 6i C 1

5h5; 1i � 4

25h4; 2i

�� 3120

�:

The number on the right hand side is the weighted degree of each cohomology class.Then

F1 D SpanCfh1; 1i; �1; : : : ; �10gand

F2 D SpanCfh1; 1ig:Using the first order annihilators

a�;1@

@xC a�;2 @

@yC a� ; � D 1; 2

with

.a1; a1;1; a1;2/ D .�48x � 60y; 12x2 C 15xy; 9xy C 12y2/

.a2; a2;1; a2;2/ D .144y2 � 60x � 1200y;�36xy2 � 9y3 C 15x2 C 300xy;� 27y3 C 9x2 C 240y2/;

one can verify that the following twelve classes constitute a basis of space F DHomDX

. zM;Bpt /:�0

h1;1i�;

�h1;1i0

�;

� �10

� 1320�10

�;

� �9

� 710�9C 1

100�10

�;

� �8

� 1720�8C 3

500�9� 3625�10

�;

� �7

� 910�7C 1

50�8� 2625�9C 8

3125�10

� � �6

� 1920�6C 1

100�7� 1125�8C 4

3125�9� 1615625�10

�;

� �5

� 2120�5� 1

100�6C 1125�7� 4

625�8C 1615625�9� 64

78125�10

�;

� �4

� 1110�4C 3

100�5C 3500�6� 3

625�7C 123125�8� 48

78125�9C 192390625�10

�;

Algebraic local cohomologies and local b-functions 111� �3

� 2320�3C 1

50�4� 2125�5� 2

625�6C 83125�7� 32

15625�8C 128390625�9� 512

1953125�10

�;

� �2

� 1310�2C 9

500�3� 9625�4C 36

3125�5C 3615625�6� 144

78125�7C 576390625�8� 2304

9765625�9C 921648828125�10

�;

and ��1

�;

where

� D � 2720�1 C 3

100�2 � 3

625�3 C 12

3125�4 � 48

15625�5 � 48

78125�6 C 192

390625�7

� 768

1953125�8 C 3072

48828125�9 � 12288

244140625�10

We have in J.s/ a second order annihilator P D s2 C As C B where

A D � 1

8000

1

1 � 16125y.�500y@x C 300y@y C 1216y � 7700/;

B D � 1

8000

1

1 � 16125y...�64y C 500/x2 C 125yx C 36y3/@2x

C .�36x2 C .�96y2 C 720y/x � 32y3 C 100y2/@x@yC ..�368y C 2425/x � 72y2 C 125y/@xC .24x2 � 29yx � 36y3 C 260y2/@2yC .�105x � 264y2 C 1795y/@y/:

Then the upper ten minors of the representation matrix of s on the basis above form atriangle whose diagonal elements are the weighted degrees of the corresponding alge-braic local cohomology classes. Thus we only need to compute the minimal polynomialof the lower two minors. For the cohomology classes, e1 D

�0h1;1i�

and e2 D� h1;1i

0

�we have �

0 1

�B �Ae1 D

� 920C 11

20

�e1 � e2;�

0 1

�B �Ae2 D 9

20

11

20e1:

Thus, the representation matrix of the action of s on .e2 e1/ is given by�0 �1

99400 1

�.

The characteristic polynomial of this matrix is .� � 920/.� � 11

20/. Then we have

b.s/ D .s C 1/.10s C 7/.10s C 9/.10s C 11/.10s C 13/.20s C 9/.20s C 11/ .20s C 13/.20s C 17/.20s C 19/.20s C 21/.20s C 23/.20s C 27/:


This observation leads to the following results, the proofs will be published else-where.

Lemma 4.1. Let f be a semiquasihomogeneous function with isolated singularity atthe origin with weight vector .w1; : : : ; wn/ of weighted degree dw.f / D 1. Set

X DnX

jD1wjxj

@

@xj� 1:

We can construct a first order annihilator of f in DX of the form

aX CQwith a 2 .f W A/ and dw.Q/ > dw.a/.

Lemma 4.2. There is a second order annihilator of f s with a weighted degree greaterthan or equal to minf�jwi � wj j W i; j D 1; : : : ; ng.

Let B1 and B2 be the basis of vector spaces F1 and F2, respectively. Then for thecase of L.f / D 2, we have the following.

Proposition 4.3. Let u 2 B1 nB2 be an algebraic local cohomology class in F with.f W A/u ¤ 0. The algebraic local cohomology class v of

�uv

�in F is of the form

v D dw.u/uC Qu;where Qu is in F1 with dw. Qu/ � dw.u/ or 0.

Let�uv

�be in F with .f W A/u ¤ 0. Since s acts on F,�

0 1

�B �A�

u

v

D�

v

�Bu � Av

D

dw.u/uC Qudw.u/.dw.u/uC Qu/C Qv

!

D dw.u/�u

v

C� QuQv

with� QuQv� 2 F holds. Thus, the corresponding diagonals of the representation matrix

of the action of s on F are the weighted degrees of the corresponding algebraic localcohomology classes in B1 n B2.

Next, let u 2 B2. Both�0u

�and

�u0

�are in F . Then�

0 1

�B �A�

u

0

D ˇ

�0

u

C�0

u00


where ˇ is a constant and u00 is in F2 with dw.u00/ � dw.u/ or 0, and�0 1

�B �A�

0

u

D�u

0

C�

0

�AuD�u

0

C ˛

�0

u

C�0

Nu;

where ˛ is a constant and Nu is in F2 with dw. Nu/ � dw.u/ or 0. Thus, we arrive at thefollowing result.

Theorem 4.4. For a semiquasihomogeneous singularity withL.f / D 2, letZ1 be theset of the weighted degrees of algebraic local cohomology classes in B1 nB2. Denoteby Z2 the set of the eigenvalues of action s on the subset²�

u10

; : : : ;

�u�2

0

;

�0

u1

; : : : ;

�0

u�2

³of F, with ui 2 B2, i D 1; : : : ; �2. Then the set of roots of the b-function is given byZ1 [Z2.

Example 4. Let f D x6C y7C x3y5. This is a semiquasihomogeneous singularitywith weight vector

�16; 17

�of weighted degree 1. Placed in order of weighted degrees,

the basis of F is as follows:� h5;5i� 57 h2;7i

� 6542 .h5;5i� 5

7 h2;7i/� 25196 h5;2iC 125

1372 h2;4i

;

� h5;4i� 59

42 h5;4iC 649 h2;6i� 3

49 h5;1iC 15343 h2;1i

;

� h4;5i� 57 h1;7i

� 2921 .h4;5i� 5

7 h1;7i/� 549 h4;2iC 25

343 h1;4i

;

� h3;6i� 12 h6;1i

� 1914 .h3;6i� 1

2 h6;1i/� 998 h4;4i

;

� h5;3i� 53

42 h5;3iC 998 h2;5iC 45

1372 h2;2i

;

� h4;4i� 26

21 h4;4iC 649 h1;6i� 3

49 h4;1iC 15343 h1;3i

;

� h3;5i� 17

14 h3;5i� 15196 h3;2i

;

� h2;6i� 25

21 h2;6iC 114 h5;1i� 5

98 h2;3i

;

� h5;2i� 47

42 h5;2iC 349 h2;4i

;

� h4;3i� 23

21 h4;3iC 998 h1:5iC 45

1372 h1;2i

;

� h3;4i� 15

14 h3;4i� 349 h3;1i

;

� h2;5i� 22

21 h2;5i� 598 h2;2i

;

�h1;6i

� 4342 h1;6iC 1

28 h4;1i� 5196 h1;3i

;

� h5;1i� 41

42 h5;1iC 398 h2;3i

;

� h4;2i� 20

21 h4;2iC 349 h1;4i

;

� h3;3i� 13

14 h3;3i

;

� h2;4i� 19

21 h2;4i

;

�h1;5i

� 3742 h1;5i� 5

196 h1;2i;

�h4;1i

� 1721 h4;1iC 3

98 h1;3i;

� h3;2i� 11

14 h3;2i

;

� h2;3i� 16

21 h2;3i

;

� h1;4i� 31

42 h1;4i

;

� h3;1i� 9

14 h3;1i

;

� h2;2i� 13

21 h2;2i

;

� h1;3i� 25

42 h1;3i

;

� h1;2i� 19

42 h1;2i

;

�h2;1i0

;

�1;1i0

;

�0

h2;1i

;

�0

h1;1i

:


Then we have

Z1 D˚ � 19

42;�25

42;�13

21;� 9

14;�31

42;�16

21;�11

14;�17

21;�37

42;�19

21;�13

14;�20

21;�41

42;

� 4342;�22

21;�15

14;�23

21;�47

42;�25

21;�17

14;�26

21;�53

42;�1914;�29

21;�59

42;�65

42

�:

There is a second order annihilator P D s2 C As C B in J.s/ with

A D � 1

210.55x@x � 204/ � 1

5 14.1 � 514y3/

.5x@x C 3y@y C 36/;

B D 1

4410.70x2@2x C 45xy@x@y � 338x@x � 90y2@2y � 702y@y/

� 1

4410 14.1 � 514y3/

.�735x2@2x C 189xy@x@y � 1743x@x

� 378y2@2y � 4914y@y � 459x3@2y/:For the cohomology classes

e1 D�

0

h1; 1i; e2 D

�0

h2; 1i; e3 D

�h1; 1i0

; e4 D

�h2; 1i0

we have �

0 1

�B �Ae1 D 5

6e1 C e3;�

0 1

�B �Ae2 D 7

6e2 C e4;�

0 1

�B �Ae3 D �143

882e1;�

0 1

�B �Ae4 D �145

441e2:

Thus the representation matrix of the action of s on .e4 e3 e2 e1/ is given by

s.e4 e3 e2 e1/ D .e4 e3 e2 e1/

0BB@0 0 1 0

0 0 0 1

�145441

0 76

0

0 �143882

0 56

1CCA :Then the characteristic polynomial of the above representation matrix of s is�

� � 1342

�� 11

21

�� 29

42

�� 10

21

�:

Therefore the set Z2 is given by Z2 D f�1342 ;�1121 ;�2942 ;�1021g.


Thus the roots of the b-function are given by the following thirty negative rationalnumbers:

�1342�1942�2042�2242�2542�2642�2742�2942�3142�3242

�3342�3742�3442�3842�3942�4042�4142�4342�4442�4542

�4642�4742�5042�5142�5242�5342�5742�5842�5942�6542:

References

[1] Y. Nakamura and S. Tajima, On weighted-degrees for algebraic local cohomologies associ-ated with semiquasihomogeneous singularities, in Singularities in geometry and topology2004. Proceedings of the 3 rd Franco–Japanese Conference held at Hokkaido University,Sapporo, September 13–18, 2004. ed. by J.-P. Brasselet and T. Suwa, Advanced Studiesin Pure Mathematics 46, Mathematical Society of Japan, Tokyo 2007, 105–117. 104, 105

[2] H. Nakayama, Algorithms of computing local b function by an approximate divisionalgorithm in yD, J. Symb. Comp. 4 (2009), 449–462. 109

[3] M. Noro, et al, Risa/Asir, http://www.math.kobe-u.ac.jp/Asir 104

[4] T. Oaku, An algorithm of computing b-function, Duke Math. J. 87 (1997), 115–132. 109

[5] K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math 14(1971), 123–142. 105, 107

[6] S. Tajima and Y. Nakamura, Algebraic local cohomology classes attached to quasi-homogeneous isolated hypersurface singularities, Publ. Res. Inst. Math. Sci 41 (2005),1–10.

[7] S. Tajima and Y. Nakamura, Annihilating ideals for an algebraic local cohomology class,J. Symb. Comp. 44 (2009), 435–448.

[8] S. Tajima, Y. Nakamura, and K. Nabeshima, Standard bases and algebraic local cohomol-ogy for zero dimensional ideals, in Singularities–Niigata-Toyama 2007. Proceedings ofthe 4 th Franco-Japanese Symposium and the workshop held in Niigata, August 27–31,2007, ed. by J.-P. Brasselet, S. Ishii, T. Suwa, and M. Vaquie, Advanced Studies in PureMathematics 56, Mathematical Society of Japan, Tokyo 2009, 341–361. 109

[9] T. Yano, On the theory of b-functions, Publ. Res. Inst. Math. Sci. 14 (1978), 111–202.103, 105, 107

http://www.math.kobe-u.ac.jp/Asir

A note on the Chern–Schwartz–MacPherson class

Toru Ohmoto�

Department of MathematicsFaculty of Science, Hokkaido University

Sapporo 060-0810, Japanemail: [email protected]

Abstract. This is a note about the Chern–Schwartz–MacPherson class for certain algebraicstacks which has been introduced in [17]. We also discuss other singular Riemann–Roch typeformulas in the same manner.

1 Introduction

In this note we state a bit detailed account about MacPherson’s Chern class trans-formation C� for quotient stacks defined in [17], although all the instructions havealready been made in that paper. Our approach is also applicable for other additivecharacteristic classes, e.g. Baum–Fulton–MacPherson’s Todd class transformation [3](see [9] and [4] for the equivariant version) and more generally Brasselet–Schürmann–Yokura’s Hirzebruch class transformation [5] (see Section 4 below). Throughout wework over the complex number field C or a base field k of characteristic 0.

We begin with recalling C� for schemes and algebraic spaces. These are spaceshaving trivial stabilizer groups. In following sections we will deal with quotient stackshaving affine stabilizers, in particular, ‘(quasi-)projective’ Deligne–Mumford stacksin the sense of Kresch [15].

1.1 Schemes

For the category of quasi-projective schemes U and proper morphisms, there is aunique natural transformation from the constructible function functor to the Chowgroup functor, C� W F.U /! A�.U /, so that it satisfies the normalization property

C�.1U / D c.T U /Z ŒU � 2 A�.U / if U is smooth.

This is called the Chern–MacPherson transformation, see MacPherson [16] in com-plex case (k D C) and Kennedy [13] in more general context of ch.k/ D 0. Here the

�Work partially supported by JSPS grant No. 21540057.


118 Toru Ohmoto

naturality means the commutativity f�C� D C�f� of C� with pushforward of propermorphisms f. In particular, for proper pt W U ! pt.D Spec.k//, the (0-th) degreeof C�.1U / is equal to the Euler characteristic of U W pt�C�.1U / D �.U / (as for thedefinition of �.U / in algebraic context see [13] and [12]).

As a historical comment, Schwartz [21] firstly studied a generalization of thePoincaré–Hopf theorem for complex analytic singular varieties by introducing a topo-logical obstruction class for certain stratified vector frames, which in turn coincideswith MacPherson’s Chern class [6]. Therefore,C�.U / D C�.1U / is usually called theChern–Schwartz–MacPherson class (CSM class) of a possibly singular variety U.

To grasp quickly what the CSM class is, there is a convenient way due to Aluffi [1]and [2]. Let U be a singular variety and � W U0 ,! U a smooth open dense reducedsubscheme. By means of resolution of singularities, we have a birational morphismp W W ! U so that W D U0 is smooth and D D W � U0 is a divisor with smoothirreducible componentsD1; : : : ;Dr having normal crossings. Then by induction on rand properties of C� it is shown that

C�.1U0/ D p�

� c.T W /Q.1CDi / Z ŒW �

�2 A�.U /:

(Here c.T W /=Q.1 C Di / is equal to the total Chern class of dual to �1W .logD/

of differential forms with logarithmic poles along D). By taking a stratification U Dj Uj , we have C�.U / D P

j C�.1Uj/. Conversely, we may regard this formula as

an alternative definition of CSM class, see [1].

1.2 Algebraic spaces

We extendC� to the category of arbitrary schemes or algebraic spaces (separated and offinite type). To do this, we may generalize Aluffi’s approach, or we may trace the sameinductive proof by means of Chow envelopes (cf. [14]) of the singular Riemann–Rochtheorem for arbitrary schemes [10].

Here is a short remark. An algebraic space X is a stack over Sch=k, under étaletopology, whose stabilizer groups are trivial. Precisely, there exists a schemeU (calledan atlas) and a morphism of stacks u W U ! X such that for any scheme W andany morphism W ! X the (sheaf) fiber product U �X W exists as a scheme, andthe map U �X W ! W is an étale surjective morphism of schemes. In addition,ı W R D U �X U ! U �k U is quasi-compact, called the étale equivalent relation.Denote by gi W R ! U (i=1,2) the projection to each factor of ı. The Chow groupA�.X/ is defined using an étale atlas U (see Section 6 in [8]). In particular, letting

g12� D g1� � g2�;

the sequence

A�.R/g12� �� A�.U /

u� �� A�.X/ �� 0

A note on the Chern–Schwartz–MacPherson class 119

is exact (see Kimura [14], Theorem 1.8). Then the CSM class of X is given by

C�.X/ D u�C�.U /:

In fact, if U 0 ! X is another atlas for X with the relation R0, then we take the thirdU 00 D U �X U 0 with R00 D R �X R0, where p W U 00 ! U and q W U 00 ! U 0 are étaleand finite. Chow groups of atlases modulo Im .g12�/ are mutually identified throughthe pullback p� and q�, and in particular,

p�C�.U / D C�.U 00/ D q�C�.U 0/;

that is checked by using resolution of singularities or the Verdier–Riemann–Roch [24]for p and q. Finally we put

C� W F.X/ �! A�.X/

by sending1W 7�! ��C�.W /

for integral algebraic subspaces W�,! X and extending it linearly, and the naturality

for proper morphisms is proved again using atlases. This is somewhat a prototype ofC� for quotient stacks described below.

2 Chern class for quotient stacks

2.1 Quotient stacks

Let G be a linear algebraic group acting on a scheme or algebraic space X. If theG-action is set-theoretically free, i.e. stabilizer groups are trivial, then the quotientX ! X=G always exists as a morphism of algebraic spaces (see [8] Proposition 22).Otherwise, in general we need the notion of quotient stack.

The quotient stack X D ŒX=G� is a (possibly non-separated) Artin stack overSch=k, under fppf topology (see, e.g., Vistoli [23], Gómez [11] for the detail). Anobject of X is a family of G-orbits in X parametrized by a scheme or algebraicspace B, that is, a diagram

Bq � P p��! X;

where P is an algebraic space, q is a G-principal bundle and p is a G-equivariantmorphism. A morphism of X is a G-bundle morphism � W P ! P 0 such that

p0 ı � D p;

where B 0 q0

P 0 p0

! X is another object. Note that there are possibly many non-trivialautomorphisms P ! P over the identity morphism id W B ! B , which form thestabilizer group associated to the object (e.g. the stabilizer group of a “point” (B D pt)is non-trivial in general). A morphism of stacks B ! X naturally corresponds to an

120 Toru Ohmoto

object B P ! X , that follows from Yoneda lemma. In particular there is amorphism (called atlas)

u W X �! X

corresponding to the diagramXq G�X p! X , being q the projection to the second

factor and p the group action. The atlas u recovers any object of X by taking fiberproducts B P D B �X X ! X .

Let f W X ! Y be a proper and representable morphism of quotient stacks,i.e. for any scheme or algebraic space W and morphism W ! Y, the base changeX �Y W ! W is a proper morphism of algebraic spaces. Take presentations X DŒX=G�, Y D ŒY=H�, and the atlases u W X ! X, u0 W Y ! Y. There are two aspectsof f.

Equivariant morphism. Put B D X �Y Y , which naturally has a H -action so thatŒB=H� D ŒX=G�, v W B ! X is a new atlas, and Nf W B ! Y is H -equivariant:

BNf ��

v

��

Y

u0

��X

f�� Y:

(2.1)

Change of presentations. Let P D X �X B; then the following diagram is con-sidered as a family of G-orbits in X and simultaneously as a family of H -orbits inB , i.e. p W P ! X is a H -principal bundle and G-equivariant and q W P ! B is aG-principal bundle and H -equivariant:

Pq ��

p

��

B

v

��X

u�� X:

(2.2)

A simple example of such f is given by proper ' W X ! Y with an injectivehomomorphismG ! H such that '.g:x/ D g:'.x/ andH=G is proper. In this case,P D H �k X and B D H �G X with p W P ! X the projection to the second factor,q W P ! B the quotient morphism.

2.2 Chow group and pushforward

For schemes or algebraic spaces X (separated, of finite type) with G-action, theG-equivariant Chow group AG� .X/ has been introduced in Edidin–Graham [8], andthe G-equivariant constructible function FG.X/ in [17]. They are based on Totaro’s


algebraic Borel construction. Take a Zariski open subset U in an `-dimensional linearrepresentation V of G so that G acts on U freely. The quotient exists as an algebraicspace, denoted by UG D U=G. Also G acts X � U freely, hence the mixed quotientX �G ! XG D X �G U exists as an algebraic space. Note thatXG ! UG is a fiberbundle with fiber X and group G. Define

AGn .X/ D AnC`�g.XG/;

where g D dimG, and

FG.X/ D F.XG/; `� 0:

Precisely saying, we take the direct limit over all linear representations of G; see [8]and [17] for the detail.AGn .X/ is trivial for n > dimX but it may be non-trivial for negative n. Also note

that the group FGinv.X/ of G-invariant functions over X is a subgroup of FG.X/.Let us recall the proof that these groups are actually invariants of quotient stacks X.

Look at diagram (2.2) above. Let g D dimG and h D dimH . Note that G �H actson P. Take open subsetsU1 andU2 of representations ofG andH, respectively, whereì D dimUi i D 1; 2, so that G and H act on U1 and U2 freely respectively. Put

U D U1 ˚ U2;on which G � H acts freely. We denote the mixed quotients for spaces arising indiagram (2.2) by

PG�H D P �G�H U;

XG D X �G U1;and

BH D B �H U2:

Then the projection p induces the fiber bundle

Np W PG�H �! XG

with fiber U2 and group H , and q induces

Nq W PG�H �! BH

with fiber U1 and group G. Thus, the pullback Np� and Nq� for Chow groups are iso-morphic,

AnC`1.XG/ ' AnC`1C`2

.PG�H / ' AnC`2.BH /:

Taking the limit, we have the canonical identification

AGnCg.X/p�

��!' AG�HnCgCh.P /

q�

��' AHnCh.B/

122 Toru Ohmoto

(see [8], Proposition 16). Note that .q�/�1 ı p� shifts the dimension by h � g. Alsofor constructible functions, put the pullback p�˛ D ˛ ı p, then we have

FG.X/ ' FG�H .P / ' FH .B/via pullback p� and q� (see Lemma 3.3 in [17]). We thus define

A�.X/ D AG�Cg.X/; F.X/ D FG.X/; and Finv.X/ D FGinv.X/;

through the canonical identification.Given proper representable morphisms of quotient stacks f W X ! Y and any

presentations X D ŒX=G�, Y D ŒY=H�, we define the pushforward

f� W A�.X/ �! A�.Y/

byf H� ı .q�/�1 ı p� W AGnCg.X/! AHnCh.Y /

and alsof� W F.X/ �! F.Y/

in the same way. By the identification .q�/�1 ı p�, everything is reduced to theequivariant setting (see diagram (2.1)).

Lemma 2.1. The above F and A� satisfy the following properties.

(i) For proper representable morphisms of quotient stacks f, the pushforward f� iswell-defined.

(ii) Let f1 W X1 ! X2, f2 W X2 ! X3 and f3 W X1 ! X3 be proper representablemorphisms of stacks so that f2 ı f1 is isomorphic to f3, then f2� ı f1� isisomorphic to f3� (f3� D f2� ı f1� using a notational convention in [11],Remark 5.3).

Proof. Look at the diagram below, where Xi D ŒXi=Gi � (i D 1; 2; 3); we may regardX1 D ŒX1=G1� D ŒB1=G2� D ŒB3=G3�; and so on:

P1 X1

B1 X1

P 0 P3

X2 X2 X3:

B 0 B3

P2 B2 X3

f1

��

��

��

��

��

��

��

��

��

��

��

��

��

�� f3

��

�� f2 ��

��

��

��

��

��

��


(i) Put f D f1, then the well-definedness of the pushforward f1� (in both of Fand A�) is easily checked by taking fiber products and by the canonical identification.

(ii) Assume that there exists an isomorphism of functors

˛ W f2 ı f1 �! f3

(i.e. a 2-isomorphism of 1-morphisms). Then two G3-equivariant morphisms Nf2 ı Nf1and Nf3 from B3 to X3 coincide up to isomorphisms of B3 and of X3 which areencoded in the definition of ˛, hence theirG3-pushforwards coincide up to the chosenisomorphisms.

2.3 Chern–MacPherson transformation

We assume that X is a quasi-projective scheme or algebraic space with action of G.Then XG exists as an algebraic space, hence C�.XG/ makes sense. Take the vectorbundle

T UG D X �G .U ˚ V /over XG , i.e. the pullback of the tautological vector bundle .U � V /=G over UGinduced by the projection XG ! UG . Our natural transformation

CG� W FG.X/ �! AG� .X/

is defined to be the inductive limit of

TU;� D c.T UG/�1 Z C� W F.XG/! A�.XG/

over the direct system of representations of G, see [17] for the detail.Roughly speaking, the G-equivariant CSM class CG� .X/ .D CG� .1X // looks like

“c.TBG/�1ZC�.EG �G X/”, where EG �G X ! BG means the universal bundle(as ind-schemes) with fiber X and group G, that has been justified using a differentinductive limit of Chow groups; see Remark 3.3 in [17].

Lemma 2.2. (i) With the same notations as in diagram (2.2) in Section 2.1, the fol-lowing diagram commutes:

FG.X/

CG�

��

p�

' �� FG�H .P /

CG�H�

��AG�Cg.X/ p�

' �� AG�H�CgCh.P /:

(ii) In particular, C� W F.X/! A�.X/ is well-defined.

(iii) C�f� D f�C� for proper representable morphisms f W X ! Y.

124 Toru Ohmoto

Proof. (i) This is essentially the same as Lemma 3.1 in [17] which shows the well-definedness of CG� . Apply the Verdier–Riemann–Roch [24] to the projection of theaffine bundle

Np W PG�H �! XG (with fiber U2),

then we have the following commutative diagram

F.XG/

C�

��

Np�

�� F.PG�H /

C�

��A�C`1

.XG/ Np�� A�C`1C`2

.PG�H /;

where Np�� D c.T Np/ Z Np� and T Np is the relative tangent bundle of Np. The twistingfactor c.T Np/ in Np�� is canceled by the factors in TU1;� and TU;�. In fact, since

T Np D Nq�T U2H ;

T Nq D Np�T U1G ;

and

T UG�H D P �G�H .T .U1 ˚ U2//D T Np ˚ T Nq;

we have

TU;� ı Np�.˛/ D c.T UG�H /�1 Z C�. Np�˛/

D c.T Np ˚ T Nq/�1c.T Np/Z Np�C�.˛/

D c.T Nq/�1 Z Np�C�.˛/

D Np�.c.T U1G/�1 Z C�.˛//

D Np� ı TU1;�.˛/:

Taking the inductive limit, we conclude that

CG�H� ı p� D p� ı CG� :Thus (i) is proved.

The claim (ii) follows from (i).

By (ii), we may considerC� as theH -equivariant Chern–MacPherson transforma-tion CH� given in [17], thus (iii) immediately follows from the naturality of CH� .

The above lemmas show the following theorem (cf. [17], Theorem 3.5).


Theorem 2.3. Let C be the category whose objects are (possibly non-separated) Artinquotient stacks X having the form ŒX=G� of separated algebraic spaces X of finitetype with action of smooth linear algebraic groups G; morphisms in C are assumedto be proper and representable. Then for the category C we have a unique naturaltransformation

C� W F.X/ �! A�.X/

with integer coefficients such that it coincides with the ordinary MacPherson trans-formation when restricted to the category of quasi-projective schemes.

2.4 Degree

Let g D dimG. The G-classifying stack BG D Œpt=G� has (non-positive) virtualdimension �g, hence

A�n.BG/ D AG�nCg.pt/ D An�gG .pt/ D An�g.BG/

for any integer n (trivial for n < g). We often use this identification. In particular,A�g.BG/ D A0.BG/ D Z.

Let X D ŒX=G� in C with X projective and equidimensional of dimension n.Then we can take the representable morphism pt W X ! BG. We have

G �X q ��

p

��

Xxpt ��

u

��

pt

��X

u�� X

pt�� BG:

Here are some remarks.

(i) For a G-invariant function ˛ 2 Finv.X/ D FGinv.X/, it is obvious that .q�/�1 ıp�.˛/ D ˛, hence we have

pt�.˛/ D xpt�.q�/�1p�.˛/ D xpt�˛ DZX

˛ D �.X I˛/;

which is called the integral, or weighted Euler characteristic of the invariantfunction ˛. In particular, by the naturality, pt�C�.˛/ D C�.pt�˛/ D �.X I˛/.More generally, in [17] we have defined theG-degree of equivariant constructiblefunction ˛ 2 F.X/ by

pt�.˛/ 2 FG.pt/ D F.BG/;which is a “constructible” function over BG. Then pt�C�.˛/ D C�.pt�˛/ 2A�.BG/ is a polynomial or power series in universal G-characteristic classes.

126 Toru Ohmoto

(ii) For invariant functions ˛ 2 Finv.X/ and for i < �g and i > n � g, the i-thcomponentCi .˛/ is trivial. A possibly nontrivial highest degree termCn�g.˛/ 2An�g.X/ .D AGn .X// is a linear sum of the G-fundamental classes ŒXi �G ofirreducible components Xi (the virtual fundamental class of dimension n � g) .As a notational convention, let 1.0/

Xdenote the constant function 1X 2 FGinv.X/ D

Finv.X/ for a presentation X D ŒX=G�. In particular, if X is smooth, then

C�.1.0/X/ D CG� .1X / D cG.TX/Z ŒX�G 2 AG�Cg.X/ D A�.X/:

(iii) From the viewpoint of the enumerative theory in classical projective algebraicgeometry (e.g. see [19]), a typical type of degrees often arises in the formZ

pt�.c.E/Z C�.˛// 2 A0.BG/for some vector bundle E over X and a constructible function ˛ 2 Finv.X/.

3 Deligne–Mumford stacks

It would be meaningful to restrict C� to a subcategory of certain quotient stackshaving finite stabilizer groups, which form a reasonable class of Deligne–Mumfordstacks (including smooth DM stacks).

Theorem 3.1. Let CDM be the category of Deligne–Mumford stacks of finite type whichadmits a locally closed embedding into some smooth proper DM stack with projectivecoarse moduli space: morphisms in CDM are assumed to be proper and representable.Then for CDM there is a unique natural transformation

C� W F.X/ �! A�.X/

satisfying the normalization property

C�.1X / D c.TX/Z ŒX�for smooth schemes.

This is due to Theorem 5.3 in Kresch [15] which states that a DM stack in CDM

is in fact realized by a quotient stack in C . In [15], such a DM stack is called to be(quasi-)projective.

Remark 3.2. (i) In the above theorem, the embeddability into smooth stack (or equiv-alently the resolution property in [15]) is required, that seems natural, since originalMacPherson’s theorem requires such a condition; see [16] and [13]. In order to extendC� for more general Artin stacks with values in Kresch’s Chow groups, we need tofind some technical gluing property.


(ii) We may admit proper non-representable morphisms of DM stacks if we userational coefficients. In fact for such morphisms the pushforward of Chow groups withrational coefficients is defined [23].

3.1 Modified pushforwards

The theory of constructible functions for Artin stacks has been established by Joycein [12]. Below let us work with Q-valued constructible functions and Chow groupswith Q-coefficients. For stacks X in CDM, each geometric point x W pt D Spec k ! X

has a finite stabilizer group Aut.x/.D Isox.x; x//. Then the group of constructiblefunctions˛ in the sense of [12] is canonically identified with the subgroupFinv.X/Q DFGinv.X/Q of invariant constructible functions ˛ over X in the following way (the barindicates constructible functions over the set of all geometric points X.k/). For eachk-point x W pt! X, the value of ˛ over the orbit x �X X is given by jAut.x/j ˛.x/,that is

F.X.k//Q ' Finv.X/Q .� F.X/Q/; ˛ $ ˛ D 1X ��˛;

where � is the projection to X.k/, ˛ ˇ is the canonical multiplication on F.X/Q,.˛ ˇ/.x/ D ˛.x/ˇ.x/, and

1X D jAut.�.�//j 2 Finv.X/Q:

It is shown by Tseng [22] that if X is a smooth DM stack, C�.1X/ coincideswith (pushforward of the dual to) the total Chern class of the tangent bundle of thecorresponding smooth inertia stack.

From a viewpoint of classical group theory, it would be natural to measure howlarge of the stabilizer group is by comparing it with a fixed group A, that leads us todefine a Q-valued constructible function over X.k/. Here the group A is supposed tobe, e.g., a finitely generated Abelian group (we basically consider A D Zm, Z=rZ,etc). Accordingly to [17] and [18], we define the canonical constructible functionmeasured by group A which assigns to any geometric point x the number of grouphomomorphisms of A into Aut.x/:

1AX.x/ DjHom .A;Aut.x// jjAut.x/ j 2 Q:

The corresponding invariant constructible function is denoted by 1AX2 Finv.X/Q, or

often by 1AXIG 2 FGinv.X/Q when a presentation X D ŒX=G� is specified. Namely,

the value of 1AXIG on the G-orbit expressed by x W pt ! X is jHom .A;Aut.x// j.The function for A D Z is nothing but 1X in our convention, and for A D f0g it is1.0/

XD 1. If A D Z2, the function counts the number of mutually commuting pairs

in Aut.x/, hence its integral corresponds to the orbifold Euler number (in physicist’ssense); see [18].

128 Toru Ohmoto

DefineT AX W F.X/Q �! F.X/Q

by the multiplicationT AX .˛/ D 1AXIG ˛:

This is a Q-algebra isomorphism, for 1AXIG is an unit in F.X/Q. A new pushforwardis introduced for proper representable morphisms f W X ! Y in CDM by

f A� W F.X/Q �! F.Y/Q;

˛ 7�! .T AY/�1 ı f� ı T AX .˛/:

Obviously, gA� ı f A� D .g ı f /A� . The following theorem says that there are severalvariations of theories of integration with values in Chow groups for Deligne–Mumfordstacks.

Theorem 3.3. Given a finitely generated Abelian group A, let F A denote the newcovariant functor of constructible functions for the category CDM, given by

FA.X/Q D F.X/Qand the pushforward by f A� . Then,

CA� D C� ı T AX W F A.X/Q �! A�.X/Qis a natural transformation.

Proof. It is straightforward that

f� ı CA� D f� ı C� ı T AX D C� ı f� ı T AXD C� ı T AY ı .T AY /�1 ı f� ı T AX D CA� ı f A� :

4 Other characteristic classes

The method in the preceding sections is applicable to other characteristic classes (overC or a field k of characteristic 0).

As the most general additive characteristic class for singular varieties, the Hirze-bruch class transformation

Ty� W K0.Var=X/ �! A�.X/˝QŒy�

was recently introduced by Brasselet–Schürmann–Yokura [5]. For possibly singularvarietiesX (and proper morphisms between them), Ty� is a unique natural transforma-tion from the Grothendieck group K0.Var=X/ of the monoid of isomorphism classesof morphisms V ! X to the rational Chow group of X with a parameter y such that


it satisfies that

Ty�ŒXid�! X� D ftdy.TX/Z ŒX�; for smooth X ,

where ftdy.E/ denotes the modified Todd class of vector bundles:

ftdy.E/ D rYiD1

� ai .1C y/1 � e�ai .1Cy/ � aiy

�;

when c.E/ D QriD1.1 C ai /; see [5] and [20]. Note that the associated genus is

well-known Hirzebruch’s �y-genus, which specializes to: the Euler characteristic ify D �1; the arithmetic genus if y D 0; and the signature if y D 1. Hence, Ty� gives ageneralization of the �y-genus to homology characteristic class of singular varieties,which unifies the following singular Riemann–Roch type formulas in canonical ways:

y D �1 the Chern–MacPherson transformation C� (see [16] and [13]);

y D 0 Baum–Fulton–MacPherson’s Todd class transformation (see [3]);

y D 1 Cappell–Shaneson’s homology L-class transformation L� (see [7]).

For a quotient stack X D ŒX=G� 2 C in Theorem 2.3, we denote byK0.C=X/ theGrothendieck group of the monoid of isomorphism classes of representable morphismsof quotient stacks to the stack X. To each element ŒV ! X� 2 K0.C=X/, we take aG-equivariant morphism V ! X where V D V �X X with natural G-action so thatV D ŒV=G�, and associate a class of morphisms of algebraic spaces ŒVG ! XG � 2K0.Var=XG/. We then define

Ty� W K0.C=X/ �! A�.X/˝QŒy�

by assigning to ŒV ! X� the inductive limit (over all G-representations) offtdy�1.T UG/Z Ty�ŒVG �! XG � 2 A�.XG/˝QŒy�:

This is well-defined, because theVerdier–Riemann–Roch forTy� holds (see [5], Corol-lary 3.1) and the same proof of Lemma 2.2 can be used in this setting. Note that ineach degree of grading the limit stabilizes, thus the coefficient is a polynomial in y.So we obtain an extension of Ty� to the category C , and hence also to CDM.

It turns out that at special values y D 0;˙1, Ty� corresponds to:

y D �1 the G-equivariant Chern–MacPherson transformation [17], i.e. C� asdescribed in Section 2 above;

y D 0 the G-equivariant Todd class transformation [8] and [4], given by thelimit of td�1.T UG/Z ;

y D 1 the G-equivariant singular L-class transformation given by the limit of.L�/�1.T UG/ZL�, where L� is the (cohomology) Hirzebruch–ThomL-class.

Applications will be considered in another paper.

130 Toru Ohmoto

References

[1] P.Aluffi, Limits of Chow groups, and a new construction of Chern–Schwartz–MacPhersonclasses, Pure Appl. Math. Q. 2 (2006), 915–941 118

[2] P. Aluffi, Classes de Chern des variétés singullières, revisitées, C. R. Acad. Sci. Paris 342(2006), 405–410. 118

[3] P. Baum,W. Fulton, and R. MacPherson, Riemann–Roch for singular varieties, Inst. HautesÉtudes Sci. Publ. Math. 45 (1975), 101–145. 117, 129

[4] J. Brylinski and B. Zhang, Equivariant Todd classes for toric varieties, preprint 2003arXiv:math/0311318v1 117, 129

[5] J. P. Brasselet, J. Schürmann and S.Yokura, Hirzebruch classes and motivic Chern classesfor singular spaces, J. Topol. Anal. 2 (2010), 1–55. 117, 128, 129

[6] J. P. Brasselet and M. H. Schwartz, Sur les classes de Chern d’un ensemble analytiquecomplexe, Astérisque 82–83 (1981), 93–148. 118

[7] S. Cappell and J. Shaneson, Stratifiable maps and topological invariants, J. Amer. Math.Soc. 4 (1991), 521–551. 129

[8] D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998),595–634. 118, 119, 120, 121, 122, 129

[9] D. Edidin andW. Graham, Riemann–Roch for equivariant Chow groups, Duke Math. J. 102(2000), 567–594. 117

[10] W. Fulton and H. Gillet, Riemann–Roch for general algebraic varieties, Bull. Soc. Math.France 111 (1983), 287–300. 118

[11] T. L. Gómez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 1–31. 119,122

[12] D. Joyce, Constructible functions on Artin stacks, J. London Math. Soc. (2) 74 (2006),583–606. 118, 127

[13] G. Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Alge-bra 18 (1990), 2821–2839. 117, 118, 126, 129

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[15] A. Kresch, On the geometry of Deligne–Mumford stacks, in Algebraic geometry–Seattle2005. Part 1. Papers from the AMS Summer Research Institute held at the University ofWashington, Seattle, WA, July 25–August 12, 2005, ed. by D. Abramovich, A. Bertram,L. Katzarkov, R. Pandharipande and M. Thaddeus, Proceedings of Symposia in PureMathematics 80, Part 1, American Mathematical Society, Providence, RI, 2009, 259–271.117, 126

[16] R. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974),421–432. 117, 126, 129

[17] T. Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions,Math. Proc. Cambridge Phil. Soc. 140 (2006), 115–134. 117, 120, 121, 122, 123, 124,125, 127, 129

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http://arxiv.org/abs/math/0311318v1


[19] A. Parusinski and P. Pragacz, Chern–Schwartz–MacPherson classes and the Euler char-acteristic of degeneracy loci and special divisors, J. Amer. Math. Soc. 8 (1995), 793–817.126

[20] J. Schürmann and S. Yokura, A survey of characteristic classes of singular spaces, inSingularity theory – Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedingsof the 2005 Marseille Singularity School and Conference, CIRM, Marseille, France, 24January – 25 February 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon andD. Trotman, World Scrientific, Singapore 2007, 865–952. 129

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On mixed projective curves

Mutsuo Oka

Department of Mathematics, Tokyo University of Science26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601, Japan


Abstract. Let f .z; Nz/ be a strongly polar homogeneous polynomial of n variables z D .z1; : : : ;

zn/. This polynomial defines a projective real algebraic variety V D fŒz� 2 CPn�1 jf .z; Nz/ D0g in the projective space CPn�1. The behavior is different from that of the projective hy-persurface. The topology is not uniquely determined by the degree of the variety even if V isnon-singular. We study a basic property of such a variety.

1 Introduction

Letf .z; Nz/ be a polar weighted homogeneous mixed polynomial with z D .z1; : : : ; zn/in Cn. Namely there exist integers .q1; : : : ; qn/ and .p1; : : : ; pn/ and positive integersdr ; dp such that

f .t ı z; t ı Nz/ D tdrf .z; Nz/; t ı z D .tq1z1; : : : ; tqnzn/; t 2 RC;

f .� ı z; � ı z/ D �dpf .z; Nz/; � ı z D .�p1z1; : : : ; �pnzn/; � 2 C; j�j D 1:

This gives a RC � S1 action by

.t; �/ ı z D .tq1�p1z1; : : : ; tqn�pnzn/; .t; �/ 2 RC � S1:

The integers dr and dp are called the radial and the polar degree respectively and wedenote them as dr D rdegf and dp D pdeg f .

We say that f .z; Nz/ is strongly polar weighted homogeneous if pj D qj for j D1; : : : ; n. Then the associated RC � S1 action on Cn is in fact the C� action which isdefined by

.z; / D ..z1; : : : ; n/; / 7�! ı z D .z1p1 ; : : : ; znpn/; 2 C�:

Assume that f .z; Nz/ D P�;� c�;�z� Nz� is a strongly polar weighted homogeneous

polynomial with radial degree dr and polar degree dp . Here � and� are multi-integersand the weights j�j; j�j are defined by

PniD1 �i and

PniD1 �i as usual. Then the

following equalities are satisfied:

j�j C j�j D dr ; j�j � j�j D dp if c�;� ¤ 0:


134 Mutsuo Oka

Put r D j�j. Then it is easy to see that j�j D dp C 2r .We say f .z; Nz/ is strongly polar homogeneous if further the weights satisfy the

equalities qj D pj D 1 for any j .A strongly polar weighted homogeneous polynomialf .z; Nz/ satisfies the equality:

f ..t; �/ ı z; .t; �/ ı z/ D tdr�dpf .z; Nz/; .t; �/ 2 RC � S1: (1.1)

Assume that f .z; Nz/ is a strongly polar weighted homogeneous polynomial withradial degree dr and polar degree dp respectively and let P D .p1; : : : ; pn/ be theweight vector. Let zV be the mixed affine hypersurface

zV D f �1.0/ D fz 2 Cn j f .z; Nz/ D 0g:Let ' W 2n�1 n K ! S1 be the Milnor fibration with K D zV \ S2n�1 and let F bethe fiber. Recall that '.z/ D f .z; Nz/=jf .z; Nz/j. Thus F is defined by

F D '�1.1/ D fz 2 S2n�1 nK j f .z; Nz/ 2 R; f .z; Nz/ > 0gWe can equivalently consider the global fibration f W Cn� zV ! C�. Then the Milnorfiber is identified with the hypersurface f �1.1/. The monodromy map h W F ! F (ineither case) is defined by

h.z/ D�

exp�2p1� i

dp

�z1; : : : ; exp

�2pn� idp

�zn

�:

We consider also the weighted projective hypersurface V defined by

V D f.z1 W z2 W W zn/ 2 CP.P /n�1 j f .z; Nz/ D 0gwhere CP.P /n�1 is the weighted projective space defined by the equivalence inducedby the above C� action:

z � w () 9 2 C�; w D ı z:

It is well known that CP.P /n�1 is an orbifold with at most cyclic quotient singularities.By (1.1), z 2 f �1.0/ and z0 � z, then z0 2 f �1.0/. Thus the hypersurface

V D fŒz� 2 CPn�1.P / j f .z/ D 0g is well-defined. Consider the quotient map� W S2n�1 ! CP.P /n�1 or � W Cn n fOg ! CP.P /n�1. For the brevity’s sake, wedenote the restrictions �jF W F ! CP.P /n�1 n V and �jK W K ! V by the same� . We are interested in the topology of V and the relation with the Milnor fibration.

In this paper, we consider only the case of strongly polar homogeneous polynomi-als. It turns out that the topology of the smooth projective mixed hypersurface V isnot an invariant of the degree dr ; dp . However we will show that the degree of V isequal to the polar degree dp (Theorem 4.1).

In §5, we study the case n D 3. In this case, let g be the genus of the mixed curveV and put q D dp . Then it is known that the following inequality (known as Thom’sconjecture and proved by Kronheimer and Mrowka [4]) holds:

g � .q � 1/.q � 2/2

:

On mixed projective curves 135

We also give examples of mixed projective curves in CP2 which shows that g isunbounded when q is fixed.

This is a continuation of our papers [8] and [9] and we use the same notations.

2 Milnor fibration and the Hopf fibration

2.1 Canonical orientation

It is well known that a complex analytic smooth variety has a canonical orientationwhich comes from the complex structure (see for example [3], p. 18). Let zV D f �1.0/be a mixed hypersurface. Take a point a 2 zV . We say that a is a mixed singular point ofzV , if a is a critical point of the mapping f W Cn ! C. Otherwise, a is a mixed regularpoint. Note that a point a 2 zV to be a regular point as a point of a real analytic variety is anecessary condition but not a sufficient condition for the regularity as a point on a mixedvariety. Recall that a is a mixed singular point if and only if dfa W TaCn ! Tf .a/Cis not surjective. This is equivalent to the existence of a complex number ˛ 2 C withj˛j D 1 such that

df .a; Na/ D ˛df .a; Na/i.e.

@f

@zj.a; Na/ D ˛ @f

@ Nzj .a; Na/; j D 1; : : : ; n

(see [7]).

Proposition 2.1. There is a canonical orientation on the smooth part of a mixedhypersurface.

Proof. Take a mixed regular point a 2 zV. The normal bundle N of zV � Cn hasa canonical orientation such that dfa W Na ! Tf .a; Na/C is an orientation preserving

isomorphism. This gives a canonical orientation on zV such that the ordered union ofthe oriented frames fv1; : : : ; v2n�2; n1; n2g of TaCn is the orientation of Cn if andonly if fv1; : : : ; v2n�2g is an oriented frame of Ta zV where fn1; n2g is an orientedframe of normal vectors.

Consider a strongly polar homogeneous hypersurface zV and let V be the corre-sponding mixed projective hypersurface.

Proposition 2.2. Let a 2 zV n fOg. Then a 2 zV is a mixed singular point of zV if andonly if �.a/ 2 V is a mixed singular point.

Proof. Assume that a D .a1; : : : ; an/ 2 zV and an ¤ 0 for simplicity. Let uj Dzj =zn; 1 � j � n � 1 be the affine coordinates of the chart Un D fzn ¤ 0g. Then

136 Mutsuo Oka

V \ Un is defined by

V \ Un D fu 2 Cn�1 j g.u; Nu/ D 0gwhere u D .u1; : : : ; un�1/ and g.u; Nu/ is defined by

g.u; Nu/ D f .u0; Nu0/; u0 D .u; 1/:Putting q C 2r D rdeg f and q D pdeg f as in §1, we observe that

g.u; Nu/ D f .z; Nz/=.zqCrn Nzrn/: (2.1)

Write an D rn exp.�ni/ in polar form. Consider the hyperplane section zUn D Cn \fzn D ang and Qf D f j zUn

. Then we have the commutative diagram:

zU

�

��

Qf �� C

ˇ

��U D Cn�1

g�� C;

where ˇ is the multiplication with rdrn exp.dp�ni/. This follows from (2.1). Put ˛ D

�.a/ 2 Un \ V . Then the above diagram says that d Qfa W Ta zUn ! TOC is surjectiveif and only if dg˛ W T˛Un ! TOC is surjective. On the other hand, TaCn is a directsum of Ta zUn and the tangent space of the RC � S1 orbit at a and the latter spaceis in the kernel of dfa W TaCn ! TOC, as zV is invariant by the RC � S1-action.This shows that the surjectivities of the two tangential maps dfa W TaCn ! TOC anddg˛ W T˛Un ! TOC are equivalent. Thus a 2 zV is mixed singular if and only if˛ 2 V is mixed singular.

Now we consider the canonical orientation of V. First we recall that the orientationof C� is given by the frame f @

@r; @@�gwhere .r; �/ are the polar coordinates of C�. The

orientation of CPn�1 as a complex manifold and the orientation of CPn�1 comingfrom the Hopf bundle using the local bundle structure Uj � C� is the same. Usingthe orientation of the affine hypersurface zV and the local product structure of therestriction of the Hopf bundle over V, we have a canonical orientation on (the smoothpart of ) V. One can easily see that the orientation as the local mixed hypersurfaceV \ Un D g�1.0/ � Un is the same with the above orientation.

2.2 Milnor fiber

Consider the Hopf fibration � W S2n�1 ! CPn�1 and its restriction to the Milnorfiber F D fz 2 S2n�1 j f .z; Nz/ 2 R; f .z; Nz/ > 0g. As f is polar weighted, it is easyto see that � W F ! CPn�1 n V is a cyclic covering of order dp and the group of the


covering transformation is generated by the monodromy map

h W F �! F;

z 7�! exp�2�idp

� z:

Thus we have

Proposition 2.3. (i) �.F / D dp�.CPn�1 n V /.(ii) �.CPn�1 n V / D n � �.V / and �.V / D n � �.F /=dp .

(iii) We have the exact sequence

1 �! �1.F /�]��! �1.CPn�1 n V / �! Z=dpZ! 1:

Corollary 2.4. If dp D 1, the projection � W F ! CPn�1 n V is a diffeomorphism.

The monodromy map h W F ! F gives a free Z=dpZ action on F . Thus, usingthe periodic monodromy argument in [5], we get the following result.

Proposition 2.5. The zeta function of h W F ! F is given by

�.t/ D .1 � tdp /��.F /=dp :

In particular, if dp D 1, h D idF and �.t/ D .1 � t/��.F /.

3 Topology of mixed projective hypersurface

We are interested in the topology of the mixed projective hypersurface. Assume thatf .z; Nz/ is a strongly polar homogeneous polynomial of radial degree dr and of polardegree q. Let V be the corresponding projective hypersurface V D ff .z; Nz/ D 0g �CPn�1. In the case of smooth complex algebraic hypersurfaces, the topology of F orV are determined by the degree q. In the case of mixed hypersurfaces, we will seelater that the degree q do not determine the topology of the Milnor fibering of f orthe topology of V.

138 Mutsuo Oka

3.1 Isolated singularity case

We consider a mixed strongly polar homogeneous polynomialf .z; Nz/ of polar degree qand we assume that zV D f �1.0/ has an isolated mixed singularity at the origin.

Assume thatF D f �1.1/has a homotopy type of a bouquet of spheres of dimensionn � 1.

Proposition 3.1. Under the above assumption, the mixed projective hypersurface hasthe following homology groups:

Hj .V / D8<:Z; j even, j � 2.n � 2/; j ¤ n � 2;0; j odd, j < 2.n � 2/; j ¤ n � 2:

The middle homology group Hn�2.V / is determined by �.F / up to the torsion part.

Proof. First, by the assumption, zHj .F / D 0 for j � n � 2 and Hn�1.F / D Z�

where� D .�1/n�1.�.F / � 1/:

By the Wang sequence of the Milnor fibration

: : : �! HjC1.S2n�1 nK/! Hj .F /h��id��! Hj .F / �! Hj .S

2n�1 nK/! : : :

we see thatHj .S

2n�1 nK/ D 0; j � n � 2As for Hn�1.S2n�1 nK/, it is isomorphic to the cokernel of

h� � id W Hn�1.F / �! Hn�1.F /

and the rank of this cokernel can be computed from the zeta function and the charac-teristic polynomial Pn�1.t/ which are related by

.tdp � 1/��.F /=dp D �.t/ D Pn�1.t/.�1/n�2

.t � 1/ :

We leave the calculation to the reader. By the Alexander duality, we get

zHj .K/ D 0; j < n � 2:Now the assertion follows from the Gysin sequence of the Hopf fibration � W K ! V :

: : : �! Hj .K/ �! Hj .V / �! Hj�2.V / �! Hj�1.K/ �! : : :

The argument is exactly same as that for a projective hypersurface (see [6])

Remark 3.2. Assume that a mixed function f .z; Nz/ is strongly non-degenerate. It isan open problem if (i) F is .n � 2/-connected, and (ii) F has a homotopy type ofCW-complex of dimension n � 1.


3.2 Solutions and points in CP1

Let us consider the case n D 2. Let M.q C 2r; qI 2/ be the set of mixed polar ho-mogeneous polynomials with radial degree qC 2r and polar degree q. Let f .z; Nz/ bea non-degenerate strongly polar homogeneous polynomial in M.q C 2r; qI 2/ wherez D .z1; z2/. For brevity, we assume that q; r > 0. We are interested to compute thenumber of points in V D fŒz� 2 CP1 j f .z; Nz/ D 0g. This number is equal to thenumber of link components of S3\ zV where zV D f �1.0/ � C2 and we denoted thisnumber by lkn. zV / in [8]. In general, f takes the form

f .z; Nz/ DX�;�

c�;�z�1

1 z�2

2 Nz�1

1 Nz�2

2

where �1 C �2 D �1 C �2 C q and �1 C �2 D r . We may assume that there are nopoints of V with z2 D 0. Thus we may consider the coordinate chart fz2 ¤ 0g withz D z1=z2 as the coordinate. To know the exact number of points of V, we need toknow the number of complex solutions of the mixed polynomialn

z 2 C jX�1;�1

c0�1;�1

z�1 Nz�1 D 0o; �1 C �1 � q C 2r; �1 � r

where c0�1;�1

D P�2;�2

c�;�. In fact, the number of solutions is not so easy to becomputed as in the case of complex polynomials.

Example 3.3. Consider the equation:

�2z2 Nz C tz2 C 1 D 0; t 2 C:

This example is considered in Example 59 of our previous paper [8]. We can seethat for a “small” t , we have only one solution. For example t D 0, z D 1

3p2

. For a

“large” t , we have three solutions. (For real numbers, t is “small” if �3 < t < 1.) Forexample, put t D 3. Then we get z D a and 1=9˙p26i=9 where a is the real rootof �2a3 C 3a2 C 1 D 0.

This example tells us that the number of solutions depends on the coefficients.However we have the following observation.

Proposition 3.4. Assume that f .z; Nz/ 2 M.q C 2r; qI 2/ and let us set V D fŒz� 2CP1 j f .z; Nz/ D 0g and F D f �1.1/ � C2. Then ˛ WD ]V can take (at least)q; qC 2; : : : ; qC 2r . Here ]V is the number of points in V. The corresponding Eulercharacteristic of F is �.F / D q.2 � ˛/.

Proof. We consider the basic two strongly polar homogeneous polynomials:

fq;j WD zqCj1 Nzj1 C zqCj

2 Nzj2 2M.q C 2j; qI 2/k` WD .z`1 � ˇz`2/. Nz`1 � Nz`2/ 2M.2`; 0I 2/; ˇ; 2 C�

140 Mutsuo Oka

By [7], Theorem 10, fq;j is strongly polar homogeneous and lkn. zV .fq;j // D q.

k` is obviously strongly polar homogeneous of degree 0 and lkn. zV .k`// D 2`. Thusfq;jkr�j for 0 � j � r is strongly polar homogeneous polynomial in M.qC2r; qI 2/and non-degenerate as long as V.fq;j / \ V.k`/ D ;. As 2`C .q C 2j / D q C 2r ,

lkn. zV .fq;jkr�j // D q C 2.r � j / with j D 1; : : : ; r . The latter assertion followsfrom Lemma 64 of [8].

Conjecture 3.5. The set of possible values of ˛ for f 2 M.q C 2r; qI 2/ is exactlyfq; q C 2; : : : ; q C 2rg.

4 Degree of mixed projective hypersurfaces

Suppose that f .z; Nz/ 2M.qC 2r; qIn/ be a strongly polar homogeneous polynomialand let

V D fŒz� 2 CPn�1 j f .z; Nz/ D 0g:We assume that the singular locus †V of V is either empty or codimR†V � 2. Wehave observed that V n †V � CPn�1 is canonically oriented so that the union ofthe oriented frame of TPV , say fv1; : : : ; v2.n�2/g, and the frame of the normal bundlefw1; w2gwhich is compatible with the local defining complex function gj on the affinechartUj D fzj ¤ 0g is the oriented frame of CPn�1. (Recall thatgj is a mixed functionof the variablesui D zi=zj ; i ¤ j defined bygj .u; Nu/ D f .z; Nz/=zqCr

j Nzrj .) Thus it hasa fundamental class ŒV � 2 H2n�4.V IZ/ by Borel and Haefliger [1]. The topologicaldegree of V is the integer d so that ��ŒV � D dŒCPn�2� where � W V ! CPn�1 is theinclusion map and ŒCPn�2� is the homology class of a canonical hyperplane CPn�2.

The main result of this paper is he following theorem.

Theorem 4.1. The topological degree of V is equal to the polar degree q. Namely thefundamental class ŒV � corresponds to qŒCPn�2� 2 H2.n�2/.CPn�1/ by the inclusionmapping ��.

Proof. Suppose that f is a non-degenerate mixed polynomial in M.q C 2r; qIn/.Take a generic 1-dimensional complex line L which is isomorphic to CP1. Then thedegree is given by the intersection number ŒV � ŒL�. Now, changing the coordinates ifnecessary, we may assume that

L W zj D aj1z1 C aj2z2; j D 3; : : : ; n: (4.1)

Substituting (4.1) in f .z; Nz/ to eliminate the variables z3; : : : ; zn, we see that theintersection V \ L is described by

g.z1; z2; Nz1; Nz2/ D 0; Œz1 W z2� 2 L D CP1:


As g is still polar homogeneous in z1; z2 under the restriction to L, g is written as

g.w; xw/ D f .w; xw/jL DX�;�

c�;�z�1

1 z�2

2 Nz�1

1 Nz�2

2 ; w D .z1; z2/;

where the summation are for the multi-integers � D .�1; �2/; � D .�1; �2/ such that

j�jCj�j D q C 2r; j�j � j�j D q; j�j D �1 C �2; j�j D �1 C �2:Thus the polynomial g.z1; z2; Nz1; Nz2/ is a polar homogeneous polynomial of polardegree q. Taking a linear change of coordinates if necessary, we may assume thatthe intersections are in the affine space z2 ¤ 0. This implies that g has a monomialzqCr1 Nzr1 with a non-zero coefficient. Use the affine coordinatew D z1=z2 for the affine

coordinate chart fz2 ¤ 0g. Then g takes the form

g.w; Nw/ D c0wqCr C c1wqCr�1 C C cqCr

where cj is a polynomial in Nw such that deg Nw cj � r and by the assumption, we havethat c0 D Pr

iD0 c0i Nwi with c0r ¤ 0. Let f˛1; : : : ; ˛mg D fw Ng.˛; N / D 0g. We cansee easily that

I.V;LI j / D 1

2�

Zjw� j jD"

Gauss.g/d�;

where w � j D " exp.i�/ and Gauss.g/.w; Nw/ D � 0 with � 0 D arg .g.w; Nw// and" is a sufficiently small positive number. In fact, the orientation of V is defined sothat a frame fv1; : : : ; v2n�4g at j is positive if and only if fv1; : : : ; v2n�4; n1; n2gare positive where n1; n2 are frames of the normal bundle of V oriented by f . On theother hand, f @

@x; @@yg is also a frame of the normal bundle where w D x C iy. The

orientations fn1; n2g and f @@x; @@yg are compatible if and only if the Gauss map at j

has the positive rotation.Topologically the intersection number is the mapping degree of the Gauss mapping,

considered as

Gauss.g/ W fjw � j j D "g Š S1 �! S1:

Take a sufficiently large positive numberR. Then by a standard argument, we see that

mXjD1

1

2�

Zjw� j jD"

Gauss.g/d� D 1

2�

ZjwjDR

Gauss.g/d�:

The right hand side is equal to the mapping degree of

Gauss.g/ W fjwj D Rg Š S1 �! S1

which is equal to q by the next lemma which completes the proof.

142 Mutsuo Oka

4.1 Residue formula for a monic mixed polynomial

Let g.w; Nw/ DPa;b ca;bwa Nwb be a mixed polynomial. Put

d D maxfaC b j ca;b ¤ 0gand we call d the radial degree of g. We say that g is a monic mixed polynomial ofdegree d if g has a unique monomial of radial degree d .

Lemma 4.2. Assume that g.w; Nw/ is a monic mixed polynomial of degree d which iswritten as

g.w; Nw/ D c0. Nw/wqCr C c1. Nw/wqCr�1 C C cqCr ;

cj . Nw/ 2 CŒ Nw�; rdeg Nwcj � r; j D 0; : : : ; q C r;c0. Nw/ D c0r Nwr C C c00; c0r ¤ 0;

with d D q C 2r . Then

1

2�

ZjwjDR

Gauss.g/d� D q:

Proof. Consider the family gt .w; Nw/ D .1� t/ g.w; Nw/C t h.w; Nw/with h.w; Nw/ Dc0r w

qCr Nwr . For a sufficiently large R, this gives a homotopy of the two Gauss mapsof gjjwjDR and h.w; Nw/jjwjDR. The rotation number of the Gauss map h.w; Nw/jjwjDRis obviously q. This proves the assertion.

5 Mixed projective curves

In this section, we study basic examples in the projective surface CP2. Thus we assumethat n D 3.

5.1 Milnor fibers

Let f .z; Nz/ be a strongly polar weighted homogeneous polynomial in three variablesz D .z1; z2; z3/. Let F D f �1.1/ � C3 be the Milnor fiber.

Proposition 5.1. Assume that f is 1-convenient (see [8] for the definition) non-degenerate, polar weighted homogeneous polynomial with an isolated mixed singu-larities at the origin and we assume that f is either of a join type or of a simplicialtype which are described below.


join type: f1.z; Nz/ D g.z1; z2; Nz1; Nz2/C za3Cb3

3 Nzb3

3 ;

simplicial type:

8ˆ<ˆ:

f2.z; Nz/ D za1Cb1

1 Nzb1

1 z2 C za2Cb2

2 Nzb2

2 z3 C za3Cb3

3 Nzb3

3 ;

f 02.z; Nz/ D za1Cb1

1 Nzb1

1 Nz2 C za2Cb2

2 Nzb2

2 Nz3 C za3Cb3

3 Nzb3

3 ;

f3.z; Nz/ D za1Cb1

1 Nzb1

1 z2 C za2Cb2

2 Nzb2

2 z3 C za3Cb3

3 Nzb3

3 z1;

f 03.z; Nz/ D za1Cb1

1 Nzb1

1 Nz2 C za2Cb2

2 Nzb2

2 Nz3 C za3Cb3

3 Nzb3

3 Nz1;with ai ; bi > 0; i D 1; 2; 3 where g.z1; z2; Nz1; Nz2/ is a convenient non-degeneratepolar weighted homogeneous polynomial. Then the Milnor fibers F.fi /; i D 1; : : : ; 3and F.f 0

i /; i D 2; 3 are simply connected and they have homotopy types of bouquetsof spheres S2_ _S2. Let �g be the Milnor number of g. The Euler characteristicsand the Milnor numbers are given as follows:

�.F.f1// D .a3 � 1/�g C 1;�.f1/ D .a3 � 1/�g

�.F.f2// D �.F.f 02// D a1a2a3 � a2a3 C a3;

�.f2/ D �.f 02/ D �.F.f2// � 1;

�.F.f3// D a1a2a3 C 1;�.f3/ D a1a2a3;

�.F.f 03// D a1a2a3 � 1

�.f 03/ D a1a2a3 � 2;

Proof. We consider first F1 D f �11 .1/ where

f1.z; Nz/ D g.z1; z2; Nz1; Nz2/C za3Cb3

3 Nzb3

3

where g.z1; z2; Nz1; Nz2/ is a convenient non-degenerate polar weighted homogeneouspolynomial. For two variables case, the Milnor fiberFg of g.z1; z2/ has the homotopytype of a bouquet of S1 as it is a connected open Riemann surface (see [8], Proposition38). Let�g be the Milnor number (that is the first Betti number) ofFg . Then the Milnorfiber F1 of f1 is homotopic to the join Fg ��a where �a is the set of a-th roots ofunity (see [2]). This join is obviously homotopic to a bouquet of�g.a�1/ S2 spheres.

Consider F2 D f �12 .1/ or F 0

2 D f �12 .1/. The Euler characteristic can be easily

computed from the additivity of the Euler characteristics, applied on the toric stratifi-cation

F2 D F �f1;2;3g2 q F �f2;3g

2 q F �f3g2 ;

F 02 D F 0�f1;2;3g

2 q F 0�f2;3g2 q F 0�f3g

2 ;

144 Mutsuo Oka

and Theorem 10 of [7], where F �I2 is defined by F2 \C�I and

C�I D fz 2 C3 j zi ¤ 0; i 2 I; zj D 0; j … I gfor I � f1; 2; 3g.

The Euler characteristics of F3 D f �13 .1/ and F 0

3 D f 03

�1.1/ can be computed in

the exact same way. The assertion on the homotopy types are now obtained simulta-neously as follows. First F �f1;2;3g

j and F 0j

�f1;2;3g are CW-complex of dimension 2 byTheorem 10 of [7]. Secondly Fj and F 0

j are simply connected by the 1-convenienceassumption (see [7]). Using the above decomposition and Mayer–Vietoris exact se-quences, we see that the (reduced) homology groups are non-trivial only on dimen-sion 2 and no torsion on H2.Fj / and H2.F 0

j / for j D 2; 3. Thus by the Whiteheadtheorem (see for example [10]), we conclude thatFj andF 0

j are homotopic to bouquetsof two dimensional spheres.

5.2 Projective mixed curves

We consider projective curves of degree q:

C D fŒz1 W z2 W z3� 2 CP2 j f .z1; z2; z3/ D 0g;where f is a strongly polar homogeneous polynomial with pdeg f D q. We haveseen that the topological degree of C is q by Theorem 4.1. The genus g of C is not aninvariant of q. Recall that for a differentiable curve C of genus g, embedded in CP2,with the topological degree q, we have the following Thom’s inequality, which wasconjectured by Thom and proved by, for example, Kronheimer–Mrowka [4]:

g � .q � 1/.q � 2/2

where the right side number is the genus of algebraic curves of degree q, given bythe Plücker formula. Recall that for a mixed strongly polar homogeneous polyno-mial, the genus and the Euler characteristic of the Milnor fiber are related as follows(cf. Proposition 2.3(ii)).

Proposition 5.2. We have

2 � 2g D 3 � �.F /q

(5.1)

where

F D f.z1; z2; z3/ 2 C3 j f .z1; z2; z3; Nz1; Nz2; Nz3/ D 1g:

Now we will see some examples which shows that �.F / is not an invariant of q.


I. Simplicial polynomials. We consider the following simplicial polar homogeneouspolynomials of polar degree q:

fs1.z; Nz/ D zqCr1 Nzr1 C zqCr

2 Nzr2 C zqCr3 Nzr3;

fs2.z; Nz/ D zqCr�11 Nzr1z2 C zqCr�1

2 Nzr2z3 C zqCr3 Nzr3;

fs3.z; Nz/ D zqCr�11 Nzr1z2 C zqCr�1

2 Nzr2z3 C zqCr�13 Nzr3z1;

fs4.z; Nz/ D zqCrC11 Nzr1 Nz2 C zqCrC1

2 Nzr2 Nz3 C zqCr3 Nzr3;

fs5.z; Nz/ D zqCrC11 Nzr1 Nz2 C zqCrC1

2 Nzr2 Nz3 C zqCrC13 Nzr3 Nz1:

Let Fsi be the Milnor fiber of fsi and let Csi be the corresponding projective curvesfor i D 1; : : : ; 5. First, the Euler characteristic of the Milnor fibers and the genera aregiven by Proposition 5.1 and Proposition 5.2 as follows:

�.Fsi / D q3 � 3q2 C 3q; g.Csi / D.q � 1/.q � 2/

2; i D 1; 2; 3;

�.Fs4/ D q.q2 C q C 1/; g.Cs4/ Dq.q C 1/

2;

�.Fs5/ D q.q2 C 3q C 3/; g.Cs4/ D.q C 2/.q C 1/

2:

In [9], we have shown that Cs1 and Cs2 are isomorphic to algebraic plane curvesdefined by the associated homogeneous polynomials of degree q:

gs1.z/ D zq1 C zq2 C zq3gs2.z/ D zq�1

1 z2 C zq�12 z3 C zq3 :

We also expect that Cs3 is isotopic to the algebraic curve

zq�11 z2 C zq�1

2 z3 C zq�13 z1 D 0;

as the genus of Cs3 suggests it (see also [9]).

II. We consider the following join type polar homogeneous polynomial:

hj .z; Nz/ D gj .w; Nw/C zqCr3 Nzr3;

gj .w; Nw/ D .wqCj1 Nwj1 C wqCj

2 Nwj2 /.wr�j1 � ˛wr�j

2 /. Nwr�j1 � ˇ Nwr�j

2 /;

where 0 � j � r and where ˛; ˇ 2 C� are generic. The Milnor fiber Fgjof gj

is connected. The Euler characteristic of �.F �gj/ (F �

gjD Fgj

\ C�2) is given by�.F �

gj/ D �rg q where rg is the link component number of g D 0which is qC2.r �

j /. Thus �.Fgj/ D �.F �

gj/ C 2q where the last terms come from �.Fgj

/�I / with

146 Mutsuo Oka

I D f1g or f2g. Thus �.Fgj/ D q.q � 2C 2.r � j // and g D .q�1/.q�2C2.r�j //

2. We

observe that the genus can take the following values by taking j D r; : : : ; 0:

.q � 1/.q � 2/2

;.q � 1/q

2; : : : ;

.q � 1/.q C 2r � 2/2

:

As we can take the positive number r arbitrary large, we have the following result.

Proposition 5.3. There exist differentiable curves embedded in CP2 with a fixeddegree q � 2 whose genera are given as

fg0 C k.q � 1/ j k D 0; 1; : : : g; g0 D .q � 1/.q � 2/2

:

In particular, taking q D 2, we obtain the following corollary.

Corollary 5.4. For any smooth surface S of genus g, there is an embedding S � CP2

such that the degree of S is 2.

References

[1] A. Borel and A. Haefliger, La classe d’homologie fondamentale d’un espace analytique,Bull. Soc. Math. France 89 (1961), 461–513. 140

[2] J. L. Cisneros–Molina, Join theorem for polar weighted homogeneous singularities, in Sin-gularities II. Geometric and topological aspects. Proceedings of the International Schooland Workshop on the Geometry and Topology of Singularities held in honor of the 60 th

birthday of Lê Dung Tráng in Cuernavaca, January 8–26, 2007, ed. by J.-P. Brasselet,J. L. Cisneros-Molina, D. Massey, J. Seade, and B. Teissier, Contemporary Mathemat-ics 475, American Mathematical Society, Providence, RI, 2008, 43–59 143

[3] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, NewYork1978; reprinted it Wiley Classics Library, John Wiley & Sons, New York 1994. 135

[4] P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projectiveplane. Math. Res. Lett. 1 (1994), 797–808. 134, 144

[5] J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies 61,Princeton University Press, Princeton 1968. 137

[6] M. Oka, On the cohomology structure of projective varieties, in Manifolds–Tokyo 1973.Proceedings of the International Conference on Manifolds and Related Topics in Topologyheld in Tokyo, April 10–April 17, 1973, ed. byA. Hattori, University of Tokyo Press, Tokyo1975, 137–143. 138

[7] M. Oka, Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31(2008), 163–182. 135, 140, 144

[8] M. Oka, Non-degenerate mixed functions. Kodai Math. J. 33 (2010), 1–62. 135, 139, 140,142, 143


[9] M. Oka, On mixed Brieskorn variety, in Topology of algebraic varieties and singularities.Papers from the Conference on Topology of Algebraic Varieties, in honor of AnatolyLibgober’s 60 th birthday, held in Jaca, June 22–26, 2009, ed. by J. I. Cogolludo-Agustínand Eriko Hironaka, Contemporary Mathematics 538. American Mathematical Societyand Real Sociedad Matemática Española, Providence, R.I., and Madrid 2011, 389–399.135, 145

[10] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., NewYork etc. 1966; correctedreprint Springer, New York and Berlin, 1981. 144

Invariants of splice quotient singularities

Tomohiro Okuma

Department of Education, Yamagata UniversityYamagata 990-8560, Japan


Abstract. This article is a survey of results on analytic invariants of splice quotient singular-ities induced by Neumann and Wahl. These singularities are natural and broad generalizationof quasihomogeneous surface singularities with rational homology sphere links. The “leadingterms” of the equations are constructed from the resolution graph. Some analytic invariants ofsplice quotients can explicitly be computed from their graph.

1 Introduction

Splice quotient singularities were introduced by Neumann and Wahl ([21], [22], [23],[18], and [30]). These singularities form a rich class of complex surface singularitieswith rational homology sphere (QHS for short) links. This class contains rationalsingularities, minimally elliptic singularities and weighted homogeneous singulari-ties with QHS links. We consider a resolution graph � of a normal complex surfacesingularity with QHS link with certain conditions. Then from � we can construct,according to Neumann–Wahl algorithm, the equations of a family of surface singu-larities in which each fiber has the resolution graph �; these singularities are calledsplice quotients. For example, if a resolution graph of a rational singularity is given,we can explicitly write down equations for a rational singularity with that graph.

From the point of the definition of the splice quotient, it is very natural to expect thatsome fundamental analytic invariants of them can be computed from the resolutiongraph. The purpose of this paper is to survey known results on some invariants ofsplice quotients. Neumann and Wahl [19] proved the “end curve theorem”, that is,the splice quotients are characterized by the existence of “end curve functions” (cf.[27] and [3]). The results on the dimension of cohomology groups of certain invertiblesheaves depend on the existence of the end curve functions.

The author is grateful to the referee for pointing out mistakes and helpful sugges-tions.


150 Tomohiro Okuma

2 Splice quotients

The system of equations for a splice type singularity is originally associated with aweighted tree called a splice diagram, which is obtained from the resolution graph ofa surface singularity with QHS link satisfying the “semigroup condition” (see [22]and [23] for details). In this section, we introduce the splice type singularities andsplice quotient singularities in terms of “monomial cycles” (see [26] for details). Theequivalence of these two definitions is verified in [22], §13. We will see later that thenotion of monomial cycles is useful for connecting the combinatorics of the resolutiongraph to analytic objects.

Let .X; o/ be a normal complex surface singularity whose link † is a QHS. Wemay assume that X is homeomorphic to the cone over †. By definition, a topologi-cal invariant of .X; o/ is an invariant of †. There uniquely exists a finite morphismq W .Xu; o/! .X; o/ of normal surface singularities that induces an unramified Galoiscovering Xu n fog ! X n fog with Galois group H1.†;Z/. The morphism

q W Xu �! X

is called the universal abelian covering of X .Let � W zX ! X be the minimal good resolution with the exceptional divisor

E D ��1.o/. Let fEvgv2V denote the set of irreducible components of E and let �denote the weighted dual graph of E. It is known that � and † determine each other(see [17]). The assumption on † is equivalent to that every Ev is a rational curve and� is a tree. Hence � and the intersection matrix I.E/ D .Ev Ew/ have the sameinformation. Since I.E/ is negative-definite, for each v 2 V there exists an effectiveQ-cycle E�

v such that E�v Ew D �ıvw for every w 2 V . Let

L DXv2V

ZEv and L� DXv2V

ZE�v :

We call an element of the group L (resp. L˝Q) a cycle (resp. Q-cycle). We have anatural isomorphism (cf. [8], §2, and [25], §2)

H D L�=L �! H1.†;Z/:

Thus H1.†;Z/ is a finite group of order jdet I.E/j.Let ıv denote the number of irreducible components of E intersecting Ev, i.e.,

ıv D .E � Ev/ Ev. A curve Ev is called an end (resp. a node) if ıv D 1 (resp.ıv � 3). Let E (resp. N ) denote the set of indices of ends (resp. nodes). A connectedcomponent of E � Ev is called a branch of Ev. Let Cfzg D Cfzw Iw 2 Eg be theconvergent power series ring in #E variables.

Definition 2.1. An element of a semigroup M D Pw2E Z�0E�

w , where Z�0 isthe set of nonnegative integers, is called a monomial cycle. For a monomial cycleD DPw2E ˛wE

�w , we associate a monomial z.D/ DQw2E z

˛ww 2 Cfzg.

Invariants of splice quotient singularities 151

For every v;w 2 V , we define positive integers ev , lvw , and mvw as follows:

.lvw/ D jH j.�I.E//�1;ev D jH j= gcd flvw j w 2 Vg;

mvw D evlvw=jH j:

Definition 2.2. For any v 2 V , we define the v-degree of a monomialQw2E z

˛ww

to bePw2E ˛wmvw . The leading form of f 2 Cfzg with respect to the v-weight is

called the v-leading form of f and denoted by LFv.f /. The v-degree of LFv.f / iscalled the v-order of f. Note that v-deg z.D/ D �evD E�

v . In Section 3, we shallsee the geometric nature of the degree.

Definition 2.3. We say that E (or �) satisfies the monomial condition if for anynode Ev and any branch C of Ev , there exists a monomial cycleD such thatD �E�

v

is an effective cycle supported on C . For such D, the monomial z.D/ is called anadmissible monomial belonging to the branch C.

Definition 2.4. Assume that the monomial condition is satisfied. LetEv be an arbitrarynode with branchesC1; : : : ; Cıv

, and letMi denote an admissible monomial belongingto Ci . Let Fv be the set of polynomials f1; : : : ; fıv�2 defined by

fi DıvXjD1

cijMj

with

.cij / D

0BBB@1 0 0 a1 b10 1 0 a2 b2:::

:::: : :

::::::

0 0 1 aıv�2 bıv�2

1CCCA;where ai ; bi 2 C� and aibj �aj bi ¤ 0 (i ¤ j ). We call

Sv2N Fv a Neumann–Wahl

system associated with E.

Example 2.5. Let us consider the following graph corresponding to a QHS link:

�

�

2

1

� �

6 7

�38

4

93

510

��

�

�

�

�

��

��

where the positive integers i indicate Ei and the weights �2 are omitted. This graphsatisfies the monomial condition. In fact, the following polynomials form a Neumann–

152 Tomohiro Okuma

Wahl system:

F6 D fz21 C z22 C z23z25g;F8 D fz21 C z33 C z35 ; z24 C z33 � z35g:

The intersection form L� � L� ! Q defined by I.E/ induces a pairing

� W H � L� �! Q=Ze��! C�;

where e.x/ D exp.2�p�1x/. We denote by‚.D/ 2 OH D Hom.H;C�/ the charac-

ter determined by �. ;D/. The groupH acts on the power series ring Cfzg as follows.For any .h;D/ 2 H �M, we define h z.D/ 2 Cfzg by

h z.D/ D �.h;D/z.D/:Note that Fv consists of ‚.E�

v /-eigenfunctions.

Definition 2.6 (see [22], §7). Consider a set

F D ffvjvj v 2 N ; jv D 1; : : : ; ıv � 2g � Cfzg:

If the setSv2N fLFv.fvjv

/ j jv D 1; : : : ; ıv � 2g is a Neumann–Wahl system asso-ciated with E, then F is called a system of splice diagram functions, and the sin-gularity .Y; o/ � .C#E ; o/ define by F is called a splice type singularity. This isan isolated complete intersection surface singularity. Furthermore if every fvjv

is a‚.E�

v /-eigenfunction, then the singularity .Y=H; o/ is a normal surface singularityand called a splice quotient singularity.

Definition 2.7. We say that zX satisfies the end curve condition if for each w 2 E

there exist a function uw on zX and an irreducible curve Hw � zX , not contained inE, such that ew.E�

w CHw/ D div.uw/. Then Hw E D Hw Ew D 1.

If the end curve condition is satisfied, then, taking functions sw D u1=eww on Xu,

we obtain an H -equivariant C-algebra homomorphism

W Cfzg ! OXu;o; .zw/ D sw :Theorem 2.8 (End Curve Theorem [19]). If zX satisfies the end curve condition, thenX is a splice quotient singularity; in fact, the homomorphism is surjective, andits kernel is generated by a system of splice diagram functions with H -action. Theconverse is also true.

3 Filtrations

We assume that .X; o/ satisfies the end curve condition. Let us recall that the universalabelian cover q W Xu ! X fits into the following commutative diagram, where p is


finite and unramified over zX nE, � is a partial resolution (cf. [25], §3.2):

zXu p ��

�

��

zX

�

��Xu

q�� X:

Then F D p�1.E/ is the �-exceptional set on zXu. Let v 2 V and Fv D p�1.Ev/.For each n 2 Z�0, we denote by I vn the ideal f .f / j v-ord.f / � ng � OXu;o. LetG.v/ denote the associated graded ring

Ln�0 I vn =I vnC1. Since p�Ev D evFv by [25],

§3.4, it follows from the definition of the v-degree that every function f 2 I vn satisfiesdiv zXu.f / � nFv.

Proposition 3.1 ([22], §2.6 [26], §3, [24], §4). For every v 2 V ,

(i) I vn D .��O zXu.�nFv//o, for n � 0, and

(ii) the ring G.v/ is a reduced complete intersection ring, isomorphic to CŒz�=I v,where I v is the ideal generated by fLFv.f / j f 2 F g.

Definition 3.2. For v 2 V and � 2 yH , let

H��;v.t/ D

1

jH jXh2H

��1.h/Yw2V

.1 � �.h;E�w/ t

mvw /ıw�2:

Proposition 3.3 ([26]). The rational function H��;v.t/ is the Hilbert series of the

�-eigenspace G.v/� with respect to the H -action, i.e.,

H��;v.t/ D

Xn�0

.dimG.v/�n/tn:

In [12], Némethi proves the Campillo–Delgado–Gusein-Zade type identity for theHilbert series of the multi-variable filtration associated with the exceptional divisorsof the minimal good resolution. This result implies the proposition above, and further-more that h1 of invertible sheaves associated with exceptional divisors are topological.

4 Some analytic invariants

In general the fundamental invariants geometric genus and multiplicity are not topo-logical invariants. However for splice quotients these are explicitly computed fromthe weighted dual graph. Assume that zX satisfies the end curve condition. Let H1be a subgroup of H and .X1; o/ D .Xu=H1; o/. Every abelian cover of X which isunramified over X n fog is of this type.

154 Tomohiro Okuma

4.1 The geometric genus

The geometric genus of .X; o/ is denoted by pg.X; o/. We define an invariant of thetriple .�; v; �/.

Definition 4.1. Suppose H��;v.t/ D p.t/C r.t/=q.t/, p; q; r 2 CŒt �, deg r < deg q.

Then c��;v D p.1/. We omit � in the notation when � D 1.

Let v 2 V . Suppose that C1; : : : ; Cıvare the branches of Ev. Let .Xi ; xi / denote

the normal surface singularity obtained by contracting Ci .

Theorem 4.2 ([26]). We have the following results.

(i) Each .Xi ; xi / is also a splice quotient.

(ii) pg.X; o/ D c�;v CPıv

iD1 pg.Xi ; xi /.

Corollary 4.3. The geometric genus of a splice quotient can be computed from � .If � is a star-shaped graph with node v, then pg.X; o/ D c�;v.

Example 4.4. Suppose that .X; o/ is a splice quotient with graph in Example 2.5.Then H�;8.t/ D t5 C t C fractional part. Since every branch of E8 corresponds to arational singularity, pg.X; o/ D c�;v D 2.

There is a formula for hi of �-eigensheaves of OXu similar to Theorem 4.2(2), andthus pg.X1; o/ can be computed from � and H1 (cf. [26]).

4.2 The Seiberg–Witten invariant

In this subsection we mention results on the Seiberg–Witten invariant of the links ofsingularities. First let us recall

Casson Invariant Conjecture (Neumann–Wahl [20]). If .V; o/ is an isolated com-plete intersection surface singularity with Z-homology sphere link, then � D �=8,where � is the Casson invariant of the link and � the signature of the Milnor fiber of.V; o/.

In [20], Neumann and Wahl proved the conjecture for Brieskorn complete inter-sections (Fukuhara–Matsumoto–Sakamoto [5] independently proved it) applying theadditivity properties with respect to splice decomposition and the result for Brieskornhypersurface proved by Fintushel and Stern [4], and also proved it for suspensionhypersurface singularities.

This conjecture is generalized as follows. If .X; o/ satisfies the assumption of theconjecture, by Laufer–Durfee formula, we obtain �� D 8pg CK2 C s, where K is


the canonical divisor and s the number of irreducible components of the exceptionaldivisor on a good resolution. Thus the equality � D �=8 is equivalent to

pg C �C K2 C s8

D 0:Note that smoothability of the singularity is not needed in this formulation. Némethi–Nicolaescu [10] considered the Seiberg–Witten invariant sw.†/ in order to generalizethe Casson Invariant Conjecture. This was a natural generalization, since sw.†/ D�.†/whenH1.†;Z/ D 0. They computed sw.†/ from the graph and formulated thefollowing:

Seiberg–Witten Invariant Conjecture ([10], cf. [2]). Let K D K zX and s D #V . Ifthe complex structure of .X; o/ is “nice”

pg.X; o/C sw.†/C K2 C s8

D 0;

The original conjecture was formulated for Q-Gorenstein singularities with QHSlinks, however the counterexamples are given in [7]. Thus the problem is to identifyclasses of singularities satisfying this identity. Némethi and Nicolaescu proved thisconjecture for some classes (see [10], [11], [15], [13], and [14]), including splicequotients with star-shaped graph. This guarantees the first step of the induction for theproof of Seiberg–Witten invariant conjecture for splice quotients.

For the singularity .Xi ; xi / in Subsection 4.1, we define the invariants †i , si , K2iin a similar way as †, s, K2.

Theorem 4.5 (Braun–Némethi [2]).

sw.†/C K2 C s8

D �c�;v CıvXiD1

�sw.†i /C K2i C si

8

�:

This theorem and the pg -formula in Subsection 4.1 shows that the Seiberg–Witteninvariant conjecture is true for splice quotients.

We note that Némethi [8] formulated the conjecture in more general situation andBraun and Némethi [2] verified it for splice quotients.

Remark. There exists a non splice quotient which satisfies the Seiberg–Witten invari-ant conjecture; in fact, there exists an equisingular deformation of a splice quotientsuch that general fibers are not splice quotients (see [7]).

4.3 The multiplicity

We denote the multiplicity of .X; o/ by mult.X; o/. Let mX denote the maximal idealof OX;o. For any function f 2 mX n f0g, if div zX .��f / DP

v2V avEv C C , where

156 Tomohiro Okuma

C has no component of E, then .f /E DPv2V avEv . For any set D of effective Q-cycles, we denote by gcd D the maximal Q-cycle (if it exists) such that gcd D � Dfor any D 2 D . The cycle

Z zX D gcd f.��f /E j f 2 mX n f0ggis called the maximal ideal cycle on zX .

Theorem 4.6 (Wagreich [29]). If O zX .�Z zX / is generated by its global sections, thenmult.X; o/ D �Z2zX .

Let M denote the semigroup of the monomial cycles andX1 D Xu=H1. Let MH1

denote the set of H1-invariant monomial cycles, i.e.,

MH1 D fD 2M j ‚.H1;D/ D f1gg:Let p W zX1 ! zX be as in the commutative diagram in Section 3, but zX1 is a goodresolution ofX1. For a Q-cycleD DP aiEi , let coefEi

.D/ denote ai . The followingtheorems are proved in [24].

Theorem 4.7. Let Z D gcd MH1 . Assume the following:

(i) for every x 2 E, there exists D 2 MH1 such that D D Z on a neighborhoodof x;

(ii) for every i 2 E with Z Ei < 0, there exists F DPj2Enfig ajE�j 2MH1 such

that coefEi.F / D coefEi

.Z/.

Then p�Z is the maximal ideal cycle on zX1, and O zX1.�p�Z/ is generated by global

sections. Hence mult.X1; o/ D jH=H1j.�Z2/.

Theorem 4.8. There exists a modification � W xX ! zX such that the conditions ofTheorem 4.7 are satisfied on xX , which can be obtained by finitely many blowing upsat “bad points.” The graph of NX can be obtained from � and H1.

Corollary 4.9. The multiplicity of a splice quotient singularity and its universalabelian cover can be computed from the weighted dual graph.

In [12], Némethi has obtained a formula for the multiplicity of splice quotients.His formula does not need resolutions of the base points of O zX .�Z zX /, and the con-tributions from the base points are expressed in terms of Newton diagrams obtainedfrom the graph.

Example 4.10. We consider a splice quotient with the weighted dual graph of Ex-ample 2.5. Let hD1; : : : ;Dki denote the subgroup of H generated by the class ofD1; : : : ;Dk 2 L�. We see thatH D hE�

1 ; E�2 ; E

�3 i and jH j D 12. (The Hermite nor-

mal form of I.E/ shows generators of H and their relations.) We denote an element

Invariants of splice quotient singularities 157P10iD1 aiEi by the sequence .a1 a10/. Then0BBBBBBB@

E�1

E�2

E�3

E�5

E�8

1CCCCCCCAD

0BBBBBBBB@

52

2 1 32

1 4 72

3 2 2

2 52

1 32

1 4 72

3 2 2

1 1 43

1 23

2 2 2 53

43

1 1 23

1 43

2 2 2 43

53

3 3 2 3 2 6 6 6 4 4

1CCCCCCCCA:

Suppose H1 D f0g. Then X1 D Xu and Z D 13E�8 . By Theorem 4.7, we have

mult.Xu; o/ D jH j.�Z2/ D 12 .2=3/ D 8:The leading form (with respect to deg zi D 1) of the Neumann–Wahl system in Exam-ple 2.5 forms a regular sequence of degree 2; 2; 2. This also implies mult.X1; o/ D 8.

IfH1 D hE�1 i, thenZ D 1

3E�8 and mult.X1; o/ D jH=H1j.�Z2/ D 6.2=3/ D 4:

Next suppose H1 D H. Then we have Z D E�3 C E�

5 . Since 2E�1 2 MH1

and coefEi.2E�

1 / D coefEi.Z/ for i D 3; 5, it follows from Theorem 4.7 that

mult.X; o/ D �Z2 D 4.

4.4 The embedding dimension

It is known that the embedding dimension of rational and weakly elliptic Gorensteinsingularities are topological (see [1], [6], and [9]). However in general the embeddingdimension e:d:.X; o/ is not topological even if .X; o/ is a weighted homogeneoussingularity.

Assume that .X; o/ is a weighted homogeneous singularity, zX the minimal goodresolution, andE0 the node. Let � D ı0. The complex structure of .X; o/ is determinedby the weighted dual graph and the configuration of the points E0 \ .E �E0/ in E0.Hence e:d:.X; o/ is obviously topological if � D 3.

Theorem 4.11 ([16], §5-6). (1) If � � 5, then the Hilbert seriesHmX=m2X

of the graded

artinian module mX=m2X is topological, hence so is e:d: .X; o/.

(2) If e:d: .X; o/ is topological, then so is HmX=m2X

.

Example 4.12 ([16], §7.1.1). Assume that � D 6, E D E0 C E1 C C E6, and.�E20 ; : : : ;�E26 / D .2; 2; 2; 3; 3; 7; 7/. Then .X; o/ is defined by

y43 � y32y5 C .p1 C p2/y23y5 C p1p2y25 ;y71 � y32 C .p1 C p2 � p3 � p4/y23 C .p1p2 � p3p4/y5

158 Tomohiro Okuma

where p1; : : : ; p4 2 C�. Hence

e:d: .X; o/ D´3 if p1p2 � p3p4 6D 0,

4 otherwise:

In case �E20 � �, we have an explicit expression of HmX=m2X

in terms of theSeifert invariant (see [16], §6); this is an extension of Van Dyke’s result [28] to thecase �E20 � � C 1. However we have not obtained explicit expression of HmX=m

2X

for other classes.

References

[1] M.Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.157

[2] G. Braun and A. Némethi, Surgery formula for Seiberg–Witten invariants of negativedefinite plumbed 3-manifolds, J. Reine Angew. Math. (2010), 189–208. 155

[3] Gabor Braun, Geometry of splice-quotient singularities, preprint 2008.arXiv:0812.4403 149

[4] R. Fintushel and R. J. Stern, Instanton homology of Seifert fibred homology three spheres,Proc. London Math. Soc. (3) 61 (1990), 109–137. 154

[5] S. Fukuhara, Y. Matsumoto, and K. Sakamoto, Casson’s invariant of Seifert homology3-spheres, Math. Ann. 287 (1990), 275–285. 154

[6] H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257–1295. 157

[7] I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariantsof superisolated singularities, J. Algebraic Geom. 14 (2005), 543–565. 155

[8] A. Némethi, Line bundles associated with normal surface singularities, preprint 2003.arXiv:math/0310084 150, 155

[9] A. Némethi, “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math. 137(1999), 145–167. 157

[10] A. Némethi and L. I. Nicolaescu, Seiberg-Witten invariants and surface singularities,Geom. Topol. 6 (2002), 269–328. 155

[11] A. Némethi and L. I. Nicolaescu, Seiberg-Witten invariants and surface singularities II.Singularities with good C�-action, J. London Math. Soc. (2) 69 (2004), 593–607. 155

[12] A. Némethi, The cohomology of line bundles of splice-quotient singularities, preprint 2008arXiv:0810.4129 153, 156

[13] A. Némethi, On the Ozsváth–Szabó invariant of negative definite plumbed 3-manifolds,Geom. Topol. 9 (2005), 991–1042.http://www.msp.warwick.ac.uk/gt/2005/09/p023.xhtml 155

[14] A. Némethi, Graded roots and singularities, in Singularities in Geometry and Topology.Proceedings of the Trieste Singularity Summer School and Workshop, ICTP, August 15–

http://arxiv.org/abs/0812.4403

http://arxiv.org/abs/math/0310084


http://www.msp.warwick.ac.uk/gt/2005/09/p023.xhtml


September 3, 2005. ed. by J-P. Brasselet, D. T. Lê, and M. Oka, World Scientific, Hack-ensack, NJ, 2007, 394–463. 155

[15] A. Némethi and L. I. Nicolaescu, Seiberg–Witten invariants and surface singularities:splicings and cyclic covers, Selecta Math. (N.S.) 11 (2005), 399–451. 155

[16] A. Némethi and T. Okuma, The embedding dimension of weighted homogeneous surfacesingularities, J. Tapol. 3 (2010), 643–667. 157, 158

[17] W. D. Neumann, A calculus for plumbing applied to the topology of complex surface sin-gularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299–344.150

[18] W. D. Neumann, Graph 3-manifolds, splice diagrams, singularities, in Singularity the-ory – Dedicated to Jean-Paul Brasselet on his 60 th birthday. Proceedings of the 2005Marseille Singularity School and Conference, CIRM, Marseille, France, 24 January – 25February 2005, ed. by D. Chériot, N. Dutertre, C. Murolo, A. Pichon and D. Trotman,World Scrientific, Singapore 2007, 787–817. 149

[19] W. D. Neumann and J. Wahl, The end curve theorem for normal complex surface singu-larities, J. Eur. Math. Soc. (JEMS) 12 (2010), 471–503. 149, 152

[20] W. D. Neumann and J. Wahl, Casson invariant of links of singularities, Comment. Math.Helv. 65 (1990), 58–78. 154

[21] W. D. Neumann and J. Wahl, Universal abelian covers of surface singularities, in Trends insingularities, ed. by A. Libgober and M. Tibar, Trends in Mathematics, Birkhäuser, Basel2002, 181–190. 149

[22] W. D. Neumann and J. Wahl, Complete intersection singularities of splice type as universalabelian covers, Geom. Topol. 9 (2005), 699–755.http://www.msp.warwick.ac.uk/gt/2005/09/p017.xhtml 149, 150, 152, 153

[23] W. D. Neumann and J. Wahl, Complex surface singularities with integral homology spherelinks, Geom. Topol. 9 (2005), 757–811.http://www.msp.warwick.ac.uk/gt/2005/09/p018.xhtml 149, 150

[24] T. Okuma, The multiplicity of abelian covers of splice quotient singularities, preprint2010. arXiv:1002.2048 153, 156

[25] T. Okuma, Universal abelian covers of rational surface singularities, J. London Math.Soc. (2) 70 (2004), 307–324. 150, 153

[26] T. Okuma, The geometric genus of splice-quotient singularities, Trans. Amer. Math.Soc. 360 (2008), 6643–6659. 150, 153, 154

[27] T. Okuma,Another proof of the end curve theorem for normal surface singularities, J. Math.Soc. Japan 62 (2010), 1–11. 149

[28] F. Van Dyke, Generators and relations for finitely generated graded normal rings of di-mension two, Illinois J. Math. 32 (1988), 115–150. 158

[29] P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419–454. 156

[30] Jonathan Wahl, Topology, geometry, and equations of normal surface singularities, inSingularities and computer algebra. Selected papers of the conference, Kaiserslautern,Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60 th birthday.ed. by Ch. Lossen and G. Pfister, London Mathematical Society Lecture Note Series 324,Cambridge University Press, Cambridge 2006, 351–371. 149




A note on the toric dualitybetween the cyclic quotient surface singularities

An;q and An;n�q

Oswald Riemenschneider

Mathematisches Seminar der Universität HamburgBundesstrasse 55, 20146 Hamburg, Germany


Abstract. In my lecture at the Franco–Japanese Symposium on Singularities I gave an intro-duction to the work of Martin Hamm [3] concerning the explicit construction of the versaldeformation of cyclic surface singularities. Since that part of his dissertation is already docu-mented in a survey article (cf. [8]), I concentrate in the present note on some other aspect of [3]:the toric duality of the total spaces of the deformations over the monodromy coverings of theArtin components for the singularities An;q and An;n�q which themselves are toric duals ofeach other. Our exhibition is based – as in Hamm’s dissertation – on the algebraic aspects, i.e.the algebras and their generators of these total spaces. We prove Hamm’s remarkable dualityresult in this note first in detail for the hypersurface case q D n � 1 in which the interplaybetween algebra and geometry of the underlying polyhedral cones is rather obvious, especiallywhen bringing also the “complementarity” of An;q and An;n�q into the game. We then treatthe dual case q D 1 of cones over the rational normal curves once more in order to develop thenecessary ideas for transforming the generators in such a way that it becomes transparent howto compute the dual, even in the general situation (which we explain in the last section by anexample).

1 Introduction

The association � 7! �0 D �=.� � 1/ establishes a one-to-one correspondence forrational numbers in the open interval .1;1/. Therefore, it gives rise to a bijection onthe set of all rational cones

� D �� D f.x; y/ 2 R2 W 0 � y � �xg � R2; � > 1;

resp. to a bijection on the set of all (additive, finitely generated) semigroups

N D N� D �� \ .Z˚Z/ D �� \ .N˚N/ D f.j; k/ 2 N2 W 0 � k � �j g � N2

with � > 1. Writing � D n=q with 1 � q < n and n; q coprime, we also set�n;q D �� D �n=q and Nn;q D N�, respectively.


162 Oswald Riemenschneider

The subalgebra

An;q D CŒ�n;q� D CŒsj tk W .j; k/ 2 Nn;q� � CŒs; t �

is finitely generated and defines a two-dimensional affine toric variety Xn;q (for thegeneral theory and these examples see e.g. [4]). It is well known thatAn;q is isomorphicto the invariant ring

CŒu; v��n;q ;

where the finite cyclic group �n;q is generated by the diagonal matrix diag.�n; �qn/

with �n an n-th primitive root of unity, i.e.

An;q D CŒsj tk W .j; k/ 2 Nn;q� Š CŒujvk; j C qk 0 mod n� D CŒu; v��n;q

and

Xn;q Š C2=�n;q :

The quotient space Xn;q has exactly one (normal) singularity at the (image of the)origin (under the natural projection C2 ! C2=�n;q D Xn;q). By abuse of language,we call this theAn;q-singularity resp. the singularity of typeAn;q or just the singularityAn;q .

Examples. (1) q D n � 1. The semigroup Nn;n�1 is generated by .1; 0/; .1; 1/ and.n � 1; n/ and thus, the ring CŒ�n;n�1� is generated by the monomials x0 D s; x1 Dst; x2 D sn�1tn with the generating relation x0x2 D xn1 . Similarly, the invariant ringis generated by un; uv; vn with the same generating relation. Hence, singularities oftype An;n�1 are the two-dimensional simple hypersurface singularities of type An�1.

(2) q D 1. The semigroup Nn;1 is generated by .1; 0/; .1; 1/; .1; 2/; : : : .1; n/, andthe ring CŒ�n;1� is therefore generated by the monomials x0 D s; x1 D st; x2 Dst2; : : : ; xn D stn. Equations for the corresponding affine variety are given in deter-minantal form, i.e. by the vanishing of the 2 � 2-minors of the matrix

x0 x1 : : : xn�2 xn�1x1 x2 : : : xn�1 xn

!:

In homogeneous coordinates of Pn, these equations define the so called rational normalcurve of degree n. Hence, the singularity of type An;1 is isomorphic to the coneover this rational curve, i.e. to the closure in CnC1 of its preimage under the naturalprojection CnC1 n f0g ! Pn. Note that this again is in accordance to the descriptionas an invariant ring, since CŒu; v��n;1 will obviously be generated by the monomialsun; un�1v; un�2v2; : : : ; uvn�1; vn.

The purpose of this note is to make the correspondence Xn;q D X� 7! X�0 DXn;n�q geometrically visible. In particular, we want to elucidate Martin Hamm’sresult that the total spaces of the versal simultaneously resolvable deformations ofXn;q and Xn;n�q are dual to each other as affine toric varieties.

A note on the toric duality 163

2 Generators for Nn;q and Hirzebruch–Jungcontinued fractions

It is more natural to start with the complement of �n;q in the half space R�RC whichwe denote by

�cn;q D f.x; y/ 2 R2 W nx � qy; y � 0g:If we expand the ratio n=q into its Hirzebruch–Jung continued fraction

n

qD b1 � 1 b2 � � 1 br ;

then the partial fractions

b1 � 1 b2 � � 1 b� ; � D 1; : : : ; r;form a strongly decreasing sequence of rational numbers.1 If we write them in theform

P�

Q�

with P�;Q� relatively prime,

it is well known that the numerators and denominators are built by the followinginductive rules:

P�1 D 0; P0 D 1; P� D b�P��1 � P��2;

Q�1 D �1; Q0 D 0; Q� D b�Q��1 �Q��2:

In particular, successive quotients P��1=Q��1 and P�=Q� are Farey neighbors, i.e.

det

P��1 P�

Q��1 Q�

!D 1:

From this, one can easily deduce the following lemmata.

Lemma 2.1. The semigroup N cn;q D �cn;q \ .Z �N/ is minimally generated by the

elements .Q�; P�/; � D �1; 0; 1; : : : ; r .

Lemma 2.2. For each rational number P=Q between successive quotients P�=Q�

and P�C1=Q�C1 the denominator Q is greater or equal to Q� CQ�C1.

Now, the linear mapping x

y

!7�!

�1 1

0 1

! x

y

!D y � xy

!;

i.e.� D y

x7�! �

� � 1 ;1From now on, the Greek letter � denotes an index running from 1 to r rather than a rational number as in

the introduction.


maps obviously �cn;q bijectively onto �n;n�q and Z � Z onto itself, hence gives asemigroup isomorphism from N c

n;q onto Nn;n�q .Interchanging the roles of q and n � q, this implies the following result.

Lemma 2.3. Expand n=.n � q/ in its Hirzebruch–Jung continued fraction:n

n � q D a1 � 1 a2 � � 1 am :

Then, the semigroup Nn;q is finitely generated by the elements

.j�; k�/; � D 0; : : : ; mC 1;where

.j0; k0/ D .1; 0/;

.j1; k1/ D .1; 1/;and

.j�C1; k�C1/ D a�.j�; k�/ � .j��1; k��1/; � D 1; : : : ; m:

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Figure 1. Four different manifestations of N7;4.

n n

n � q q

q � nn

q

n


3 Duality of the singularities An;q and An;n�q

The dual of the cone we get by rotating �cn;q clockwise by 90ı equals �n;q . Let us statethis remark as follows.

Lemma 3.1. We have�?n;q Š �cn;q:

This identity has been found and conceptually proven in more generality by PatrickPopescu-Pampu (see [5]). Since we haveN c

n;q Š Nn;n�q by our considerations before,this implies the following result.

Lemma 3.2. The singularities An;q and An;n�q are dual to each other as affine toricvarieties, i.e., if An;q is defined by a lattice L and a cone � � L˝ R, then An;n�q isgiven by the dual lattice L_ and the cone �? � L_ ˝ R.

Corollary 3.3. The ordinary double point A1 D A2;1 is selfdual (and the unique onewith that property).

Lemma 3.1 suggests to leave the realm of affine toric varieties by regarding thefan we get by subdividing the closed upper half plane along the line qy � nx D 0.This gives a space xXn;q which contains a projective line P1 such that at the origin ofthis projective line, xXn;q possesses a singularity of type An;q , whereas at 1, xXn;qpossesses a singularity of type An;n�q (and no others). In particular, xXn;q is a partialcompactification of Xn;q .

Figure 2. The fan for xXn;q .

n

q


Notice that the action of the cyclic group �n;q on C � C can be extended to thepartial compactification P1 � C, i.e. to the trivial line bundle on P1. If u; v are thecoordinates of C2, we put u; v for the coordinates at infinity, i.e.

u D u�1; v D v:Hence, the action of the standard generator of �n;q extends uniquely at1 to

u 7�! ��1n u; v 7�! �qnv

such that, by using the inverse of the standard generator, we see that the action at1is of type �n;n�q . We leave it as an exercise to the reader to show the following result.

Lemma 3.4. We havexXn;q Š .P1 �C/=�n;q :

4 Quasi-determinantal equations and the Artin component

Lemma 2.3 is fundamental for finding equations of the singularities of type An;q(see [6]). We put

x� D sj� tk� ; � D 0; 1; : : : ; m;mC 1:Then, obviously, we have

x��1x�C1 D xa�� ; � D 1; : : : ; m:

However, these are not all equations unless m D 1. In general, one has to look at theso called quasi-matrix0BB@

x0 x1 x2 : : : xm�1 xm

xa1�21 x

a2�22 : : : x

am�2m

x1 x2 x3 : : : xm xmC1

1CCAand to form all 2 � 2-quasi-determinants.

By perturbing the entries of this quasi-matrix, one gets automatically deformationsof the given singularity. A nice way to do this is to set

x.`/� D x� C t .`/� ; ` D 0; : : : ; a� � 1; � D 1; : : : ; m;and to form the 2 � 2-quasi-determinants of the quasi-matrix0BB@x0 x

.a1�1/1 : : : x

.am�1/m

x.1/1 : : : x

.a1�2/1 x

.1/2 : : : x

.a2�2/2 : : : x

.1/m : : : x

.am�2/m

x.0/1 x

.0/2 : : : xmC1

1CCA:


In order to minimize the cardinality of deformation parameters, one can choose t .0/� D0 (i. e. x.0/� D x�) or better, since more symmetrically,

a��1X`D0

t .`/� D 0 ; � D 1; : : : ; m :

The resulting deformation has an intrinsic meaning. There exists a versal deforma-tion zXn;q ! T

.vers/n;q of the minimal resolution zXn;q of Xn;q which has a smooth base

space T .vers/n;q of dimension

rX�D1

.b� � 1/

(notice that the numbers �b� are invariants of the minimal resolution: The selfinter-section numbers of the components of the exceptional divisor). By a general result ofJ. Wahl and the author, this deformation can fiberwise be blown down to a deformationof the singularity Xn;q itself. It has been shown in [6] that the resulting deformationis exactly the one given explicitly above. Consequently,

rX�D1

.b� � 1/ DmX�D1

.a� � 1/: (�)

(for a direct proof of this equality, see also [6] or [4], Corollary 1.23). Denote thisdeformation by

Yn;q �! T .vers/n;q :

Obviously, on this deformation operates in a natural manner the group W D Wn;qwhich is the product of the symmetric groups Sa1�1� �Sam�1. Moreover, it acts

on T .vers/n;q by reflections such that the induced deformation

X.Art/n;q D Yn;q=Wn;q �! T .vers/

n;q =Wn;q D S .Art/n;q

has a smooth base space of the same dimension. It is well known that the (reduced)base space .S .vers/

n;q /red of the versal deformationX.vers/n;q ! S

.vers/n;q ofXn;q has in general

several (smooth) components. There exists exactly one such that the induced defor-mation is isomorphic to X.Art/

n;q ! S.Art/n;q . We call this the Artin deformation, resp.

the Artin component. This deformation is the versal deformation space for deforma-tions of Xn;q which possess a simultaneous resolution after finite base change. The

deformation Yn;q ! T.vers/n;q is also versal for deformations of Xn;q which can be

resolved simultaneously without base change. We call it the (directly) resolvable Artindeformation.


5 The grand monodromy covering

The existence of a grand monodromy covering has been conjectured by the author andwas proven by S. Brohme [1] and M. Hamm [3]. In the following, the pair .n; q/ isfixed and will be suppressed as an index.

Theorem 5.1. There exists a finite union Tred of linear subspaces Tk of a suitable CN

that is invariant under the action of a (larger) product zW of symmetric groups whichacts canonically on CN such that

S.vers/red D Tred= zW :

The components Sj of S .vers/red are exactly the images of the irreducible components Tk

of Tred under the canonical projection Tred ! S.vers/red , and two irreducible components

of Tred will be mapped onto the same component Sj only if they are translates of eachother under the action of zW.

In the induced diagram of deformations

Yred��

��

X(vers)red

��

Yk

��

��

��

Xy��

��

��Tk��

��

�� Sj ��

��

��

��

Tred= zW

�� S (vers)red

the deformation Yred ! Tred can explicitly be constructed, and the inner squares areexactly the monodromy coverings of the deformations on the components of S .vers/

red .

Conjecture 5.2. Enlarging Tred by some embedded components with linear support toa non reduced spaceT , it should be possible to extend the familyYred ! Tred explicitlyto a zW -equivariant family Y ! T such that the quotient by eW is isomorphic to theversal deformation X.vers/ ! S .vers/.

Remark. It had been already remarked by Christophersen [2] that the total defor-mation spaces Yk can be equipped with an affine toric structure. M. Hamm givesin [3] a precise description of these toric structures and a kind of toric recipe how thecomponents intersect.


6 Duality of the Artin components

We shortly explain the toric description of the resolvable Artin deformation via gen-erators in a special example.

Example. For n=.n � q/ D 56=15 D 4 � 1 4 � 1 4 we can abbreviate thequasi-determinantal equations by the scheme0BB@

.1; 0/ .1; 1/ .3; 4/ .11; 15/

.1; 1/ .1; 1/ .3; 4/ .3; 4/ .11; 15/ .11; 15/

.1; 1/ .3; 4/ .11; 15/ .41; 56/

1CCAin which the symbol .j; k/ stands either for the element .j; k/ 2 N2 or for the mono-mial sj tk .

In order to describe the toric structure of the resolvable Artin deformation, we haveto insert 3C3C3 D 9 further deformation variables. This has to be done systematicallyin the following way:

[email protected] 0; 0; 0I 0; 0; 0I 0; 0; 0I 0/ .1I 0; 0; 0I 1; 0; 0I 0; 0; 0I 0/

.1I 0; 1; 0I 0; 0; 0I 0; 0; 0I 0/ .1I 0; 0; 1I 0; 0; 0I 0; 0; 0I 0/ : : :

.1I 1; 0; 0I 0; 0; 0I 0; 0; 0I 0/ .3I 1; 1; 1I 1; 0; 0I 0; 0; 0I 0/.3I 1; 1; 1I 0; 0; 0I 1; 0; 0I 0/

: : : .3I 1; 1; 1I 0; 1; 0I 0; 0; 0I 0/ .3I 1; 1; 1I 0; 0; 1I 0; 0; 0I 0/ .11I 4; 4; 4I 0; 1; 1I 1; 0; 0I 0/.11I 4; 4; 4I 0; 1; 1I 0; 0; 0I 1/

: : : .11I 4; 4; 4I 0; 1; 1I 0; 1; 0I 0/ .11I 4; 4; 4I 0; 1; 1I 0; 0; 1I 0/.41I 15; 15; 15I 0; 4; 4I 0; 1; 1I 1/

1CCA:In general, this procedure defines uniquely

mC 2CmXjD1

.a� � 1/

vectors in Nd , where

d D 2CmXjD1

.a� � 1/;

which we denote by v0; v.0/1 ; : : : ; v

.a1�1/1 ; : : : according to the second matrix at the

beginning of Section 4. Writing them as column vectors into a matrix, one sees imme-diately that this matrix has maximal rank d such that the kernel of the corresponding


homogeneous linear system of equations has exactly dimension m which is 2 lessthan the embedding dimension of the singularity. By construction, there are indeedmlinearly independent elements in the kernel, namely

v0 C v.0/2 D v.0/1 C C v.a1�1/1 ; v

.a1�1/1 C v.0/3 D v.0/2 C C v.a2�1/

2 ; etc:;

from which all the other relations like

v0 C v.0/3 D v.0/1 C C v.a1�2/1 C v.1/2 C C v.a2�1/

2

follow.Writing the elements of the matrix symbolically as .j; k1; k2; : : :/ and projecting

down each element to .j; k/with k D k1Ck2C , we see immediately that our con-struction yields a toric variety which contains the given singularity as a distinguishedsubspace and which, in fact, is isomorphic to the total space of the resolvable Artindeformation of the given singularity. In particular, it is automatically normal.

Because of identity (�), the total spaces of the resolvable Artin deformations ofAn;q and An;n�q have the same dimension. Martin Hamm’s result says much more.

Theorem 6.1 (M. Hamm). The total spaces of the resolvable Artin deformations ofAn;q and An;n�q are dual to each other as affine toric varieties.

In other words: given the generators in case An;q , which are determined by thesequence a1; : : : ; am, the corresponding rational cone in Rd will minimally be de-scribed (up to isomorphism) by inequalities defined by the generators in case An;n�qand hence by the sequence b1; : : : ; br .

Remark. Hamm’s proof rests on a clever matrix construction which is based onanother simple manifestation of the duality in question observed by the author [6] andcalled “Riemenschneider’s dot diagram” by several authors (see also Section 10).

In the following three sections we shall prove Hamm’s result – after treating thesimplest case A1 – for the special cases q D n � 1 and q D 1.

7 The case A1

Let us start with the simplest case, the singularityA1. According to Hamm’s result, notonly A1 is selfdual as a toric variety, but also its resolvable Artin deformation is. Ofcourse, this duality can not anymore be realized by a complement in some kind of “quar-ter” space. However, something else happens.As one can check easily (see also the nextsection) the convex rational cone with generators .1; 0; 0/; .1; 1; 0/; .1; 0; 1/; .1; 1; 1/is given by the inequalities

x � y1 � 0 ; x � y2 � 0 :


Using the “complementary” description of the algebra of the singularity A1 withgenerators .�1; 0/, .0; 1/, .1; 2/, we have for the resolvable Artin deformation thegenerators .�1; 0; 0/, .0; 1; 0/; .0; 0; 1/; .1; 1; 1/, and the convex rational cone gener-ated by these points is equal to the union of the octant x � 0; y1 � 0; y2 � 0 andthe set f.x; x; x/C .0; y1; y2/ W x; y1; y2 � 0g, in other words, it is equal to the conesatisfying the inequalities

y1 � xC ; y2 � xC; xC D max.x; 0/:

Hence this cone is “complementary” to the cone x � y1 � 0; x � y2 � 0 in thesense, that they intersect in the line segment .x; x; x/; x � 0, only, and that theirconvex union is the quarter space y1 � 0; y2 � 0.

Figure 3

y1 y1

�x �x

y2 y2

In fact each one is (isomorphic to) the dual of the other: it is again very easy to seethat the dual of the cone in the complementary description is just the cone

f.x; y1; y2/ W y1; y2 � 0; y1 C y2 � �x � 0g:

Replacing x by�x, this cone is generated by .0; 1; 0/, .0; 0; 1/, .1; 1; 0/, and .1; 0; 1/,but one has

0B@1 1 1 1

0 1 0 1

0 0 1 1

1CA D0B@0 1 1

0 0 1

1 0 0

1CA0B@0 0 1 1

1 0 1 0

0 1 0 1

1CAso that the dual of the cone in the complementary description is isomorphic to theoriginal one. (For unproven statements please see the next section.) It is perhaps moresuggestive to visualize the “complementarity” of both cones by drawing x–slices only.


Figure 4

y1

y2

xC

xC

8 The hypersurface case

In order to prove Hamm’s result for q D n � 1, we write down the generators for.n; n � 1/ and .n; 1/. We first do this in a “small” case, say n D 5.

Case .5; 4/. We [email protected] 0; 0; 0; 0I 0/ .1I 0; 0; 0; 0I 1/

.1I 0; 1; 0; 0I 0/ .1I 0; 0; 1; 0I 0/ .1I 0; 0; 0; 1I 0/.1I 1; 0; 0; 0I 0/ .4I 1; 1; 1; 1I 1/

1CCACase .5; 1/. We have .1I 0I 0I 0I 0I 0/ .1I 0I 1I 0I 0I 0/ .1I 0I 1I 1I 0I 0/ .1I 0I 1I 1I 1I 0/ .1I 0I 1I 1I 1I 1/.1I 1I 0I 0I 0I 0/ .1I 1I 1I 0I 0I 0/ .1I 1I 1I 1I 0I 0/ .1I 1I 1I 1I 1I 0/ .1I 1I 1I 1I 1I 1/

!:

Denoting the standard basis of RnC1 by e0; e1; : : : ; en, the “formats” of the singularitiesAn;n�1 and An;1 can thus be written in the form (An;n�1):0B@ e0 e0 C en

e0 C e2 ; e0 C e3 ; : : :e0 C e1 .n � 1/e0 C .e1 C C en/

1CAand (An;1):

e0 e0 C e2 e0 C .e2 C e3/ e0 C e1 e0 C .e1 C e2/ e0 C .e1 C e2 C e3/


e0 C .e2 C e3 C C en/ e0 C .e1 C e2 C e3 C C en/

!:

In order to get a somewhat simpler result, we first transform the scheme .5; 4/ inan obvious way to the “complementary” situation:0BB@.�1I 0; 0; 0; 0I 0/ .0I 0; 0; 0; 0I 1/

.0I 0; 1; 0; 0I 0/ .0I 0; 0; 1; 0I 0/ .0I 0; 0; 0; 1I 0/.0I 1; 0; 0; 0I 0/ .1I 1; 1; 1; 1I 1/

1CCA:For general n, this becomes the much nicer format .An;n�1/c :0B@�e0 en

e2 ; e3 ; : : : ; en�1e1 e0 C .e1 C C en/

1CA:From this, it is evident that (an isomorphic copy of) the convex cone for An;n�1 witharbitrary n � 2 is given by the union of the “quarter space”

�j � 0; k` � 0 ; ` D 1; : : : ; n;and the set of all points

.j; j; : : : ; j /C .0; k1; : : : ; kn/ ; j � 0; k1; : : : ; kn � 0:Thus, it can be described in (�; �1; : : : ; �n)-space by the inequalities

�` � �C D max.�; 0/ ; ` D 1; : : : ; n;and therefore the situation is an obvious generalization of the picture on the right handside of Figure 3.

More precisely, these are exactly the following 2n inequalities:

�` � �; �` � 0; ` D 1; : : : ; n:Hence, the dual cone will be generated by the 2n vectors e`; e` � e0; ` D 1; : : : ; n.From the format

e1 e2 en

e1 � e0 e2 � e0 en � e0

!it follows that these generators satisfy the same relations as do the generators for theresolvable Artin deformations of the singularity An;1. Therefore, it should be possibleto transform them by a unimodular matrix. In fact, we have for n D 5 (after replacinge0 by �e0):


0BBBBBBBBBB@

0 1 0 1 0 1 0 1 0 1

1 1 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0

0 0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 0 0 1 1

1CCCCCCCCCCA

D

0BBBBBBBBBB@

0 1 0 0 0 0

1 0 �1 0 0 0

0 0 1 �1 0 0

0 0 0 1 �1 0

0 0 0 0 1 �10 0 0 0 0 1

1CCCCCCCCCCA

0BBBBBBBBBB@

1 1 1 1 1 1 1 1 1 1

0 1 0 1 0 1 0 1 0 1

0 0 1 1 1 1 1 1 1 1

0 0 0 0 1 1 1 1 1 1

0 0 0 0 0 0 1 1 1 1

0 0 0 0 0 0 0 0 1 1

1CCCCCCCCCCA:

It is an easy exercise to write down the unimodular transformation for arbitrary nfrom the “dual” set of generators e`; e0 C e`; ` D 1; : : : ; n, in case An;n�1 to theset of generators in case of An;1. In order to have better control we write the 2ngenerators in a matrix as above, replacing once more e0 by �e0. Then, we transformek 7! e1 C e2 C C ek; k D 1; : : : ; n, and we get a new matrix of generators

e1 e1 C e2 e1 C e2 C C ene0 C e1 e0 C e1 C e2 e0 C e1 C e2 C C en

!(��)

It remains to interchange e0 and e1. This then finishes the proof of Martin Hamm’sduality result in the special case q D n � 1. �

Remark. Obviously, the unimodular transformation from the “complementary” sit-uation to the original one in case .n; n � 1/ is given by

� D �x C y1 C C yn; �` D y`; ` D 1; : : : ; n:After a few elementary considerations this implies that the original convex cone in thiscase has a slightly more complicated description:

x �1;:::;nX�¤`

y�; y` � 0; ` D 1; : : : ; n:

For n D 3, the x-slices look as follows (x � 0; for x < 0 they are, of course, empty).


Figure 5

y1

y2

y3

x

x

x

In particular, for n D 2, we have (as already used in the preceding section)

x � y1 � 0 ; x � y2 � 0:In order to check the validity of this description, we compute the generators of the

inequalities and the relations between them. Clearly, the generators are e1; : : : ; en and

fj D e0 C1;:::;nX`¤j

e` ; j D 1; : : : ; n:

The relations e1 e2 en

f1 f2 fn

!are exactly those for the singularity An;1.

9 The case An;1

It is after the results of the last section not necessary to go also in the opposite directionbut quite amusing and helpful for understanding the general situation.

Let us first remark that we get another interesting system of generators for An;1 bytransforming the entries of the matrix (��) into the following one:

e0 C e1 e0 C e1 C e2 e0 C e1 C e2 C C ene0 e0 C e2 e0 C e2 C C en

!:


For n D 5, we have concretely .1I 1I 0I 0I 0I 0/ .1I 1I 1I 0I 0I 0/ .1I 1I 1I 1I 0I 0/.1I 0I 0I 0I 0I 0/ .1I 0I 1I 0I 0I 0/ .1I 0I 1I 1I 0I 0/

.1I 1I 1I 1I 1I 0/ .1I 1I 1I 1I 1I 1/

.1I 0I 1I 1I 1I 0/ .1I 0I 1I 1I 1I 1/

!:

Defining inequalities for the corresponding cone are, as one can check:

0 � y1 � x ; 0 � yn � yn�1 � � y2 � x :Therefore, the dual cone is generated by the vectors

e0 � e1; e1; e0 � e2; e2 � e3; : : : ; en�1 � en; en :The format 0@ e0 � e1 en

e2 � e3 ; e3 � e4; : : : ; en�1 � ene0 � e2 e1

1Ashows that the dual cone in fact belongs to the case An;n�1.

In particular, this new cone for An;1 is contained in the set 0 � y` � x; with` D 1; : : : ; n. Recall that a cone for An;n�1 is equal to y` � xC D max.x; 0/, with` D 1; : : : ; n:

So, this again is a manifestation of the interplay between duality and complemen-tarity.

Figure 6

y3

y1

y2

x

x

x


Another way from An;1 to An;n�1 starts with the format e0 C e1 e0 C e2 : : : e0 C ene1 e2 : : : en

!which one obtains from former formats of An;1 by unimodular transformations (andpossibly interchanging e0 and e1). It is not difficult at all to show that the cone generatedby these vectors can be described as

fxe0 C y1e1 C C ynen W yj � 0; j D 1; : : : ; n; y1 C C yn � x � 0g:Thus the dual cone is generated by e0; e1; : : : ; en;�e0 C e1 C C en, and this isobviously isomorphic to the cone of An;n�1.

The search for a kind of natural format which explains the duality in questionin a more systematic way leads in case An;1 to the following (the reader may checkhim/herself the correctness of transformations):

e1 e2 : : : en

e0 e0 � e1 C e2 : : : e0 � e1 C en

!:

10 The general case

In this last section we sketch by an example a way how one may understand Hamm’sduality result quite simply by introducing the right formats (which, on the other hand,destroy the motives how they were originally introduced).

We look at the case .n; q/ with the a-sequence .4; 2; 3/ such that by the author’sdot diagram

the corresponding b-sequence is .2; 2; 4; 2/.According to the a-sequence we start with a quasi-matrix with fixed entries0BB@

e1 e4 e5 e7

e2 e3 e6

e0 � � �

1CCA;where the crosses have to be filled in the correct way. In this case, they have to besuccessively from left to right

e0 � e1 C e2 C e3 C e4; e0 � e1 C e2 C e3 C e5; e0 � e1 C e2 C e3 C e6 C e7:


These are, of course, again generators of the case we discuss, and therefore the cor-responding cone in the 8-dimensional space generated by e0; : : : ; e7 is defined by allelements

7XjD0

j ej ;

where˛0D j C k C l Cm; ˛1D a � k � l �m;˛2D b C k C l Cm; ˛3D c C k C l Cm;˛4D d C k; ˛5Df C l;˛6Dg Cm; ˛7DhCm;

when the eleven non negative coefficients a; b; c; d; f; g; h; j; k; `;m are associatedto the generators of the cone via the matrix0@ a d f h

b c g

j k ` m

1A:From these inequalities, one deduces for the ˛ that j � 0 for j ¤ 1 and that

˛0 C ˛1 � 0 ; ˛1 C ˛2 � 0 ; ˛1 C ˛3 � 0and

˛1 C ˛4 C ˛5 C ˛6 � 0 ; ˛1 C ˛4 C ˛5 C ˛7 � 0:So, we have twelve inequalities between the ˛, and one can easily check that theyform a minimal set of generators. Therefore, the dual cone will be generated by thevectors

e0; e2; : : : ; e7;

e0 C e1; e1 C e2; e1 C e3;e1 C e4 C e5 C e6; e1 C e4 C e5 C e7:

After replacing e1 by e1 � e0 and then interchanging e0 and e1, we are left with thenew generators

e0; e1; : : : ; e7;

e0 � e1 C e2; e0 � e1 C e3;e0 � e1 C e4 C e5 C e6; e0 � e1 C e4 C e5 C e7;

which fit exactly into the corresponding format for the b-sequence .2; 2; 4; 2/:0@ e1 e2 e3 e6 e7e4 e5

e0 � � � �

1A:


References

[1] S. Brohme, Monodromieüberlagerung der versellen Deformation zyklischer Quo-tientensingularitäten, Dissertation, Hamburg 2002. URN: urn:nbn:de:gbv:18-6733http://ediss.sub.uni-hamburg.de/volltexte/2002/673/ 168

[2] J. Christophersen, On the components and discriminant of the versal base space of cyclicquotient singularities, in Singularity Theory and its applications. Part I. Geometric aspectsof singularities. Papers from the symposium held at the University of Warwick, Coventry,1988–1989, ed. by D. Mond and J. Montaldi, Lecture Notes in Mathematics 1462. Springer,Berlin 1991, 81–92. 168

[3] M. Hamm, Die verselle Deformation zyklischer Quotientensingularitäten: Gleichungenund torische Struktur, Dissertation, Hamburg 2008. URN: urn:nbn:de:gbv:18-37828http://ediss.sub.uni-hamburg.de/volltexte/2008/3782/ 161, 168

[4] T. Oda, Convex Bodies and Algebraic Geometry. An introduction to the theory of toricvarieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Folge 15, Springer, Berlinetc. 1985. 162, 167

[5] P. Popescu-Pampu, The geometry of continued fractions and the topology of surfacesingularities, in Singularities in geometry and topology 2004. Proceedings of the 3 rd

Franco–Japanese Conference held at Hokkaido University, Sapporo, September 13–18,2004. ed. by J.-P. Brasselet and T. Suwa, Advanced Studies in Pure Mathematics 46,Mathematical Society of Japan, Tokyo 2007, 119–195. 165

[6] O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischenGruppen), Math. Ann. 209 (1974), 211–248. 166, 167, 170

[7] J. Stevens, On the versal deformation of cyclic quotient singularities, in Singularity theoryand its applications. Part I. Geometric aspects of singularities. Papers from the symposiumheld at the University of Warwick, Coventry, 1988–1989, ed. by D. Mond and J. Montaldi.Lecture Notes in Mathematics 1462, Springer, Berlin 1991, 302–319.

[8] J. Stevens, The versal deformation of cyclic quotient singularities, preprint 2009.arXiv:0906.1430 161

http://ediss.sub.uni-hamburg.de/volltexte/2002/673/

http://ediss.sub.uni-hamburg.de/volltexte/2008/3782/


Nearby cycles and characteristic classesof singular spaces

Jörg Schürmann�

Mathematisches Institut, Universität MünsterEinsteinstr. 62, 48149 Münster, Germany


Abstract. In this paper we give an introduction to our recent work on characteristic classes ofcomplex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach,and Kagoshima. We explain the relation between nearby cycles for constructible functions orsheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodgemodules, and the specialization of characteristic classes of singular spaces like the Chern-,Todd-, Hirzebruch-, and motivic Chern-classes. As an application we get a description of thedifferences between the corresponding virtual and functorial characteristic classes of complexhypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.

1 Introduction

A natural problem in complex geometry is the relation between invariants of a singu-lar complex hypersurface X (like Euler characteristic and Hodge numbers) and thegeometry of the singularities of the hypersurface (like the local Milnor fibrations). Forthe Euler characteristic this is for example a special case of the difference betweenthe Fulton and MacPherson Chern classes of X, whose differences are the now wellstudied Milnor classes ofX (see [1], [6], [7], [8], [28], [30], [35], [36], and [46]). Theirdegrees are related to Donaldson–Thomas invariants of the singular locus (see [3]).

A very powerful approach to this type of questions is by the theory of the nearbyand vanishing cycle functors. For example a classical result of Verdier [45] says thatthe MacPherson Chern class transformation [26] and [23] commutes with specializa-tion, which for constructible functions means the corresponding nearby cycles. Herewe explain the corresponding result for our motivic Chern and Hirzebruch class trans-formations as introduced in our joint work with J.-P. Brasselet and S. Yokura [5],i.e. they also commute with specialization defined in terms of nearby cycles. Hereone can work either in the motivic context with relative Grothendieck group of vari-eties [4] and [20], or in the Hodge context with Grothendieck groups of M. Saito’smixed Hodge modules [31] and [32]. The key underlying specialization result [37] is

�Work supported by the SFB 878 “groups, geometry and actions”.


182 Jörg Schürmann

about the filtered de Rham complex of the underlying filtered D-module in terms ofthe Malgrange–Kashiwara V -filtration. But here we focus on the geometric motiva-tions and applications as given in our joint work with S. E. Cappell, L. Maxim andJ. L. Shaneson [13].

In this paper we work (for simplicity) in the complex algebraic context, since thisallows us to switch easily between an algebraic geometric language and an underlyingtopological picture. Many results are also true in the complex analytic or algebraiccontext over base field of characteristic zero. First we introduce the virtual character-istic classes and numbers of hypersurfaces and local complete intersections in smoothambient manifolds. Next we recall some of the theories of functorial characteristicclasses for singular spaces; see [26], [2], [10], [5], and [38]. Finally we explain therelation to nearby and vanishing cycles following our earlier results [35] and [36]about different Chern classes for singular spaces.

Acknowledgements. This paper is an extended version of some talks given at con-ferences in Strasbourg, Oberwolfach and Kagoshima. Here I would like to thank theorganizers for the invitation to these conferences. I also would like to thank SylvainCappell, Laurentiu Maxim and Shoji Yokura for the discussions on our joint workrelated to the subject of this paper.

2 Virtual classes of local complete intersections

Recall that we are working in the complex algebraic context. A characteristic class cl�of (complex algebraic) vector bundles over X is a map

cl� W Vect.X/ �! H�.X/˝Rfrom the set Vect.X/ of isomorphism classes of complex algebraic vector bundlesover X to some cohomology theory H�.X/˝ R with a coefficient ring R, which iscompatible with pullbacks. Here we use as a cohomology theory

H�.X/ D

8<ˆ:H 2�.X;Z/; the usual cohomology in even degrees,

CH�.X/; the operational Chow cohomology of [17],

K0.X/; the Grothendieck group of vector bundles.

We also assume that cl� is multiplicative, i.e.

cl�.V / D cl�.V 0/ [ cl�.V 00/

for any short exact sequence

0 �! V 0 �! V �! V 00 �! 0

Nearby cycles and characteristic classes of singular spaces 183

of vector bundles onX, with[given by the cup- or tensor-product. Such a characteristicclass cl� corresponds by the “splitting principle” to a unique formal power seriesf .z/ 2 RŒŒz�� with cl�.L/ D f .c1.L// for any line bundle L on X. Here c1.L/ 2H1.X/ is the nilpotent first Chern class of L, which in the case H�.X/ D K0.X/ isgiven by c1.L/ D 1 � ŒL_� 2 K0.X/ (with ./_ the dual bundle). Finally cl� shouldbe stable in the sense that f .0/ 2 R is a unit so that cl� induces a functorial grouphomomorphism

cl� W .K0.X/;˚/ �! .H�.X/˝R;[/:Let us now switch to smooth manifolds, which will be an important intermediate

step on the way to characteristic classes of singular spaces. For a complex algebraicmanifold M its tangent bundle TM is available and a characteristic class cl�.TM/

of the tangent bundle TM is called a characteristic cohomology class cl�.M/ of themanifold M . We also use the notation

cl�.M/ D cl�.TM/ \ ŒM � 2 H�.M/˝Rfor the corresponding characteristic homology class of the manifold M, with ŒM � 2H�.M/ the fundamental class (or the class of the structure sheaf) in

H�.M/ D

8<ˆ:HBM2� .M/; the Borel–Moore homology in even degrees,

CH�.M/; the Chow group,

G0.M/; the Grothendieck group of coherent sheaves.

IfM is moreover compact, i.e. the constant map k W M ! fptg is proper, one getsthe corresponding characteristic number

].M/ D k�.cl�.M// D deg.cl�.M// 2 R :

Example 2.1 (Hirzebruch 1954). The famous Hirzebruch �y-genus is the charac-teristic number, whose associated characteristic class can be given in two versions(see [21]).

(i) The cohomological version, with R D QŒy�, is given by the Hirzebruch classcl� D T �

y corresponding to the normalized power series

f .z/ D Qy.z/ D z.1C y/1 � e�z.1Cy/ � zy 2 QŒy�ŒŒz��:

(ii) The K-theoretical version, with R D ZŒy�, is given by the dual total Lambda-class cl� D ƒ_

y , with

ƒ_y ./ D ƒy../_/ D

Xi�0Œƒi ../_/� yi

corresponding to the unnormalized power series

f .z/ D 1C y � yz 2 ZŒy�ŒŒz��:


So the �y-genus of the compact complex algebraic manifold M is given by

�y.M/ DXp�0

�.M;ƒpT �M/ yp

DXp�0

�Xi�0.�1/idimCH

i .M;ƒpT �M/� yp ;

with T �M the algebraic cotangent bundle of M . The equality

�y.M/ D deg.T �y .TM/ \ ŒM �/ 2 QŒy� (gHRR)

is called the generalized Hirzebruch–Riemann–Roch theorem [21]. The correspondingpower series Qy.z/ (as above) specializes to

Qy.z/ D

8<ˆ:

1C z; for y D �1,

z

1 � e�z ; for y D 0,

z

tanh z; for y D 1.

Therefore the Hirzebruch class T �y .TM/ unifies the following important (total)

characteristic cohomology classes of TM :

T �y .TM/ D

8<ˆ:c�.TM/; the Chern class for y D �1,

td�.TM/; the Todd class for y D 0,

L�.TM/; the Thom–Hirzebruch L-class for y D 1.

(2.1)

The gRHH-theorem specializes to the calculation of the following important in-variants:

��1.M/ D e.M/ D deg.c�.TM/ \ ŒM �/; the Euler characteristic,

�0.M/ D �.M/ D deg.td�.TM/ \ ŒM �/; the arithmetic genus,

�1.M/ D sign.M/ D deg.L�.TM/ \ ŒM �/; the signature,

(2.2)

which are, respectively, the Poincaré–Hopf or Gauss–Bonnet theorem, the Hirzebruch–Riemann–Roch theorem and the Hirzebruch signature theorem.

If X is a singular complex algebraic variety, then the algebraic tangent bundle ofX doesn’t exist so that a characteristic (co)homology class of X can’t be defined asbefore. But if X can be realized as a local complete intersection inside a complexalgebraic manifold M, then a substitute for TX is available. Indeed this just meansthat the closed inclusion i W X !M is a regular embedding into the smooth algebraicmanifold M, so that the normal cone NXM ! X is an algebraic vector bundle overX (compare [17]). Then the virtual tangent bundle of X

TvirX D Œi�TM �NXM� 2 K0.X/;


is independent of the embedding in M (e.g., see Example 4.2.6 in [17]), so it is awell-defined element in the Grothendieck group of vector bundles on X . Of course

TvirX D ŒTX� 2 K0.X/in case X is a smooth algebraic submanifold.

If cl� W K0.X/! H�.X/˝R denotes a characteristic cohomology class as before,then one can associate to X an intrinsic homology class (i.e. independent of theembedding X ,!M ) defined as:

clvir� .X/ D cl�.TvirX/ \ ŒX� 2 H�.X/˝R :Here ŒX� 2 H�.X/ is again the fundamental class (or the class of the structure sheaf)of X in

H�.X/ D

8<ˆ:HBM2� .X/; the Borel–Moore homology in even degrees,

CH�.X/; the Chow group,

G0.X/; the Grothendieck group of coherent sheaves.

Here \ in the K-theoretical context comes from the tensor product with the coherentlocally free sheaf of sections of the vector bundle. Moreover, for the class cl� D ƒ_

y

one has to take R D ZŒy; .1C y/�1� to make it a stable characteristic class definedon K0.X/.

Let i W X ! M be a regular embedding of (locally constant) codimension rbetween possible singular complex algebraic varieties. Using the famous deformationto the normal cone, one gets functorial Gysin homomorphisms (compare [17], [44],and [45])

i Š W H�.M/ �! H��r.X/

and

i Š W G0.M/ �! G0.X/:

Note that i is of finite tor-dimension, so that the last i Š can also be described as

i Š D Li� W G0.M/ ' K0.Dbcoh.M// �! K0.D

bcoh.X// ' G0.X/

coming from the derived pullback Li� between the bounded derived categories withcoherent cohomology sheaves. IfM is also smooth, then one gets easily the followingimportant relation between the virtual characteristic classes clvir� .X/ of X and theGysin homomorphisms:

i Š.cl�.M// D i Š.cl�.TM/ \ ŒM �/ D cl�.NXM/ \ clvir� .X/: (2.3)

From now on we assume that

X D ff D 0g D ffi D 0 j i D 1; : : : ; ng


is a global complete intersection in the complex algebraic manifold M coming froma cartesian diagram

ff D 0g Xi ��

f0

��

M

fD.f1;:::;fn/

��f0g

i0

�� Cn:

Then NXM ' f �.Nf0gCn/ D X �Cn is a trivial vector bundle of rank n on Xso that

cl�.NXM/ D8<:1 for cl� D T �

y ; c�; td� or L�,

.1C y/n for cl� D ƒ_y .

(2.4)

Assume now thatf is proper so thatX is compact. Since the Gysin homomorphismsi Š commute with proper pushdown (compare [17], [44], and [45]), one gets by theprojection formula

]vir.X/ D f0�.clvir� .X// D f0�.cl�.NXM/�1 \ i Šcl�.M//

D cl�.Nf0gCn/�1\ i Š0.f�cl�.M//:

Taking a (small) regular value 0 ¤ t 2 Cn, in the same way from the cartesian diagram

ff D 0g Xi ��

f0

��

M

f

��

Xi 0��

ft

��

ff D tg

f0gi0

�� Cn ftgit

��

for the “nearby” smooth submanifold Xt D ff D tg, one gets the equality

].Xt / D ft�.cl�.Xt // D cl�.NftgCn/�1\ i Št .f�cl�.M//:

Note that the set of critical values of f is a proper algebraic subset of Cn, as can beseen by “generic smoothness” or from an adapted stratification of the proper algebraicmap f. Now Nf0gCn ' Cn ' NftgCn and the smooth pullback �� for the (vectorbundle) projection � W Cn ! fptg is an isomorphism

�� W R D H�.fptg/˝R ' H�Cn.Cn/˝Rwith inverse i Š0 and i Št (see [17], [44], and [45]), so that the “virtual characteristicnumber”

]vir.X/ D f0�.clvir� .X// D ].Xt / 2 Ris the corresponding characteristic number of a “nearby” smooth fiber Xt .


3 Functorial characteristic classes of singular spaces

For a more general singular complex algebraic variety X its “virtual tangent bundle”is not available any longer, so characteristic classes for singular varieties have tobe defined in a different way. For an introduction to this subject compare with oursurvey paper [39] (and see also [38] and [47]). The theory of characteristic classesof vector bundles is a natural transformation of contravariant functorial theories. Thisnaturality is an important guide for developing various theories of characteristic classesfor singular varieties. Almost all known characteristic classes for singular spaces areformulated as natural transformations

cl� W A.X/ �! H�.X/˝Rof covariant functorial theories. Here A is a suitable theory (depending on the choiceof cl�), which is covariant functorial for proper algebraic morphisms.

There is always a distinguished element IX 2 A.X/ such that the correspondingcharacteristic class of the singular space X is defined as

cl�.X/ D cl�.IX /:

Finally one has the normalization

cl�.IM / D cl�.TM/ \ ŒM � 2 H�.M/˝RforM a smooth manifold, with cl�.TM/ the corresponding characteristic cohomologyclass ofM . This justifies the notation cl� for this homology class transformation, whichshould be seen as a homology class version of the following characteristic number ofthe singular space X :

].X/ D cl�.k�IX / D deg.cl�.IX // 2 H�.fptg/˝R ' R;with k W X ! fptg a constant map. Note that the normalization implies that for Msmooth:

].M/ D deg.cl�.M// D deg.cl�.TM/ \ ŒM �/

so that this is consistent with the notion of characteristic number of the smooth manifoldM as used before.

But only few characteristic numbers and classes have been extended in this way tosingular spaces. For example the three characteristic numbers (2.2) and classes (2.1)have been generalized to a singular complex algebraic varietyX in the following way(where the characteristic numbers are only defined for X compact):

e.X/ D deg.c�.X//; with c� W F.X/! H�.X/ (y D �1)

the Chern class transformation of MacPherson [26] and [23] from the abelian groupF.X/ of complex algebraically constructible functions to homology, where one canuse the Chow group CH�./ or the Borel–Moore homology groupHBM

2� .;Z/ (in evendegrees). Here e.X/ is the (topological) Euler characteristic of X, and the distin-


guished element IX D 1X 2 F.X/ is simply given by the characteristic function ofX.Then c�.X/ D c�.1X / agrees by [9] via “Alexander duality” for compact X embed-dable into a complex manifold with the Schwartz class of X as introduced before byM.-H. Schwartz [40].

The Todd transformation in the singular Riemann–Roch theorem of Baum, Fulton,and MacPherson [2] (for Borel–Moore homology) or Fulton [17] (for Chow groups)is

�.X/ D deg.td�.X//; with td� W G0.X/! H�.X/˝Q: (y D 0)

HereG0.X/ is the Grothendieck group of coherent sheaves, with �.X/ the arithmeticgenus (or holomorphic Euler characteristic) ofX . Then td�.X/ D td�.ŒOX �/, with thedistinguished element IX D ŒOX � the class of the structure sheaf.

Finally for compact X one also has

sign.X/ D deg.L�.X//; with L� W �.X/! H2�.X;Q/ (y D 1)

the homology L-class transformation of Cappell and Shaneson [10] as formulatedin [5]. Here �.X/ is the abelian group of cobordism classes of selfdual constructiblecomplexes. Then L�.X/ D L�.ŒICX �/ is the homology L-class of Goresky andMacPherson [19], with the distinguished element IX D ŒICX � the class of their inter-section cohomology complex. So sign.X/ is the intersection cohomology signatureof X. For a rational PL-homology manifold X, these L-classes are due to Thom [43].

So all these theories have the same formalism, but they are defined on completelydifferent theories. Nevertheless, it is natural to ask for another theory of characteristichomology classes of singular complex algebraic varieties, which unifies the abovecharacteristic homology class transformations. Of course in the smooth case, this isdone by the Hirzebruch class T �

y .TM/\ ŒM � of the tangent bundle. An answer to thisquestion was given in [5] (together with some improvements in [38]). Using Saito’sdeep theory of algebraic mixed Hodge modules [31] and [32], we introduced in [5]the motivic Chern class transformations as natural transformations (commuting withproper push down) fitting into a commutative diagram:

G0.X/Œy� �� G0.X/Œy; y�1� G0.X/Œy; y�1�

K0.var=X/

mCy

��

�� M.var=X/

mCy

��

�Hdg

�� K0.MHM.X//:

MHCy

��

HereK0.MHM.X// is the Grothendieck group of algebraic mixed Hodge moduleson X , and K0.var=X/ (resp. M.var=X/ D K0.var=X/ŒL�1�) is the (localization ofthe) relative Grothendieck group of complex algebraic varieties over X (with respectto the class of the affine line L, compare e.g. [4] and [20]). The distinguished elementis given by the constant Hodge module (complex), resp. by the class of the identity


arrow,

IX D ŒQHX � 2 K0.MHM(X)/ resp. IX D ŒidX � 2 K0.var=X/;

and the canonical “Hodge realization” homomorphism �Hdg is given by

�Hdg W K0.var=X/ �! K0.MHM.X//

Œf W Y ! X� 7�! ŒfŠQHY �:

The motivic Chern class transformations mCy , MHCy capture information aboutthe filtered de Rham complex of the filtered D-module underlying a mixed Hodgemodule. The corresponding characteristic class of the space X,

mCy.X/ D MHCy.X/ 2 G0.X/Œy�;can also be defined with the help of the (filtered) Du Bois complex of X [16], andsatisfies for M smooth the normalization condition

mCy.M/ D MHCy.M/ D ƒ_y .TM/ \ ŒM � 2 G0.M/Œy�: (3.1)

The motivic Chern class transformations are a K-theoretical refinement of theHirzebruch class transformations Ty�, MHTy�, which can be defined by the (functo-rial) commutative diagram:

G0.X/Œy� �� G0.X/Œy; y�1� G0.X/Œy; y�1�

K0.var=X/ ��

mCy

��

M.var=X/

mCy

��

�Hdg

�� K0.MHM.X//;

mCy

��

with td� W G0.X/! H�.X/˝Q the Todd class transformation of Baum, Fulton, andMacPherson [2] and Fulton [17] and .1Cy/�� the renormalization given in degree iby the multiplication

.1Cy/�i W Hi .�/˝QŒy; y�1� �! Hi .�/˝QŒy; y�1; .1Cy/�1� D H�.�/˝Qloc :

This renormalization is needed to get for M smooth the normalization condition

Ty�.M/ D MHTy�.M/ D T �y .TM/ \ ŒM � 2 H�.M/˝QŒy�:

It is the Hirzebruch class transformationTy�, which unifies the (rationalized) Chernclass transformation c� ˝Q, Todd class transformation td� and L-class transforma-tion L� (compare [5]). The corresponding characteristic number

�y.X/ D deg.MHTy�.X// 2 ZŒy�

for a singular (compact) algebraic variety X captures information about the Hodgefiltration of Deligne’s ([14] and [15]) mixed Hodge structure on the rational cohomol-ogy (with compact support)H�

.c/.X IQ/ ofX. In fact, by M. Saito’s work [32] one has


an equivalence

MHM.fptg/ ' mHsp

between mixed Hodge modules on a point space, and rational (graded) polarizablemixed Hodge structures. Moreover, the corresponding mixed Hodge structure on ra-tional cohomology with compact support

H�c .X IQ/ D H�.fptgI kŠQH

X /

(withk W X ! fptg a constant map) agrees with Deligne’s one by another deep theoremof M. Saito [33]. Therefore the transformations MHCy and MHTy� can be seen as acharacteristic class version of the ring homomorphism

�y W K0.mHsp/! ZŒy; y�1�

defined on the Grothendieck group of (graded) polarizable mixed Hodge structures by

�y.ŒH�/ DXp

dim GrpF .H ˝C/ .�y/p ;

for F the Hodge filtration of H 2 mHsp . Note that �y.ŒL�/ D �y.These characteristic class transformations are motivic refinements of the (rational-

ization of the) Chern class transformation c�˝Q of MacPherson. MHTy� factorizesby [38] as

MHTy� W K0.MHM.X// �! H�.X/˝QŒy; y�1� � H�.X/˝Qloc ;

fitting into a (functorial) commutative diagram

F.X/

c�˝Q

��

K0.Dbc .X//

�stalk��

c�˝Q

��

K0.MHM.X//rat��

MHTy�

��H�.X/˝Q H�.X/˝Q H�.X/˝QŒy; y�1�:

yD�1��

Here Dbc .X/ is the derived category of algebraically constructible sheaves on X

(viewed as a complex analytic space), with rat associating to a (complex of) mixedHodge module(s) the underlying perverse (constructible) sheaf complex, and �stalk isgiven by the Euler characteristic of the stalks.

Let us go back to the case when X is a local complete intersection in some ambi-ent smooth algebraic manifold. Then it is natural to compare cl�.X/ for a functorialhomology characteristic class theory cl� as above with the corresponding virtual char-acteristic class clvir� .X/. If M is smooth, then clearly we have that

clvir� .M/ D cl�.TM/ \ ŒM � D cl�.M/:


However, if X is singular, the difference between the homology classes clvir� .X/ andcl�.X/ depends in general on the singularities of X. This motivates the followingproblem.

Problem 3.1. Describe the difference clvir� .X/ � cl�.X/ in terms of the geometry ofsingular locus of X.

The above problem is usually studied in order to understand the complicated homol-ogy classes cl�.X/ in terms of the simpler virtual classes clvir� .X/, with the differenceterms measuring the complexity of the singularities of X.

This question was first studied for the Todd class transformation td�, where thisdifference term is vanishing. More precisely one has the following result.

Theorem 3.2 (Verdier 1976). Assume that i W X ! Y is regular embedding of (lo-cally constant) codimension n. Then the Todd class transformation td� commutes withspecialization (see [44]), i.e.

i Š ı td� D td� ı i Š W G0.Y / �! H��n.X/:

Note that Y need not be smooth.

Corollary 3.3. Assume that X can be realized as a local complete intersection insome ambient smooth algebraic manifold. Then tdvir� .X/ D td�.X/. Especially, if Xis a global complete intersection given as the zero-fiber X D ff D 0g of a propermorphism f W M ! Cn on the algebraic manifold M, then the arithmetic genus

�.X/ D �vir.X/ D �.Xt /of X agrees with that of a nearby smooth fiber Xt for 0 ¤ t small and generic.

The next case studied in the literature is the L-class transformation L� for X acompact global complex hypersurface.

Theorem 3.4 (Cappell-Shaneson ’91). Assume X is a global compact hypersurfaceX D ff D 0g for a proper complex algebraic function f W M ! C on a complexalgebraic manifoldM. Fix a complex Whitney stratification of X and let V0 be the setof strata V with dimV < dimX. Assume, for simplicity, that all V 2 V0 are simply-connected (otherwise one has to use suitable twisted L-classes, see [11] and [12]).Then

Lvir� .X/ � L�.X/ DXV 2V0

�.lk.V // L�. xV /; (3.2)

where �.lk.V // 2 Z is a certain signature invariant associated to the link pair of thestratum V in .M;X/.

This result is in fact of topological nature, and holds more generally for a suitablecompact stratified pseudomanifold X, which is PL-embedded into a manifold M inreal codimension two (see [11] and [12] for details).


If cl� D c� is the Chern class transformation, the problem amounts to comparingthe Fulton–Johnson class cFJ� .X/ D cvir� .X/ (see, e.g. [17] and [18]) with the homol-ogy Chern class c�.X/ of MacPherson. The difference between these two classes ismeasured by the so-called Milnor class M�.X/ of X, which is studied in many refer-ences like [1], [6], [7], [8], [28], [30], [35], [36], and [46]. This is a homology classsupported on the singular locus of X, and for a global hypersurface it was computedin [30] (see also [35], [36], [46], and [28]) as a weighted sum in the Chern–MacPhersonclasses of closures of singular strata of X, the weights depending only on the normalinformation to the strata. For example, ifX has only isolated singularities, the Milnorclass equals (up to a sign) the sum of the local Milnor numbers attached to the singularpoints. In the following section we explain our approach [35] and [36] through nearbyand vanishing cycles (for constructible functions), which recently was adapted to themotivic Hirzebruch and Chern class transformations [13] and [37].

4 Nearby and vanishing cycles

Let us start to explain some basic constructions for constructible functions in thecomplex algebraic context (compare [34], [35], and [36]). Here we work in the classicaltopology on the complex analytic space X associated to a separated scheme of finitetype over Spec.C/.

Definition 4.1. A function ˛ W X ! Z is called (algebraically) constructible if itsatisfies one of the following two equivalent properties:

(i) ˛ is a finite sum ˛ D Pj nj 1Zj

, with nj 2 Z and 1Zjthe characteristic

function of the closed complex algebraic subset Zj of X;

(ii) ˛ is (locally) constant on the strata of a complex algebraic Whitney b-regularstratification of X.

This notion is closely related to the much more sophisticated notion of (alge-braically) constructible (complexes of) sheaves on X. A sheaf F of (rational) vector-spaces on X with finite dimensional stalks is (algebraically) constructible if thereexists a complex algebraic Whitney b-regular stratification as above such that therestriction of F to all strata is locally constant. Similarly, a bounded complex ofsheaves is constructible, if all it cohomology sheaves have this property, and we denoteby Db

c .X/ the corresponding derived category of bounded constructible complexeson X. The Grothendieck group of the triangulated category Db

c .X/ is denoted byK0.D

bc .X//.

Since we assume that all stalks of a constructible complex are finite dimensional,by taking stalkwise the Euler characteristic we get a natural group homomorphism

�stalk W K0.Dbc .X// �! F.X/;

ŒF � 7�! .x 7! �.Fx//:


Here F.X/ is the group of (algebraically) constructible functions on X . It is easy toshow that natural transformation �stalk is surjective.

As is well known (and explained in detail in [34]), all the usual functors in sheaf the-ory, which respect the corresponding category of constructible complexes of sheaves,induce by the epimorphism �stalk well-defined group homomorphisms on the level ofconstructible functions. We just recall these, which are important for later applicationsor definitions.

Definition 4.2. Let f W X ! Y be an algebraic map of complex spaces associated toseparated schemes of finite type over Spec.C/. Then one has the following transfor-mations.

(1) Pullback. f � W F.Y / ! F.X/I ˛ 7! ˛ ı f , which corresponds to the usualpullback of sheaves

f � W Dbc .Y / �! Db

c .X/:

(2) Exterior product. ˛ � ˇ 2 F.X � Y / for ˛ 2 F.X/ and ˇ 2 F.Y /, given by˛ � ˇ..x; y// D ˛.x/ ˇ.y/. This corresponds on the sheaf level to the exteriorproduct

�L W Dbc .X/ �Db

c .Y /! Dbc .X � Y /:

(3) Euler characteristic. SupposeX is compact and Y D fptg is a point. Then one has� W F.X/! Z, corresponding to

R�.X; / D k� W Dbc .X/ �! Db

c .fptg/on the level of constructible complexes of sheaves, with k W X ! fptg the constantproper map. By linearity it is characterized by the convention that for a compactcomplex algebraic subspace Z � X

�.1Z/ D �.H�.ZIQ//is just the usual Euler characteristic of Z.

(4) Proper pushdown. Suppose f is proper. Then one has f� D fŠ W F.X/! F.Y /,corresponding to

Rf� D RfŠ W Dbc .X/! Db

c .Y /

on the level of constructible complexes of sheaves. Explicitly it is given by

f�.˛/.y/ D �.˛jffDyg/;

and in this form it goes back to the paper [26] of MacPherson.

(5) Nearby cycles. Assume Y D C and let X0 D ff D 0g be the zero fiber. Then onehas f W F.X/! F.X0/, corresponding to Deligne’s nearby cycle functor

f W Dbc .X/ �! Db

c .X0/:


This was first introduced in [45] by using resolution of singularities (compare with [34]for another approach using stratification theory). By linearity, f is uniquely definedby the convention that for a closed complex algebraic subspace Z � X the value

f .1Z/.x/ D �.H�.Ff jZ ;xIQ//is just the Euler characteristic of a local Milnor fiber Ff jZ ;x of f jZ at x. Here thislocal Milnor fiber at x is given by

Ff jZ ;x D Z \ B�.x/ \ ff D yg ;with 0 < jyj � � � 1 and B�.x/ an open (or closed) ball of radius � around x (insome local coordinates). Here we use the theory of a Milnor fibration of a function fon the singular space Z (compare [25] and [34]).

(6) Vanishing cycles. Assume Y D C and let i W X0 D ff D 0g ,! X be the inclusionof the zero-fiber. Then one has �f W F.X/ ! F.X0/; �f D f � i�, correspondingto Deligne’s vanishing cycle functor

�f W Dbc .X/ �! Db

c .X0/ :

By linearity, �f is uniquely defined by the convention that for a closed complexalgebraic subspace Z � X the value

�f .1Z/.x/ D �.H�.Ff jZ ;xIQ// � 1 D �. zH�.Ff jZ ;xIQ//is just the reduced Euler-characteristic of a local Milnor fiber Ff jZ ;x of f jZ at x.

Remark 4.3. Let the global hypersurface X D ff D 0g be the zero-fiber of analgebraic function f W M ! C on the complex algebraic manifold M. Then thesupport of �f .1M / is contained in the singular locus Xsing of X :

supp.�f .1M // � Xsing:

And �f .1M /jXsing is (up to a sign) the Behrend function of Xsing (see [3]), an intrin-sic constructible function of the singular locus appearing in relation to Donaldson–Thomas invariants.

A beautiful result of Verdier [45] and [24] shows that for a global hypersurfaceMacPherson’s Chern class transformation c� commutes with specialization, if oneuses the nearby cycle functor f on the level of constructible functions (as opposedto the pullback functor i� for the corresponding inclusion i W X D ff D 0g ! Y ).

Theorem 4.4 (Verdier 1981). Assume that X D ff D 0g is a global hypersurface(of codimension one) in Y given by the zero-fiber of a complex algebraic functionf W Y ! C. Then the MacPherson Chern class transformation c� commutes withspecialization (see [45] and [23]), i.e.

i Š ı c� D c� ı f W F.Y / �! H��1.X/

for the closed inclusion i W X D ff D 0g ! Y.


Note that Y need not be smooth. As an immediate application one gets by (2.3)and (2.4) the following important result (compare [35] and [36]).

Corollary 4.5. Assume that X D ff D 0g is a global hypersurface (of codimensionone) in some ambient smooth algebraic manifold M, given by the zero-fiber of acomplex algebraic function f W M ! C. Then

cvir� .X/ � c�.X/ D c�. f .1M // � c�.1X / D c�.�f .1M // 2 H�.Xsing/;

since supp.�f .1M // � Xsing. Here we also use the naturality of c� for the closedinclusion Xsing ! X to view this difference term as a localized class inH�.Xsing/. Inparticular we have the following properties.

(i) cviri .X/ D ci .X/ 2 Hi .X/ for all i > dimXsing.

(ii) If X has only isolated singularities (i.e. dimXsing D 0), then

cvir� .X/ � c�.X/ DXx2Xsing

�. zH�.FxIQ//;

whereFx is the local Milnor fiber of the isolated hypersurface singularity .X; x/.

(iii) If f W M ! C is proper, then

deg.c�.�f .1M /// D deg.cvir� .X/ � c�.X// D �.Xt / � �.X/is the difference between the Euler characteristic of a global nearby smooth fiberXt D ff D tg (for 0 ¤ jt j small enough) and of the special fiberX D ff D 0g.

For a general local complete intersection X in some ambient smooth algebraicmanifold (e.g. a local hypersurface of codimension one), one doesn’t have globalequations so that the theory of nearby and vanishing cycles can’t be applied directly.Instead one has to combine them with the deformation to the normal cone leadingto Verdier’s theory of specialization functors (compare [35] and [36]). But even ifX D ff D 0g is a global complete intersection inside the ambient smooth algebraicmanifold M, given by the zero-fiber of a complex algebraic map f W M ! Cn, onedoesn’t have a theory of nearby and vanishing cycles, because a local theory of Milnorfibers for f is missing (if n > 1). But if one fixes an ordering of the components of f(or of the coordinates on Cn), then a corresponding local Milnor fibration exists forany ordered tuple

.f / D .f1; : : : ; fn/ W Z �! Cn

of complex algebraic functions on the singular algebraic variety Z (as observedin [29]).

Definition 4.6 (Nearby and vanishing cycles for an ordered tuple). Let

.f / D .f1; : : : ; fn/ W Y �! Cn


be an ordered n-tuple of complex algebraic functions on Y , with

X D ff D 0g D ff1 D 0; : : : ; fn D 0gthe zero-fiber of .f /. Then nearby cycles of

.f / D .f1; : : : ; fn/are defined by iteration as

.f / D f1ı ı fn

W F.Y /! F.X/:

By linearity, .f / is uniquely defined by the convention that for a closed complexalgebraic subspace Z � Y the value

.f /.1Z/.x/ D �.H�.F.f /jZ ;xIQ//is just the Euler-characteristic of a local Milnor fiber F.f /jZ ;x of .f /jZ at x. Herethis local Milnor fiber of .f / at x is given by

F.f /jZ ;x D Z \ B�.x/ \ ff1 D y1; : : : ; fn D yng;with 0 < jynj � � jy1j � � � 1 and B�.x/ an open (or closed) ball of radius �around x (in some local coordinates, compare [29]).

The corresponding vanishing cycles of .f / are defined by

�.f / D .f / � i� W F.Y / �! F.X/;

with i W X ! Y the closed inclusion. By linearity, �.f / is uniquely defined by theconvention that for a closed complex algebraic subspace Z � X the value

�.f /.1Z/.x/ D ��H�.F.f /jZ ;xIQ/

� � 1 D �. zH�.F.f /jZ ;xIQ//is just the reduced Euler-characteristic of a local Milnor fiber F.f /jZ ;x of .f /jZ at x.

Note that again supp.�.f /.1M // � Xsing in case the ambient space Y D M is asmooth algebraic manifold. Assume moreover that X is of codimension n so that theregular embedding i W X ! Y factorizes into n regular embeddings of codimensionone i D in ı ı i1:

X Dff1 D 0; : : : ; fn D 0g i1��! ff2 D 0; : : : ; fn D 0g i2��! : : :

: : :in�2��! ffn�1 D 0; fn D 0g in�1��! ffn D 0g in��! Y:

By the functoriality of the Gysin homomorphisms one gets

i Š D i Š1 ı ı i Šn W H�.Y / �! H��n.X/ :

Since in Verdier’s specialization theorem (4.4) the ambient space need not be smooth,we can apply it inductively to all embeddings ij (for j D n; : : : ; 1) above.


Corollary 4.7. Assume that X D ff D 0g D ff1 D 0; : : : ; fn D 0g is a globalcomplete intersection (of codimension n) in some ambient smooth algebraic mani-fold M, given by the zero-fiber of an ordered n-tuple of complex algebraic function.f / D .f1; : : : ; fn/ W M ! Cn. Then

cvir� .X/ � c�.X/ D c�.�.f /.1M // 2 H�.Xsing/;

since supp.�.f /.1M // � Xsing. Here we also use the naturality of c� for the closedinclusion Xsing ! X to view this difference term as a localized class in H�.Xsing/. Inparticular we have:

(i) cviri .X/ D ci .X/ 2 Hi .X/ for all i > dimXsing;

(ii) if X has only isolated singularities (i.e. dimXsing D 0), then

cvir� .X/ � c�.X/ DXx2Xsing

�. zH�.FxIQ//;

where Fx is the local Milnor fiber of the ordered n-tuple .f / at the isolatedsingularity x;

(iii) if .f / D .f1; : : : ; fn/ W M ! Cn is proper, then

deg.c�.�.f /.1M /// D deg.cvir� .X/ � c�.X// D �.Xt / � �.X/is the difference between the Euler characteristic of a global nearby smooth fiberXt D ff1 D t1; : : : ; fn D tng (for t D .t1; : : : ; tn/ with 0 < jtnj � � jt1jsmall enough) and of the special fiber X D ff D 0g.

As explained in Section 3, the motivic Hirzebruch and Chern class transformationsTy�, MHTy� andmCy , MHCy can be seen as “motivic or Hodge theoretical liftings”of the (rationalized) Chern class transformation c� under the comparison maps

K0.var=Y /�

Hdg��! K0.MHM.Y //rat��! K0.D

bc .Y //

�stalk��! F.Y /:

Here these Grothendieck groups have the same calculus as for constructible functionsin Definition 4.2 (1–4), respected by these comparison maps. So it is natural to tryto extend known results about MacPherson’s Chern class transformation c� to thesetransformations. In the “motivic” (resp. “ Hodge theoretical”) context this has beenworked out in [5] (resp. [38]) for

(i) the functorialty under push down for proper algebraic morphism,

(ii) the functorialty under exterior products,

(iii) the functorialty under smooth pullback given by a relatedVerdier–Riemann–Rochtheorem.

And recently we could also prove the “counterpart” of Verdier’s specializationtheorem (4.4). Let X D ff D 0g be a global hypersurface in Y given by the zero-fiber of a complex algebraic function f on Y :

X D ff D 0g i��! Yf��! C:


First note that one can use the nearby and vanishing cycle functors f and �f eitheron the motivic level of localized relative Grothendieck groups

M.var=�/ D K0.var=�/ŒL�1�

(see [4] and [20]), or on the Hodge-theoretical level of algebraic mixed Hodge modules(see [31] and [32]), “lifting” the corresponding functors on the level of algebraicallyconstructible sheaves (see [35]) and algebraically constructible functions as introducedbefore, so that the following diagram commutes:

M.var=Y / m

f; �m

f ��

�Hdg

��

M.var=X/

�Hdg

��K0.MHM.Y // 0H

f; �0H

f��

rat

��

K0.MHM.X//

rat

��K0.D

bc .Y // f ; �f

��

�stalk

��

K0.Dbc .X//

�stalk

��F.Y /

f ; �f

�� F.X/:

(4.1)

We also use the notation 0HfD H

fŒ1� and �0H

fD �H

fŒ1� for the shifted functors,

with Hf; �HfW MHM.Y / ! MHM.X/ and f Œ�1�; �f Œ�1� W Perv.Y / ! Perv.X/

preserving mixed Hodge modules and perverse sheaves, respectively. On the level ofGrothendieck groups one simply has �m

fD m

f� i� and �0H

fD 0H

f� i�.

Remark 4.8. The motivic nearby and vanishing cycles functors of [4], and [20] takevalues in a refined equivariant localized Grothendieck group M O�.var=X/ of equivari-ant algebraic varieties overX with a “good” action of the pro-finite group O� D lim�nof roots of unity. For mixed Hodge modules this corresponds to an action of the semi-simple part of the monodromy. But in the following applications we don’t need to takethis action into account. Also note that for the commutativity of diagram (4.1) one hasto use 0H

f, �0H

f(as opposed to H

f, �H

f).

Now we are ready to formulate the main new result from [37].

Theorem 4.9 (Schürmann 2009). Assume that X D ff D 0g is a global hyper-surface of codimension one given by the zero-fiber of a complex algebraic functionf W Y ! C. Then the motivic Hodge–Chern class transformation MHCy commutes


with specialization in the following sense:

.1C y/ MHCy. 0Hf .�/ / D i ŠMHCy.�/

as transformations K0.MHM.Y //! G0.X/Œy; y�1�.

Again the smoothness of Y is not needed. The appearance of the factor .1 C y/should not be a surprise, as it can already be seen in the case of a smooth hypersurfaceX inside a smooth ambient manifold Y,

.1C y/ MHCy.X/ D i ŠMHCy.Y /;

if one recalls (2.3), (2.4), and the normalization condition (3.1), with

QHX D i�QH

Y ' 0Hf .QH

Y /

in this special case. But the proof of this theorem given in [37] is far away from thegeometric applications described here. In fact it uses the algebraic theory of nearby andvanishing cycles in the context ofD-modules given by the V -filtration of Malgrange–Kashiwara, together with a specialization result about the filtered de Rham complexof the filtered D-module underlying a mixed Hodge module.

Using Verdier’s result that the Todd class transformation td� commutes with spe-cialization (see Theorem 3.2), one gets the following corollary (see [37]).

Corollary 4.10. Assume that X D ff D 0g is a global hypersurface of codimensionone given by the zero-fiber of a complex algebraic function f W Y ! C. Then themotivic Hirzebruch class transformation MHTy� commutes with specialization, thatis:

MHTy�. 0Hf .�// D i ŠMHTy�.�/ (4.2)

as transformations K0.MHM.Y //! H�.X/˝QŒy; y�1�.

Again the smoothness of Y is not needed here, but only the fact thatX D ff D 0gis a global hypersurface (of codimension one) is needed. Also the factor .1 C y/ inTheorem 4.9 canceled out by the renormalization factor .1 C y/�i on Hi .�/ usedin the definition of MHTy�, since the Gysin map i Š W H�.Y /! H��1.X/ shifts thisdegree by one.

By the definition of mf

in [4] and [20] one has that

mf .K0.var=Y // � im.K0.var=X/!M.X//;

so MHTy� ı mf maps K0.var=Y / into H�.X/ ˝ QŒy� � H�.X/ ˝ QŒy; y�1�.Together with [38], Proposition 5.2.1, one therefore gets the following commutative


diagram:

K0.var=Y /Ty �ı m

fDi ŠıTy � ��

�Hdg

��

H�.X/˝QŒy�

��K0.MHM.Y // MHTy �ı 0H

fDi ŠıMHTy �

��

�stalk

ı rat

��

H�.X/˝QŒy; y�1�

yD�1��

F.Y /c�ı f Di Šc�

�� H�.X/˝Q :

(4.3)

As before one gets the following result from Theorem 4.9 and Corollary 4.10together with (2.3) and (2.4).

Lemma 4.11. Assume that X D ff D 0g is a global hypersurface (of codimensionone) in some ambient smooth algebraic manifold M, given by the zero-fiber of acomplex algebraic function f W M ! C. Then

mC viry .X/ D mCy. mf .ŒidM �// D MHCy�. 0H

f .ŒQHM �//;

andT viry� .X/ D Ty�. mf .ŒidM �// D MHTy�. 0H

f .ŒQHM �//:

If i W X D ff D 0g !M is the closed inclusion, then one has i�.ŒidM �/ D ŒidX �and i�.ŒQH

M �/ D ŒQHX �. So by �m

fD m

f� i� and �0H

fD 0H

f� i� (on the level of

Grothendieck groups) one gets the following corollary (compare [13]).

Corollary 4.12. Assume thatX D ff D 0g is a global hypersurface (of codimensionone) in some ambient smooth algebraic manifold M, given by the zero-fiber of acomplex algebraic function f W M ! C. Then

mC viry .X/ �mCy.X/ D mCy.�mf .ŒidM �//

D MHCy�.�0Hf .ŒQH

M �// 2 G0.Xsing/Œy�;

and

T viry� .X/ � Ty�.X/ D Ty�.�mf .ŒidM �//

D MHTy�.�0Hf .ŒQH

M �// 2 H�.Xsing/˝QŒy�:(4.4)

Here we usesupp.�0H

f .QHM // � Xsing

and the naturality of our characteristic class transformations for the closed inclusionXsing ! X. In particular:


(1) T viry;i .X/ D Ty;i .X/ 2 Hi .X/˝QŒy� for all i > dimXsing;

(2) if X has only isolated singularities (i.e. dimXsing D 0), then

mC viry .X/ �mCy.X/ D

Xx2Xsing

�y. zH�.FxIQ//

D T viry� .X/ � Ty�.X/ ;

where Fx is the Milnor fiber of the isolated hypersurface singularity .X; x/;

(3) if f W M ! C is proper, then

deg.MHCy�.�0Hf .ŒQH

M �/// D �y.H�.Xt IQ// � �y.H�.X IQ//D deg.MHTy�.�0H

f .ŒQHM �///

is the difference between the �y-characteristics of a global nearby smooth fiberXt D ff D tg (for 0 ¤ jt j small enough) and of the special fiber X D ff D 0g.

Remark 4.13 (Hodge polynomials vs. Hodge spectrum). Let us explain the preciserelationship between the Hodge spectrum and the less-studied �y-polynomial of theMilnor fiber of a hypersurface singularity. Here we follow notations and sign con-ventions similar to those in [20]. Denote by mHsmon the abelian category of mixedHodge structures endowed with an automorphism of finite order, and by Kmon

0 .mHs/the corresponding Grothendieck ring. There is a natural linear map called the Hodgespectrum,

hsp W Kmon0 .mHs/ �! ZŒQ� '

[n�1

ZŒt1=n; t�1=n� ;

such thathsp.ŒH�/ D

X˛2Q\Œ0;1/

t˛�Xp2Z

dim.GrpFHC;˛/tp�: (4.5)

for any mixed Hodge structure H with an automorphism T of finite order, whereHC is the underlying complex vector space of H, HC;˛ is the eigenspace of T witheigenvalue exp.2�i˛/, and F is the Hodge filtration onHC. It is now easy to see thatthe �y-polynomial of H is obtained from hsp.ŒH�/ by equating to 1 the parametert corresponding to fractional powers ˛ 2 Q \ Œ0; 1/, and by setting the t of integerpowers be equal to �y.

As already explained before, Corollary 4.12 reduces for the value y D �1 ofthe parameter to the (rationalized version of) Corollary 4.5. Since the ambient spacein Theorem 4.9 and Corollary 4.10 need not be smooth, one can generalize in thesame way the Corollary 4.7 for a global complete intersection X D ff D 0g Dff1 D 0; : : : ; fn D 0g (of codimension n) in some ambient smooth algebraic mani-fold M, given by the zero-fiber of an ordered n-tuple of complex algebraic function


.f / D .f1; : : : ; fn/ W M ! Cn. Here we leave the details to the reader.

It is also very interesting to look at the other specializations of Corollary 4.12 fory D 0 and y D 1. Let us first consider the case when y D 0. Note that in generalT0�.X/ ¤ td�.X/ for a singular complex algebraic variety (see [5]). But if X hasonly Du Bois singularities (e.g. rational singularities, cf. [33]), then by [5] we haveT0�.X/ D td�.X/. So if a global hypersurface X D ff D 0g has only Du Boissingularities, then by Corollaries 3.3 and 4.12 we get

MHT0�.�0Hf .ŒQH

M �// D 0 2 H�.X/˝Q:

This vanishing (which is in fact a class version of Steenbrink’s cohomologicalinsignificance ofX [41]) imposes interesting geometric identities on the correspondingTodd-type invariants of the singular locus. For example, we obtain the following result.

Corollary 4.14. If the global hypersurfaceX has only isolated Du Bois singularities,then

dimCGr0FHn.FxIC/ D 0 (4.6)

for all x 2 Xsing, with n D dimX.

It should be pointed out that in this setting a result of Ishii [22] implies that (4.6) is infact equivalent to x 2 Xsing being an isolated Du Bois hypersurface singularity. Alsonote that in the arbitrary singularity case, the Milnor–Todd class

T0�.�mf .ŒidM �// D MHT0�.�0Hf .ŒQH

M �// 2 H�.Xsing/˝Q

carries interesting non-trivial information about the singularities of the hypersurfaceX .Finally, if y D 1, the formula (4.4) should be compared to the Cappell–Shaneson

topological result of (3.2). While it can be shown (compare with [27]) that the normalcontribution �.lk.V // in (3.2) for a singular stratum V 2 V0 is in fact the signature�.Fv/ (v 2 V ) of the Milnor fiber (as a manifold with boundary) of the singularityin a transversal slice to V in v, the precise relation between �.Fv/ and �

1.Fv/ is in

general very difficult to understand. For X a rational homology manifold, one wouldlike to have a “local Hodge index formula”

�.Fv/‹D �

1.Fv/;

which is presently not available. But if the hypersurfaceX is a rational homology man-ifold with only isolated singularities, then this expected equality follows from [42],Theorem 11. One therefore gets in this case (by a comparison of the different special-ization results for L� and T1�) the following conjectural interpretation of L-classesfrom [5] (see [13] for more details).

Theorem 4.15. Let X be a compact complex algebraic variety with only isolatedsingularities, which is moreover a rational homology manifold and can be realized as


a global hypersurface (of codimension one) in a complex algebraic manifold. Then

L�.X/ D T1�.X/ 2 H2�.X IQ/:

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Residues of singular holomorphic distributions

Tatsuo Suwa�

Department of Mathematics, Hokkaido UniversitySapporo 060-0810, Japan


Abstract. We present two types of residue theories for singular holomorphic distributions. Thefirst one is for certain Chern polynomials of the normal sheaf of a distribution and the residuesarise from the vanishing, by rank reason, of the relevant characteristic classes on the non-singularpart. The second one is for certain Atiyah polynomials of vector bundles admitting an action ofa distribution and the residues arise from the Bott type vanishing theorem on the non-singularpart.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

2. Holomorphic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

3. Local Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

4. Chern residues of singular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5. Atiyah classes and Cech–Dolbeault cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6. Atiyah residues of singular distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

1 Introduction

In this article we review some local invariants (residues) associated to singular holo-morphic distributions.

For singular foliations, i.e. involutive distributions, various residues are known.They usually arise from localization of some characteristic classes by the Bott van-ishing theorem, which depends on involutivity (see [22] for a systematic treatment).Note that they turned out be also closely related to local invariants of holomorphicself-maps (cf. [2] and [3]).

�Partially supported by a grant of JSPS.


208 Tatsuo Suwa

Some of those residues coincide with the ones that arise in other contexts where in-volutivity is not required.Also, if we consider other types of characteristic polynomials,we have a Bott type vanishing theorem without involutivity. Thus the correspondingresidues can be defined also for singular distributions.

Here we take up two kinds of such localizations. The first one is for the normal sheafof the distribution. It is rather primitive and comes from the vanishing by rank reason.In case we have involutivity, for a relevant characteristic polynomial, the resultingresidue coincides with the Baum–Bott residue in [6]. The second one comes fromthe fact that Atiyah forms are easier to vanish than the corresponding Chern forms sothat we have a Bott type vanishing theorem for certain Atiyah polynomials withoutinvolutivity, which leads to a localization theory of Atiyah classes of vector bundlesadmitting an action of a distribution.

In Section 2, we present some basics on singular holomorphic distributions andin Section 3, we review local Chern classes via the Chern–Weil theory adapted tothe Cech–de Rham cohomology. In this context, we also recall the Riemann–Rochtheorem for embeddings as given in [23]. Section 4 is an almost thorough revision of[21]. This is based on the observation that the localization considered there arises infact from a rather primitive fact, i.e. the Chern forms of degree greater than the rank ofthe vector bundle vanish, and involutivity has nothing to do with it. Thus we define thelocalization, by rank reason, of some characteristic classes and associated residues ofthe normal sheaf of the distribution. We show that for singular foliations they coincidewith the corresponding Baum–Bott residues, which partially answers the RationalityConjecture in [6]. We also express the residues in terms of the local Chern class ofsome sheaf supported on the singular set of the distribution. This allows us to applythe Riemann–Roch theorem for embeddings to compute the residues.

In Section 5, we review theAtiyah classes defined in the Cech–Dolbeault cohomol-ogy following [1] and [26]. These classes are originally defined in [4] using complexanalytic connections for holomorphic vector bundles. Here we use the construction in[1], which is more appropriate for localization purposes. In Section 6, we recall a Botttype vanishing theorem in [1] for some Atiyah forms, which leads to a localization the-ory for singular distributions. As an example we discuss Camacho–Sad type residuesfor the normal bundle of an invariant subvariety of a distribution. For foliations, theseare first introduced in [11] to prove the existence of separatrices for holomorphic vectorfields on the complex plane and then generalized by several authors, e.g. [17] and [18].We also discuss the localization problem on singular varieties and give an example.

I hoped to include more material on singular distribution as well as on singularcontact structures, but was not able to do so. Let me simply list [15] and [20] asliterature directly related to characteristic classes of singular holomorphic distributionsand thank J. Adachi for precious information on contact structures.

Residues of singular holomorphic distributions 209

2 Holomorphic distributions

2.1 Non-singular distributions

Let M be a complex manifold of dimension m and TM its holomorphic tangentbundle. A distribution on M is an assignment of a subspace of TxM to each point xin M; more precisely, it is defined in the following way.

Definition 2.1. A non-singular holomorphic tangential distribution on M of rank ris a holomorphic subbundle F of rank r of TM.

We callF also the tangent bundle of the distribution and the quotientNF D TM=Fthe normal bundle of the distribution. A distribution can be dually defined in terms ofcotangent bundle T �M.

Definition 2.2. A non-singular holomorphic cotangential distribution onM of coranks is a holomorphic subbundle G of rank s of T �M.

In the following we sometimes use the word distribution to refer to the abovenotions. Definitions 2.1 and 2.2 are equivalent by taking the annihilator of each other.Namely, if F is a distribution of rank r ,

G D F a D[x2Mf! 2 T �

xM j hv; !i D 0 for all v 2 TxM g

is a distribution of corank s D m � r , where h ; i denotes the paring of vector fieldsand differential forms. The above G coincides with the dual N �

F of NF and is calledthe conormal bundle of the distribution. Likewise, if G is a distribution of corank s,F D Ga is a distribution of rank r D m � s.

A foliation is a distribution F which is involutive, i.e. closed under the bracketoperation in TM.

LetF be a distribution of rank r onM and V a complex submanifold of dimensionn of M. Denoting by

� W V ,�!M

the inclusion, we identify T V with the image of its differential �� W T V �! TM jV .

Definition 2.3. We say that F is tangent to V, or leaves V invariant, if F jV � T V.

In this case F jV is a distribution of rank r on V. We state the above property interms of conormal bundle. Thus let

� W T �M jV �! T �V

be the dual of ��, which is a surjection. For a distribution G � T �M on M, we setG0 D �.GjV /. Note that it is the restriction of G to V as differential forms. Thefollowing proposition is not difficult to see.

210 Tatsuo Suwa

Proposition 2.4. Let G be a distribution of corank s on M and set F D Ga. ThenF leaves V invariant if and only if G0 is a subbundle of T �V of rank s � k, i.e. adistribution of corank s � k on V, k D m � n.

2.2 Singular distributions

Most of the material in this and subsequent subsections are similar to the ones forsingular foliations, for which we refer to [22], Chapter VI.

Let M be a complex manifold of dimension m. We denote by OM, ‚M and �M,respectively, its structure sheaf, tangent sheaf and cotangent sheaf. For simplicity, weassume that M is connected.

In general, for a coherent OM -module � , we set

Sing.�/ D f x 2M j �x is not OM;x-free gand call it the singular set of � . Locally Sing.�/ is given as follows. By definition,each point ofM has a neighborhood U such that there exists an exact sequence of theform

Or1U

'�! Or2U �! � jU �! 0:

If we represent the map ' by a matrix .'ij / of holomorphic functions on U,

Sing.�/ \ U D fx 2 U j rank.'ij .x// is not maximalg:Thus Sing.�/ is an analytic set inM. Away from Sing.�/, � is locally free. Its rank

is called the rank of � . If the maximal rank of .'ij / is r , the rank of � is r2 � r .

Definition 2.5. A singular holomorphic tangential distribution of rank r on M is acoherent sub-OM -module F of rank r of ‚M.

Note that, since ‚M is locally free, the coherence of F here simply means that itis locally finitely generated. We call F the tangent sheaf of the distribution and thequotient NF D ‚M=F the normal sheaf of the distribution so that we have the exactsequence

0 �! F �! ‚M �! NF �! 0: (2.1)

The singular set S.F / of a distribution F is defined to be the singular set of thecoherent sheaf NF :

S.F / D Sing.NF /:

Note that Sing.F / � S.F /. Away from S.F /, F defines a non-singular distribu-tion of rank r , i.e. there is a rank r subbundle F0 of TM0, M0 D M n S.F /, suchthat F jM0

D OM0.F0/.

We say that F is reduced, if for any open set U in M,

�.U;‚M / \ �.U n S.F /;F / D �.U;F /:


As in the case of singular foliations, it can be shown that, if F is reduced, thencodim S.F / � 2 and that, if F is locally free and if codim S.F / � 2, then F isreduced.

In particular, if F is locally free of rank r , in a neighborhood of each point inM itis generated by r holomorphic vector fields v1; : : : ; vr , without relations, onU. The setS.F / \ U is the set of points where the vector fields fail to be linearly independent.In this case, (2.1) gives a locally free resolution of NF and the general theory ofdeterminantal varieties tells us that dim S.F / � r � 1.

Again, singular distributions can be defined in terms of holomorphic 1-forms.

Definition 2.6. A singular holomorphic cotangential distribution of corank s on Mis a coherent sub-OM -module G of rank s of �M.

Definitions 2.5 and 2.6 are related as follows. If a rank r distribution F is given,denoting by F a the annihilator of F ,

F a D f v 2 OM j hv; !i D 0 for all v 2 F g;which may be also written as HomOM

.NF ;OM /, we set G D F a. Then G is a coranks D m � r reduced distribution and we have S.G / � S.F /. Conversely, if a coranks distribution G is given, we let F be the annihilator G a of G . Then F is a rankr D m � s reduced distribution and we have S.F / � S.G /.

Thus if we consider only reduced distributions, by the above correspondence, thetwo definitions are equivalent. Moreover in this case, the singular sets S.F / and S.G /are the same.

2.3 Singular distributions on singular varieties

LetM be a complex manifold of dimensionm and V a possibly singular analytic vari-ety inM of dimension n. We denote by �V the ideal sheaf in OM of germs of holomor-phic functions vanishing identically on V. Thus the quotient sheaf OV D OM=�V isthe sheaf of germs of holomorphic functions on V. We denote by Sing.V / the singularset of V and V 0 D V n Sing.V / the regular part. We consider the sheaf‚M .logV / oflogarithmic vector fields of V :

‚M .logV / D f v 2 ‚M j v.�V / � �V g:Note that a germ of vector field v in‚M is in‚M .logV / if and only if it is tangent

toV 0. We define the tangent sheaf‚V ofV to be the image of the sheaf homomorphism

‚M .logV /˝OV �! ‚M ˝OV:

The sheaf‚V may also be defined as the dual of the sheaf of holomorphic 1-forms�Von V. Namely, recall that there is an exact sequence

�V =�2V

�! �M ˝OV �! �V �! 0;

212 Tatsuo Suwa

where .Œf �/ D df ˝1. Then‚V coincides with HomOV.�V ;OV / so that we have

the exact sequence

0 �! ‚V �! ‚M ˝OV��! NV ;

where NV D HomOV.�V =�

2V ;OV /. The restriction of NV to V 0 is the sheaf of germs

of holomorphic sections of the normal bundle of V 0 in M.Let F be a singular distribution of rank r on M and assume that

(1) F leaves V invariant, i.e. F � ‚M .logV /, and that

(2) V 6� S.F /.The singular distribution FV on V induced from F is defined to be the image of

the sheaf homomorphism F ˝ OV �! ‚M ˝ OV. Note that from the condition (1)above, FV is a subsheaf of ‚V.

3 Local Chern classes

3.1 Chern–Weil theory for virtual bundles

For the Chern–Weil theory of characteristic classes of complex vector bundles, werefer to [6], [8], [19], and [22].

Let M be a C1 manifold and E a C1 complex vector bundle of rank ` over M.For an open set U in M, we denote by Ar.U / the complex vector space of complexvaluedC1 r-forms onU. Also, we denote byAr.U;E/ the vector space of “E-valuedr-forms” onU, i.e. C1 sections of the bundle

Vr.T cRM/�˝E onU, where .T cRM/�

denotes the dual of the complexification of the real tangent bundle TRM of M. ThusA0.U / is the ring ofC1 functions andA0.U;E/ is theA0.U /-module ofC1 sectionsof E on U.

Recall that a connection for E is a C-linear map

r W A0.M;E/ �! A1.M;E/

satisfying the Leibniz rule:

r.f s/ D df ˝ s C f r.s/ for f 2 A0.M/ and s 2 A0.M;E/:Note that E always admits a connection. If r is a connection for E, it induces a

C-linear mapr 0 W A1.M;E/ �! A2.M;E/

satisfying

r 0.! ˝ s/ D d! ˝ s � ! ^ r.s/ for ! 2 A1.M/ and s 2 A0.M;E/:The composition

K D r 0 ı r W A0.M;E/ �! A2.M;E/


is called the curvature of r. It is not difficult to see thatK isA0.M/-linear so that wemay think of it as a C1 2-form with coefficients in the bundle Hom.E;E/.

Definition 3.1. We set

c�.r/ D det.I C A/; A Dp�12�

K;

and call it the total Chern form of r.

It is shown that c�.r/ is a closed form on M. The p-th Chern form cp.r/ of r isthe component of c�.r/ of degree 2p. Thus we may write

cp.r/ D�p�12�

�p�p.K/;

where �p is the p-th elementary symmetric polynomial. We call

cp D�p�12�

�p�p

the p-th (elementary) Chern polynomial.We refer to [8] and [22] for the construction of the following “difference forms”.

Here we use the sign convention of [22].

Proposition 3.2. Suppose we have r C 1 connections r0; : : :rr for E. Then thereexists a .2p� r/-form cp.r0; : : : ;rr/, alternating in the r C 1 entries and satisfying

rX�D0

.�1/�cp.r0; : : : ; yr� ; : : : ;rr/C .�1/rdcp.r0; : : : ;rr/ D 0:

In particular, if we have two connections r0 and r1, there is a .2p � 1/-formcp.r0;r1/ satisfying

dcp.r0;r1/ D cp.r1/ � cp.r0/:Thus, if r is a connection for E, the class of cp.r/ in the de Rham cohomologyH2pdR .M/ does not depend on the choice of r.

Definition 3.3. The p-th Chern class cp.E/ of E is the class of cp.r/ in H 2pdR .M/,

where r is a connection for E. The total Chern class is

c�.E/ D 1C c1.E/C C c`.E/:

Remark 3.4. The class cp.E/ is in the image of the canonical homomorphism

H 2p.M;Z/ �! H 2p.M;C/ ' H 2pdR .M/;

see [25] for detailed discussions on this matter.

214 Tatsuo Suwa

More generally let' be a symmetric series.We may write' as a series in elementaryChern polynomials ; ' D P.c1; c2; : : : /. For a connection r for E, we set

'.r/ D P.c1.r/; c2.r/; : : : /;which is a closed form. We also have the difference forms as in Proposition 3.2and we define '.E/ to be the class of '.r/ in H�

dR.M/, which coincides withP.c1.E/; c2.E/; : : : /.

As an example, the Segre polynomials s1; s2; : : : are defined by

.1C s1 C s2 C /.1C c1 C c2 C / D 1:Then for a connection r for E, we have the (total) Segre form s�.r/ D c�.r/�1 andthe Segre class s�.E/ D c�.E/�1 of E.

If we have a complex vector bundle Ei on M, for each i D 0; : : : ; q, we mayconsider the “virtual bundle” � DPq

iD0.�1/iEi . Lettingr.i/ be a connection forEi ,i D 0; : : : ; q, we denote by r the family of connections .r.q/; : : : ;r.0// and defineits total Chern form c�.r/ by

c�.r/ DqYiD0

c�.r.i//�.i/;

where �.i/ D .�1/i . The p-th Chern form cp.r/ is the component of c�.r/ ofdegree 2p. More generally, for a symmetric series ', we write ' D P.c1; c2; : : : /

as before and set '.r/ D P.c1.r/; c2.r/; : : : /. If we have a finite number offamilies of connections r

� D .r.q/� ; : : : ;r.0/� /, � D 0; : : : ; r , we have the differenceform '.r

0 ; : : : ;rr / as in Proposition 3.2 (cf. [22], Chapter II, 8). In particular, for

two families of connections,

d'.r0 ;r

1/ D '.r1/ � '.r

0/: (3.1)

Thus the class of '.r/ in H�.M/ is well-defined. We denote it by '.�/ and call itthe characteristic class of � with respect to '. In particular, the total Chern class c�.�/is the class of c�.r/ and is also given by

c�.�/ DqYiD0

c�.Ei /�.i/:

The p-th Chern class cp.�/ is the component of c�.�/ in H 2p.M/ and is the classof cp.r/.

Now let

0 �! Eqhq�! h2�! E1

h1�! E0 �! 0 (3.2)

be a sequence of vector bundles onM and, for each i, let r.i/ be a connection for Ei .We say that the family .r.q/; : : : ;r.0// is compatible with the sequence if, for each i,


the following diagram is commutative:

A0.M;Ei /

hi

��

r.i/�� A1.M;Ei /

1˝hi

��A0.M;Ei�1/ r.i�1/

�� A1.M;Ei�1/:

We have the following “vanishing by exactness” (cf. [6], Lemma (4.22)).

Lemma 3.5. If the sequence (3.2) is exact, there exists always a family

r D .r.q/; : : : ;r.0//of connections compatible with the sequence and for such a family we have

cp.r/ 0 for p > 0:

Thus in the above situation, we have c�.r/ D 1, and, in particular, c�.�/ D 1,with � D Pq

iD0.�1/iEi . In fact, the above holds for the difference form of a finitenumber of families of connections compatible with (3.2). For a symmetric series 'without constant term, we also have a similar vanishing '.r/ D 0. From this wehave the following result.

Proposition 3.6. Suppose the sequence (3.2) is exact. Let' be a symmetric polynomialand r D .r.q/; : : : ;r.0//, a family of connections compatible with (3.2). Then

'.{r/ D '.r.0//and, in particular,

'. L�/ D '.E0/;where {r denotes the family of connections .r.q/; : : : ;r.1// for the virtual bundleL� DPq

iD1.�1/i�1Ei .Similar identities hold for the other “partitions” of the virtual bundle � and for

the difference forms of families of connections.

3.2 Characteristic classes in the Cech–de Rham cohomology

The Cech–de Rham cohomology is defined for an arbitrary covering of a manifoldM,however for simplicity here we only consider coverings of M consisting of two opensets. For the general case and details, we refer to [9], [16], [22], and [25]. In Section 5.2below we recall the Cech–Dolbeault cohomology for coverings with arbitrary numberof open sets.

216 Tatsuo Suwa

Let M be a C1 manifold of (real) dimension m0 and U D fU0; U1g an opencovering of M. We set U01 D U0 \ U1 and define the vector space Ar.U/ as

Ar.U/ D Ar.U0/˚ Ar.U1/˚ Ar�1.U01/:

Thus an element � in Ar.U/ is given by a triple � D .�0; �1; �01/. We define theoperator

D W Ar.U/ �! ArC1.U/

byD� D .d�0; d�1; �1 � �0 � d�01/:

Then we have D ı D D 0. The Cech–de Rham cohomology H rD.U/ of U is the

cohomology of the complex .A�.U/;D/. Note that there is a natural isomorphism

H rD.U/ ' H r

dR.M/: (3.3)

The Cech–de Rham cohomology is also equipped with the cup product, which isdefined on the cochain level by assigning to � in Ar.U/ and in As.U/ the cochain� Y in ArCs.U/ given by

� Y D .�0 ^ 0; �1 ^ 1; .�1/r�0 ^ 01 C �01 ^ 1/: (3.4)

The cup product is compatible with the usual one in H r .M;C/ ' H rdR.M/ via the

isomorphism (3.3).IfM is oriented and compact, we may define the integration on the Cech–de Rham

cohomology Hm0

D .U/ and the cup product followed by the integration describes thePoincaré duality:

H r .M;C/ ' H rD.U/

�! Hm0�r.U/� ' Hm0�r.M;C/:

Next we define the relative Cech–de Rham cohomology and describe the Alexan-der duality. Let S be a closed subset of M. Letting U0 D M n S and U1 an openneighborhood of S , we consider the covering U D fU0; U1g of M. We set

Ar.U; U0/ D f � D .�0; �1; �01/ 2 Ar.U/ j �0 D 0 g:Then we see that if � is in Ar.U; U0/, D� is in ArC1.U; U0/. This gives rise

to another complex .A�.U; U0/;D/ and we define the r-th relative Cech–de Rhamcohomology H r

D.U; U0/ of the pair .U; U0/ to be the cohomology of this complex.Note that there is a natural isomorphism

H rD.U; U0/ ' H r .M;M n S IC/:

Note that the cup product of a cochain in A�.U/ and a cochain in A�.U; U0/ isin A�.U; U0/ and this induces a natural H�

D.U/-module structure on H�D.U; U0/,

which is compatible with the usual H�.M/-module structure on H�.M;M n S/.SupposeM is oriented and S is compact (M may not be). Then we may define the

integrationRMW Hm

D .U; U0/! C. From (3.4) we see that the cup product induces apairing Ar.U; U0/ � Am�r.U1/ ! Am.U; U0/, which, followed by the integration,


gives a bilinear pairing

Ar.U; U0/ � Am�r.U1/ �! C:

If we further assume thatU1 is a regular neighborhood ofS , this induces theAlexanderduality

A W H r.M;M n S IC/ ' H rD.U; U0/

�! Hm�r .U1;C/� ' Hm�r .S;C/: (3.5)

The following proposition, which is rather obvious from the above construction,is fundamental in the localization theory.

Proposition 3.7. If M is compact, we have the commutative diagram

H r .M;M n S IC/ j�

��

o A

��

H r .M;C/

o P

��Hm�r .S;C/

i�� Hm�r .M;C/;

where i and j denote the inclusionsS ,!M and .M;;/ ,! .M;M nS/, respectively.

Let M be a C1 manifold, � D PqiD0.�1/iEi a virtual bundle over M and '

a symmetric series, as before. Also let U D fU0; U1g be an open covering of M.Choosing a family of connections r

� D .r.q/� ; : : : ;r.0/� / for � on U� , � D 0; 1, wehave a Cech–de Rham cochain

'.r�/ D .'.r0/; '.r

1/; '.r0 ;r

1//: (3.6)

By (3.1), this is a cocycle and defines a class in the Cech–de Rham cohomologyH�D.U/, which corresponds to the class '.�/ via the isomorphism (3.3).

Moreover, if we may choose r0 so that '.r

0/ 0, the cocycle '.r�/ definesa class in the relative cohomology H�

D.U; U0/. This idea is used in the localizationtheory of characteristic classes of virtual bundles. In the next subsection, we give suchan example.

3.3 Local Chern classes and characters

We discuss the localization theory of characteristic classes by exactness of vectorbundle sequences. For details of this and the subsequent subsections, we refer to [23].

Let M be a C1 manifold and S a closed set in M. Letting U0 DM n S and U1 aneighborhood of S in M, we consider the covering U D fU0; U1g of M, as before.

Suppose that (3.2) is a sequence complex of vector bundles overM which is exacton U0. Then we will see below that, for each p > 0, there is a natural localization

218 Tatsuo Suwa

cpS .�/ in H 2p.M;M n S/ of the Chern class cp.�/ in H 2p.M/ of the virtual bundle� DPq

iD0.�1/iEi .Let r

0 be a family of connections compatible with (3.2) on U0 and r1 an arbi-

trary family of connections for � D PqiD0.�1/iEi on U1. Then the class cp.�/ is

represented by the cocycle

cp.r�/ D .cp.r0/; c

p.r1/; c

p.r0 ;r

1//

inA2p.U/. By Lemma 3.5, we have cp.r0/ D 0 and thus the cocycle is inA2p.U; U0/

and it defines a class cpS .�/ in H 2pD .U; U0/. It is sent to cp.�/ by the canonical

homomorphism j �. It is not difficult to see that the class cpS .�/ does not depend onthe choice of the family of connections r

0 compatible with (3.2) or on the choice ofthe family of connections r

1 . If ' is a symmetric series without constant term, wemay also define the localized class 'S .�/ of '.�/.

The localized Chern character ch�S .�/ of the virtual bundle � as above can be

defined in this context. Thus in general, let r be a connection for a complex vectorbundle E of rank `. The Chern character form of r is defined by

ch�.r/ D tr.eA/; A Dp�12�

K;

where K is the curvature of r. If we set tp.r/ D tr.Ap/, it is a closed 2p-form onM and we may write

ch�.r/ D `CXp�1

tp.r/pŠ

:

The forms cp D cp.r/ and tp D tp.r/ are related by Newton’s formula:

tp � c1tp�1 C c2tp�2 � C .�1/pp cp D 0; p � 1:The class of ch�.r/ inH�

dR.M/ is the Chern character ch�.E/ of E. More gener-ally, let � DPq

iD0.�1/iEi be a virtual bundle over M and r D .r.q/; : : : ;r.0// afamily of connections for � . We set

ch�.r/ DqXiD0.�1/ich�.r.i//

and define the Chern character ch�.�/ of � to be the class of ch�.r/.Let S , U, � , r

0 and r1 be as in the beginning of this subsection. Then the class

ch�.�/ in H�D.U/ is represented by the cocycle

ch�.r�/ D .ch�.r0/; ch�.r

1/; ch�.r0 ;r

1//

in A�.U/. Noting that the alternating sum of the ranks ofEi is zero ifM nS ¤ ;, byLemma 3.5 we have ch�.r

0/ 0 and we see that the cocycle is inA�.U; U0/. Its classin the relative cohomology H�

D.U; U0/ is the localized Chern character ch�S .�/ of � .

It is sent to ch�.�/ by the homomorphism j � W H�D.U; U0/! H�

D.U/.


Remark 3.8. The local Chern characters defined as above have all the necessaryproperties and should coincide with the ones in [14]. Hence they are in the cohomologyH�.M;M nS IQ/ with Q coefficients. Also, the local Chern classes above are in theimage of H�.M;M n S IZ/ ! H�.M;M n S IC/. See also [7] for local Cherncharacters.

Now let M be a complex manifold and denote by AM the sheaf of germs ofreal analytic functions on M. Let � be a coherent OM -module and suppose that thesupport S of � is compact. Taking a relatively compact open neighborhood U of SinM, there is a complex of real analytic vector bundles on U as (3.2) such that, on thesheaf level, the sequence

0 �! AU .Eq/ �! �! AU .E0/ �! AU ˝OU� �! 0

is exact [5]. We call such a sequence a resolution of � by vector bundles. We definethe Chern character ch�.�/ of � by

ch�.�/ D ch�.�/; � DqXiD0.�1/iEi :

Then ch�.�/ does not depend on the choice of the resolution. We set U0 D U n S ,U1 D U and U D fU0; U1g. Since the sequence (3.2) is exact on U0, we have thelocal Chern character ch�

S .�/ in H�D.U; U0/ ' H�.U; U n S/ ' H�.M;M n S/.

We finish this subsection by recalling the Todd class in our context, which will beused in the subsequent sections. Let r be a connection for a complex vector bundleEof rank ` on a C1 manifold M. The Todd form of r is defined by

td.r/ D det� A

I � e�A�; A D

p�12�

K;

where K is the curvature of r. The constant term in td.r/ is 1 so that the form isinvertible. It is closed and its class in H�

dR.M/ is the Todd class td.E/ of E. Wehave the following fundamental relation (cf. [12], III, Corollary 5.4), which is one ofthe essential ingredients in the proof of the Riemann–Roch theorem for embeddingspresented and used below:X

iD0.�1/ich�.ƒir�/ D td.r/�1 c`.r/;

wherer� denotes the connection forE� dual tor andƒir� the connection forƒiE�induced byr�. We also setƒ0E� DM �C andƒ0r� D d (the exterior derivative).See [13], Theorem 10.1.1, for the above formula in cohomology.

220 Tatsuo Suwa

3.4 Riemann–Roch theorem for embeddings

Let M be a complex manifold of dimension m and V a compact analytic subvarietyof pure dimension n in M. Let k D m � n. Recall that ([10], see also [25]) we havethe Poincaré homomorphism

P W H r.V / �! H2n�r.V /

and the Thom homomorphism

T W H r .V / �! H rC2k.M;M n V /so that the following diagram is commutative:

H r .V /

P

��

T �� H rC2k.M;M n V /

o A

��H2n�r.V / id

�� H2n�r.V /

Recall also that a subvariety V of codimension k in M is a local complete in-tersection (abbreviated as LCI) in M if the ideal sheaf �V in OM of functionsvanishing on V is locally generated by k functions. In this case, the normal sheafNV D HomOV

.�V =�2V ;OV / is a locally free OV -module of rank k. We denote by

NV the associated vector bundle, which gives a natural extension of the normal bundleNV 0 in M of the regular part V 0 of V to the whole V.

We say that a subvariety V of codimension k in M is an LCI defined by a section,if there exist a holomorphic vector bundle N of rank k over M and a holomorphicsection s of N such that the local components of s locally generate �V. In this caseV is an LCI and we have NV D N jV.

Let � W V ,! M denote the embedding and let � be a coherent OV -module. Thedirect image �Š� is a coherent OM -module, which is simply � extended by zero onM n V, and we have the localized Chern character ch�

V .�Š�/ in H�.M;M n V IQ/.The following localized version of the Riemann–Roch theorem for embeddings is

proved on the level of Cech–de Rham cocycles in [23].

Theorem 3.9. Let V be a compact subvariety in M and � a coherent OV -module.Suppose

(i) V is non-singular, or

(ii) V is an LCI defined by a section and � is locally free.

Then we have

ch�V .�Š�/ D T .ch�.�/Y td�1.NV // in H�.M;M n V IQ/:

Here we emphasize that M may not be compact. See [23], Remark 3.6, for otherworks related to the above.


4 Chern residues of singular distributions

4.1 Localization by rank reason

LetM be a complex manifold of dimensionm and F a singular distribution of rank ron M. Let NF be the normal sheaf so that we have the exact sequence (2.1). LetS D S.F / be the singular set of the distribution. There is a rank r subbundle F0 ofTM0

,M0 DM nS , such that F jM0D O.F0/. If we setNF0

D TM0=F0, it is a vector

bundle of rank m � r D s and NF jM0D O.NF0

/. Observe that, for an arbitraryconnection r for NF0

,

cp.r/ 0 for p > s:

Thus for a Chern monomial ' D cp1 cpk , if pi > s for some i, '.r/ 0. Thisleads to the following result.

Theorem 4.1. In the above situation, suppose S is compact. Then there is a naturallocalization'S .NF ;F / inH 2d .M;M nS/of'.NF / inH 2d .M/,d D p1C Cpk .

Proof. Let U be a relatively compact regular neighborhood of S in M and setU0 D U n S , U1 D U , and U D fU0; U1g. Take a locally free resolution of NF

on U (we omit to write down the sheaf A of germs of real analytic functions):

0 �! Eq �! �! E0 �! NF �! 0: (4.1)

Let r be a connection for NF0on U0 and r.i/0 a connection for each Ei on U0

such that the family .r.q/0 ; : : : ;r.0/0 ;r/ is compatible with (4.1) on U0. Let r.i/1be a connection for each Ei on U1 and set r

� D .r.q/� ; : : : ;r.0/� /, � D 0; 1. If' is a polynomial as above, then by Proposition 3.6 and the above observation wehave '.r

0/ D '.r/ D 0 and the cocycle '.r�/ D .0; '.r1/; '.r

0 ;r1// is in

A2d .U; U0/.Now we claim that, if we start with another connectionD forNF0

, we get a cocycle

cohomologous to the above. In fact, let D.i/0 be a connection for each Ei on U0 such

that the family .D.q/0 ; : : : ;D

.0/0 ;D/ is compatible with (4.1) on U0. Let D.i/

1 be a

connection for each Ei on U1 and set D� D .D

.q/� ; : : : ;D

.0/� /, � D 0; 1. Again we

have '.D0/ D '.D/ D 0 and we have a cocycle '.D�/ D .0; '.D

1/; '.D0;D

1// in

A2p.U; U0/. We have

'.D�/ � '.r�/ D .0; '.D1/ � '.r

1/; '.D0;D

1/ � '.r

0 ;r1//:

On the other hand, considering the difference form '.r0 ;D

0;D

1/ for the three fam-

ilies of connections r0 , D

0 and D1, we have

'.D0;D

1/ � '.r

0 ;D1/C '.r

0 ;D0/C d'.r

0 ;D0;D

1/ D 0:

222 Tatsuo Suwa

Again by Proposition 3.6 and by rank reason '.r0 ;D

0/ D '.r;D/ D 0. Considering

also the difference form '.r0 ;r

1 ;D1/,

'.r1 ;D

1/ � '.r

0 ;D1/C '.r

0 ;r1/C d'.r

0 ;r1 ;D

1/ D 0

so that

'.D�/ � '.r�/ D D; D .0; '.r1 ;D

1/; '.r

0 ;D0;D

1/ � '.r

0 ;r1 ;D

1//:

Thus the class of '.r�/ in H 2dD .U; U0/ ' H 2d .U; U n S/ ' H 2d .M;M n S/

does not depend on the choice of connections involved.We can also show that it does not depend on the choice of resolution in a similar

manner.Thus the class of '.r�/, denoted by 'S .NF ;F /, is well-defined.

Suppose S has a finite number of connected components .S/, then via the Alexan-der isomorphism

A W H 2d .M;M n S/ �! H2m�2d .S/ DM

H2m�2d .S/;

the above localization 'S .NF ;F / defines a local invariant in H2m�2d .S;Q/ (infact, in the image of H2m�2d .S;Z/ ! H2m�2d .S;Q/) for each �. We call it theresidue of F on NF at S with respect to ' and denote by Res'.F ;NF IS/. If Mis compact, from Proposition 3.7, we have the residue formula:X

.�/�Res'.F ;NF IS/ D P.'.NF //; (4.2)

where � W S ,!M denotes the inclusion.If F is a singular foliation, i.e. ifF0 in involutive, there is an “action” ofF0 onNF0

and, if r is an “F0-connection” for NF0, we have the Bott vanishing '.r/ D 0 for a

homogeneous symmetric polynomial ' of degree d > s (cf. Section 6.1, in particularRemark 6.4 below). The Baum–Bott residue with respect to ' is defined exactly thesame way as above, using an F0-connection asr. Here we emphasize that the Baum–Bott residue is defined for an arbitrary ' whose degree is greater than s and in generalis in the homology with complex coefficients, while the above residue is defined for' containing cp with p > s in each of its terms, but is in the homology with rationalcoefficients, if the coefficients of ' are rational.

From the construction we have the following, which shows the rationality of therelevant Baum–Bott residues (cf. [6], Rationality conjecture):

Proposition 4.2. If F is a foliation, the above residue coincides with the Baum–Bottresidue.


4.2 Residues and the local Chern classes

In this subsection, we show that the residues defined in the previous subsection arerelated to the local Chern class of some sheaf supported on the singular set of thedistribution. This way, in some cases we can compute the residues using the Riemann–Roch theorem for embeddings (cf. Theorem 3.9). The contents of this subsection areessentially in [21]. Note that involutivity is not really needed there.

Let G be a reduced singular distribution of corank s on a complex manifold M ofdimension m and set F D G a and �G D �M=G . Taking the duals of

0 �! G �! �M �! �G �! 0;

we have the exact sequence

0 �! F �! ‚M��! G � �! Ext1O.�G ;O/ �! 0: (4.3)

Note that, by definition, the support of Ext1O.�G ;O/ is in S D S.G / D S.F /,

which is assumed to be compact with a finite number of connected components .S/.Comparing with (2.1), we have the exact sequence

0 �! NF �! G � �! Ext1O.�G ;O/ �! 0: (4.4)

Recall that there is a rank r D m�s subbundleF0 ofTM0 such that F jM0D O.F0/.

We setNF0D TM0=F0:

Hereafter we assume that G is locally free of rank s. Thus there is a vector bundleGof rank s on M with G D OM .G/. Note that we may think of G as a subbundle ofT �M only away from S . We express the Chern classes of NF in terms of those ofG � D OM .G

�/ and Ext1O.�G ;O/ using (4.4). In the sequel we denote Ext1

O.�G ;O/

simply by E . We have the local Chern classes cpS .E/ in H 2p.M;M n S/ for p > 0.If we denote by j � W H 2p.M;M n S/ ! H 2p.M/ the canonical homomorphism,j �cpS .E/ D cp.E/ is the p-th Chern class of the coherent sheaf E . Accordingly,we have the local Segre class spS .E/ in H 2p.M;M n S/ for each p > 0, so thatj �spS .E/ D sp.E/ is the p-th Segre class of E inH 2p.M/. For simplicity we assumethat S is connected. Basically, we have

c�.NF / D c�.G � � E/ D c�.G �/ s�.E/

with some localized components.

Definition 4.3. For each integer p, 1 � p � m, we set

cp.G � � E/ D

8<ˆ:cp.G �/C cp�1.G �/ s1.E/C C c1.G �/ sp�1.E/C sp.E/;

1 � p � s;cs.G �/ sp�s

S .E/C C c1.G �/ sp�1S .E/C spS .E/;

s < p � m:

224 Tatsuo Suwa

Note that, for p � s, cp.G � � E/ is inH 2p.M;Q/, while for p > s, cp.G � � E/

is in H 2p.M;M n S IQ/. Let

A W H 2d .M;M n S/ �! H2m�2d .S/

denote the Alexander isomorphism.

Theorem 4.4. For a Chern monomial ' D cp1 cpk with pi > s for some i, theproduct cp1.G ��E/ cpk .G ��E/ is inH 2d .M;M nS/, d D p1C Cpk , andwe have

Res'.F ;NF IS/ D A.cp1.G � � E/ : : : cpr .G � � E//;

which is inH2m�2d .S;Q/, in fact in the image ofH2m�2d .S;Z/! H2m�2d .S;Q/.

Proof. Let U and U D fU0; U1g be as in the proof of Theorem 4.1. Combining (4.4)and (4.1), we have a locally free resolution defining the local Chern class of E:

0 �! Eq �! �! E0 �! G� �! E �! 0:

We may identify NF0and G� on U0. Let r be a connection for G� on U. Let

r.i/0 be a connection for each Ei on U0 so that the family .r.q/0 ; : : : ;r.0/0 ;r/ is

compatible with (4.1) on U0. Let r.i/1 be a connection for each Ei on U1 and setyr� D .r.q/� ; : : : ;r.0/� ;r/, � D 0; 1. The point here is that we take, for G�, the same

connection r on U0 and U1. Then we have

.c�.r0/; c

�.r1/; c

�.r0 ;r

1//

D .c�.r/; c�.r/; 0/Y .s�.yr0/; s

�.yr1/; s

�.yr0 ;yr1//;

which proves the theorem.

As noted above, in some cases the residues can be computed using the Riemann–Roch theorem for embeddings. The following result is proved in [21] this way (invo-lutivity is not necessary as noted above).

Proposition 4.5. In the above situation, suppose that s D 1 and that S D fpg is anisolated point. Then we have

Rescm.F ;NF I fpg/ D .�1/m.m � 1/Š dim Ext1O.�G ;O/ in H0.fpg;Q/ D Q;

where we denoted OM;p and �G ;p simply by O and �G .

Note that if .z1; : : : ; zm/ is a coordinate system onM aroundp and! DPmiD fidzi

is a generator of G near p, then the set of common zeros of the fi ’s is fpg and we have

dim Ext1O.�G ;O/ D dim O=.f1; : : : ; fm/:

The following result is proved in [15] using a Koszul complex associated to thedistribution.


Proposition 4.6. If G is locally free of rank one, then

cm.NF / D .�1/m.m � 1/Š cm.�M ˝ G �/:

Thus the residue in Proposition 4.5 also arises from the localization of the topChern class of �M ˝ G � by a section, i.e. the section locally corresponding to W G ' O ! �M given by .1/ D ! (cf. [24]).

See [21] for more examples. In the next subsection we give an example for whichthe singular set of the distribution is non-isolated and singular.

4.3 An example

We consider the 1-form

! D z dx C z dy � y dzon C3 D f.x; y; z/g. It defines a corank one singular distribution on C3 with singularset fy D z D 0g. As generators of its annihilator, we may take the vector fields

v1 D y @@yC z @

@zand v2 D @

@x� @

@y:

We extend the distribution to the projective space. Let P3 denote the complex pro-jective space of dimension three with homogeneous coordinates � D .�0 W �1 W �2 W �3/.It is covered by four open sets W .i/,0 � i � 3, given by �i ¤ 0. We take the originalaffine space C3 as W .0/ with x D �1=�0, y D �2=�0 and z D �3=�0. Let G be thecorank one distribution on P3 naturally obtained as an extension of the above.

(0) On W .0/, G is defined by !0 D z dx C z dy � y dz as given before.

(1) On W .1/, we set x1 D �0=�1, y1 D �3=�1 and z1 D �2=�1. Then G is defined by

!1 D �y1 dx1 � x1z1 dy1 C x1y1 dz1:(2) On W .2/, we set x2 D �3=�2, y2 D �0=�2 and z2 D �1=�2. Then G is defined by

!2 D �y2 dx2 � x2z2 dy2 C x2y2 dz2:(3) On W .3/, we set x3 D �2=�3, y3 D �1=�3 and z3 D �0=�3. Then G is defined by

!3 D z3 dx3 C z3 dy3 � y3 dz3:We see that G is a reduced distribution of corank one. It is locally free and from!i D .�j =�i /

3!j in W .i/ \ W .j /, we see that it is .H�3 /

˝3 as a line bundle, whereH3 denotes the hyperplane bundle on P3.

The singular set S D S.G / is defined by �0�2 D �0�3 D �1�3 D 0 and has threeirreducible components S1 D f�2 D �3 D 0g, S2 D f�0 D �3 D 0g and S3 D f�0 D�1 D 0g, each of which is a projective line P1. We also let P1 D .0 W 1 W 0 W 0/,

226 Tatsuo Suwa

which is the intersection point of S1 and S2 and P2 D .0 W 0 W 1 W 0/, which is theintersection point of S2 and S3. Let F D G a denote the annihilator of G , which is asingular distribution of rank two on P3 with the singular set S.F / D S.G / D S .

We try to find the residues Res'.F ;NF IS/ inH�.S/ for ' D c2, c1c2 and c3. Forthis, we first find the local Chern class c�

S .E/, E D Ext1.�G ;O/, using Theorem 3.9,which in our case reads

ch�S .�ŠE/ D T .ch�.E/Y td�1.NS // in H�.P3;P3 n S IQ/: (4.5)

Note that we see below that the hypothesis (2) of the theorem is satisfied in our case.Let T0, T1, T2 and T3 be tubular neighborhoods of S1 n fP1g, S2 n fP2g, S2 n fP1g

and S3 n fP2g, respectively, sufficiently small so that Ti � W .i/ and that none of thethree distinct Ti ’s intersects. We set T D T0 [ [ T3, which is a neighborhood ofS in P3.

Let �S denote the ideal sheaf of S . Then it is generated by .f0; g0/ D .z; y/ onW .0/, .f1; g1/ D .y1; x1z1/ onW .1/, .f2; g2/ D .x2z2; y2/ onW .2/ and .f3; g3/ D.y3; z3/ on W .3/. Thus we see that S is a local complete intersection, although it isnot a complete intersection.

LetAij denote the 2�2matrix defined by .fi ; gi / D .fj ; gj /Aj i . Then the locallyfree sheaf �S=�

2S is defined by the system of transition matrices fAij g. We compute

A01 D

0B@1

x0

01

x2

1CA ; A12 D

0BB@1

z210

01

z21

1CCA ; A23 D

0BB@1

x220

01

x2

1CCA : (4.6)

LetNS be the normal bundle of S in P3, which is the vector bundle correspondingto the locally free sheaf Hom.�S=� 2S ;OS /. Since it is defined by the system ftA�1

ij g DfAj ig, by (4.6), we see that NS jS1

D H1 ˚H 21 , NS jS2

D H 21 ˚H 2

1 and NS jS3D

H 21 ˚ H1, H1 being the hyperplane bundle on P1. It is not difficult to see that we

may construct a vector bundleN of rank two on T extendingNS and a regular sectionof N which defines S .

We try to find td�1.NS /. We have

H�.S/ D H 0.S/˚H 2.S/ D Q˚H 2.S1/˚H 2.S2/˚H 2.S3/

and we may write

c�.NS / D 1C .c1.NS jS1/; c1.NS jS2

/; c1.NS jS3// D 1C .31; 41; 31/;

where we set 1 D c1.H1/. Thus we have

td�1.NS / D 1 � 12c1.NS / D 1 � 1

2.31; 41; 31/: (4.7)

We now try to find E D Ext1.�G ;O/. For this, we use the exact sequence (4.3).We have G � D OP3.H˝3

3 / and G �jW .i/ ' OW .i/ , the correspondence being given


by ' $ fi D '.!i /. We compute

�� @@x

�D z; �

� @@y

�D z; �

� @@z

�D�y; on W .0/;

�� @

@x1

�D �y1; �

� @

@y1

�D �x1z1; �

� @

@z1

�D x1y1; on W .1/;

�� @

@x2

�D �y2; �

� @

@y2

�D �x2z2; �

� @

@z2

�D x2y2; on W .2/;

�� @

@x3

�D z3; �

� @

@y3

�D z3; �

� @

@z3

�D�y3 on W .3/:

Thus we see that E D OS .H˝33 jS /, which is locally free of rank one, and

ch�.E/ D 1C c1.E/ D 1C .31; 31; 31/: (4.8)

From (4.7) and (4.8), we have

ch�.E/Y td�1.NS / D 1 � 12.31; 21; 31/:

and by (4.5),

ch�S .�ŠE/ D T

�1 � 1

2.31; 21; 31/

�:

From this we have

ch1S .�ŠE/ D 0;ch2S .�ŠE/ D .�1; �2; �3/;

ch3S .�ŠE/ D �1

2.3C 2C 3/ D �4;

where �i denotes a generator of each component of H 4.P3;P3 n S/ ' Q˚Q˚Q.Note that H 6.P3;P3 n S/ ' Q. By the Newton formula,

c1S .E/ D ch1S .iŠE/ D 0;c2S .E/ D �ch2S .iŠE/ D �.�1; �2; �3/;c3S .E/ D 2 ch3S .iŠE/ D �8:

Thus we have

d1S .E/ D �c1S .E/ D 0;d2S .E/ D �c2S .E/ D .�1; �2; �3/;d3S .E/ D �c3S .E/ D 8

228 Tatsuo Suwa

andc1.G � � E/ D c1.G �/C d1S .E/ D c1.G �/ D 33;c2.G � � E/ D c1.G �/ d1S .E/C d2S .E/ D d2S .E/;c3.G � � E/ D c1.G �/ d2S .E/C d3S .E/;

where 3 D c1.H3/. Note that c1.G � � E/ is in H 2.P3/, while c2.G � � E/ andc3.G � � E/ are in the relative cohomology H�.P3;P3 n S/.

From (4.3) we see that

Resc2.F ;NF IS/ D A.c2.G � � E/// D ŒS�;where A W H 4.P3;P3 n S/ �! H2.S/ is the Alexander isomorphism.

Resc1c2.F ;NF IS/ D A.c1.G � � E/c2.G � � E///

D ��c1.G � � E/ a A.c2.G � � E//

D 3��3 a ŒS� D 9;

Resc3.F ;NF IS/ D A.c3.G � � E//

D A.c1.G �/ d2S .E/C d3S .E//D 3��3 a ŒS�C 8 D 17;

where A W H 6.P3;P3 n S/ �! H0.S/ D Q.

5 Atiyah classes and Cech–Dolbeault cohomology

5.1 Atiyah classes

For details of this subsection, we refer to [1].LetM be a complex manifold of dimensionm andE a holomorphic vector bundle

of rank òverM.Also letr be a connection forE (cf. Section 3.1). Note thatr is a localoperator. Thus, if s.`/ D .s1; : : : ; s`/ is a frame (` C1 sections linearly independenteverywhere) of E on an open set U, we have the connection matrix � D .�ij / withentries �ij 1-forms on U defined by

r.si / DXjD1

�j i ˝ sj :

Definition 5.1. A connection r for E is of type .1; 0/ if the entries of the connectionmatrix with respect to a holomorphic frame are forms of type .1; 0/. In this case, wealso say that r is a .1; 0/-connection.


Note that the above property of r does not depend on the choice of a holomorphicframe and that a holomorphic vector bundle always admits a .1; 0/-connection.

Let r be a .1; 0/-connection for E and K its curvature. The curvature matrix� D .�ij /, which is defined by K.si / D P`

jD1 �j i ˝ sj , is related to the connectionmatrix � by � D d� C � ^ � . Thus we have the decomposition � D �2;0C �1;1 with

�2;0 D @� C � ^ � and �1;1 D N@�: (5.1)

Accordingly we have a decomposition

K D K2;0 CK1;1;whereK2;0 andK1;1 are, respectively, a .2; 0/-form and a .1; 1/-form with coefficientsin Hom.E;E/. Thus we may define, for each elementary symmetric polynomial �p ,p D 1; 2; : : : ; `, a C1 .p; p/-form �p.K

1;1/ on M, which is N@-closed, being locallyN@-exact by (5.1).


ap.r/ D�p�12�

�p�p.K

1;1/

and call it the p-th Atiyah form of r, which is a N@-closed .p; p/-form on M.

The following is proved using the construction of Chern difference forms (cf.Proposition 3.2), in fact the form ap.r0; : : : ;rr/ is the .p; p � r/-component ofcp.r0; : : : ;rr/, see [26] for details.

Proposition 5.3. Suppose we have r C 1 .1; 0/-connections r0; : : : ;rr for E. Thenthere exists a .p; p � r/-form ap.r0; : : : ;rr/, alternating in the r C 1 entries andsatisfying

rX�D0

.�1/�ap.r0; : : : ; yr� ; : : : ;rr/C .�1/r [email protected]; : : : ;rr/ D 0:

In particular, if r D 1, we have

ap.r1/ � ap.r0/ D [email protected];r1/:Thus, if r is a .1; 0/-connection for E, the class of ap.r/ in Hp;p

N@ .M/ does notdepend on the choice of r.

Definition 5.4. Thep-th Atiyah classap.E/ofE is the class ofap.r/ in the Dolbeaultcohomology Hp;p

N@ .M/, where r is a .1; 0/-connection for E.

Remark 5.5. The Atiyah form ap.r/ is the .p; p/-component of the correspondingChern form cp.r/. In particular, am.r/ D cm.r/.

230 Tatsuo Suwa

The p-th Chern class cp.E/ of E is the class of cp.r/ in H 2pdR .M/. Thus, al-

though we may not be able to compare the Atiyah and Chern classes directly on thecohomology level (unless M is compact Kähler), we may do so on the form level.

If r is a metric connection for E, i.e. a .1; 0/-connection compatible with someHermitian metric on E, then its curvature is of type .1; 1/ and thus we have ap.r/ Dcp.r/ for all p.

If M is compact Kähler, the Hodge decomposition gives a canonical injectionh W Hp;p

N@ .M/! H2pdR .M/ and we have, by the above, h.ap.E// D cp.E/.

Let ' be a homogeneous symmetric polynomial of degree d . We may write 'as a polynomial in elementary symmetric polynomials ; ' D P.�1; �2; : : : /. For a.1; 0/-connection r for E, we set

'.r/ D P.c1.r/; c2.r/; : : : /;which is a closed 2d -form and 'A.r/ D P.a1.r/; a2.r/; : : : /, which is a [email protected]; d/-form and is the .d; d/-component of '.r/. For two .1; 0/-connections r andr 0, we have the difference forms '.r;r 0/ and 'A.r;r 0/ as above and we have theclasses '.E/ in H 2d

dR .M/ and 'A.E/ in Hd;dN@ .M/.

5.2 Cech–Dolbeault cohomology

In this and subsequent sections, we recall the theory of Cech–Dolbeault cohomology.For details we refer to [26]. The treatment of relative cohomologies in Section 5.3below is slightly more general than [26].

LetM be a complex manifold of dimensionm. For an open set U ofM, we denotebyAp;q.U / the vector space of C1 .p; q/-forms on U. Let U D fU˛g˛2I be an opencovering of M, indexed by an ordered set I . We set

I .r/ D f.˛0; : : : ; ˛r/ j ˛0 < < ˛r ; ˛� 2 I gand denote by C r.U; Ap;q/ the direct product

C r.U; Ap;q/ DY

.˛0;:::;˛r /2I .r/

Ap;q.U˛0:::˛r/;

where we set U˛0:::˛rD U˛0

\ \U˛r. Thus an element � in C r.U; Ap;q/ assigns

to each .˛0; : : : ; ˛r/ in I .r/ a form �˛0:::˛rin Ap;q.U˛0:::˛r

/. The coboundary oper-ator ı W C r.U; Ap;q/! C rC1.U; Ap;q/ is defined as in the usual Cech theory. Thistogether with the operator

N@ W C r.U; Ap;q/ �! C r.U; Ap;qC1/


makes C �.U; Ap;�/ a double complex for each p D 0; : : : ; n. The simple complexassociated to this is denoted by .Ap;�.U/; xD/. Thus

Ap;q.U/ DM

q0CrDqC r.U; Ap;q

0

/

and the differential

xD D xDq W Ap;q.U/ �! Ap;qC1.U/

is given by

. xD�/˛0:::˛rD

rX�D0

.�1/��˛0::: O� :::˛rC .�1/r N@�˛0:::˛r

; (5.2)

where y means the letter under it is to be omitted. We denote the q-th cohomology of.Ap;�.U/; xD/ by Hp;q

xD .U/ and we call it the Cech–Dolbeault cohomology of U oftype .p; q/. We refer to [26] for the proof of the following.

Theorem 5.6. The restriction map Ap;q.M/ ! C 0.U; Ap;q/ � Ap;q.U/ inducesan isomorphism

Hp;qN@ .M/

�! Hp;qxD .U/:

We define the “cup product”

Ap;q.U/ � Ap0;q0

.U/ �! ApCp0;qCq0

.U/

by assigning to � inAp;q.U/ and inAp0;q0.U/ the element �Y inApCp0;qCq0

.U/

given by

.� Y /˛0:::˛rD

rX�D0

.�1/.pCq��/.r��/�˛0:::˛�^ ˛� :::˛r

:

Then this induces the cup product

Hp;qxD .U/ �Hp0;q0

xD .U/ �! HpCp0;qCq0

xD .U/

compatible, via the isomorphism of Theorem 5.6, with the product in the Dolbeaultcohomology induced by the exterior product of forms.

Now we recall the integration on the Cech–Dolbeault cohomology. Let M andU D fU˛g˛2I be as above and fR˛g˛2I a system of honey-comb cells adapted to U

(see [16] and also [22]).SupposeM is compact, then eachR˛ is compact and we may define the integrationZ

M

W Am;m.U/ �! C

by the sum ZM

� DmXrD0

� X.˛0;:::;˛r /2I .r/

ZR˛0:::˛r

�˛0:::˛r

�

232 Tatsuo Suwa

for � in Am;m.U/. Then it induces the integration on the cohomologyZM

W Hm;mxD .U/ �! C;

which is compatible, via the isomorphism of Theorem 5.6, with the usual integrationon the Dolbeault cohomology Hm;m

N@ .M/. Also the bilinear pairing

Ap;q.U/ � Am�p;m�q.U/ �! Am;m.U/ �! C

defined as the composition of the cup product and the integration induces the Kodaira–Serre duality

KS W Hp;qN@ .M/ ' Hp;q

xD .U/�! H

m�p;m�qxD .U/� ' Hm�p;m�q

N@ .M/�: (5.3)

5.3 Relative Cech–Dolbeault cohomology

Let S be a closed set in M. Let U 0 be an open neighborhood of S in M and letU0 D fU˛g˛2I 0 be an open covering ofU 0. LetU0 DM nS and consider the coveringU D fU˛g˛2I of M, where I D f0g t I 0 with the order 0 < ˛ for all ˛ in I 0.

We denote byAp;q.U; U0/ the subspace ofAp;q.U/ consisting of elements � with�0 D 0 so that we have the exact sequence

0 �! Ap;q.U; U0/ �! Ap;q.U/ �! Ap;q.U0/ �! 0:

We see that xD mapsAp;q.U; U0/ intoAp;qC1.U; U0/. Denoting byHp;qxD .U; U0/

the q-th cohomology of the complex .Ap;�.U; U0/; xD/, we have the long exact se-quence

�! Hp;q�1N@ .U0/ �! H

p;qxD .U; U0/ �! H

p;qxD .U/ �! H

p;qN@ .U0/ �! :

In view of the fact that Hp;qxD .U/ ' Hp;q

N@ .M/, we set

Hp;qN@ .M;M n S/ D Hp;q

xD .U; U0/:

Suppose S is compact (M may not be) and let fR˛g be a system of honey-combcells adapted to U. Then we may assume that each R˛ is compact for ˛ in I 0 and wehave the integration on Am;m.U; U0/ given by

ZM

� DX˛2I 0

ZR˛

�˛ CmXrD1

� X.˛0;:::;˛r /2I .r/

ZR˛0:::˛r

�˛0:::˛r

�:

This again induces the integration on the cohomologyZM

W Hm;mxD .U; U0/ �! C:


It is not difficult to see that the cup productAp;q.U/�Am�p;m�q.U/! Am;m.U/

induces a pairing Ap;q.U; U0/ � Am�p;m�q.U0/! Am;m.U; U0/, which, followedby integration, gives a bilinear pairing

Ap;q.U; U0/ � Am�p;m�q.U0/ �! C:

This induces a homomorphism

xA W Hp;qN@ .M;M n S/ D Hp;q

xD .U; U0/ �! Hm�p;m�qxD .U0/� D Hm�p;m�q

N@ .U 0/�;(5.4)

which we call the N@-Alexander homomorphism.From the above construction, we have the following result.

Proposition 5.7. If M is compact, the following diagram is commutative:

Hp;qN@ .M;M n S/ j�

��

xA��

Hp;qN@ .M/

o KS

��Hm�p;m�qN@ .U 0/�

i�

�� Hm�p;m�qN@ .M/�;

where i W U 0 ,!M denotes the inclusion.

Remark 5.8. Suppose that S is a compact complex submanifold ofM and that thereexists a holomorphic retraction r W U 0 ! S , i.e. a holomorphic map with r ı � D 1S ,where � W S ,! U 0 is the embedding. Then the following diagram is commutative:

Hp;qN@ .M;M n S/ j�

��

r�ı xA��

Hp;qN@ .M/

o KS

��Hm�p;m�qN@ .S/�

i�ı�� Hm�p;m�q

N@ .M/�;

where r� and �� denote the transposed of the pull-backs

r� W Hm�p;m�qN@ .S/ �! H

m�p;m�qN@ .U 0/

and

�� W Hm�p;m�qN@ .U 0/ �! H

m�p;m�qN@ .S/;

respectively.

5.4 Atiyah classes in Cech–Dolbeault cohomology

Let U D fU˛gbe an open covering ofM as in Section 5.2.Also, letE be a holomorphicvector bundle over M. For each ˛, we choose a .1; 0/-connection r˛ for E on U˛ ,

234 Tatsuo Suwa

and for the collection r� D .r˛/˛ , we define the element ap.r�/ in Ap;p.U/ D˚prD0C r.U; Ap;p�r/ by

ap.r�/˛0:::˛rD ap.r˛0

; : : : ;r˛r/:

Then we have xDap.r�/ D 0 by the identity in Proposition 5.3. See [26] for the proofof the following result.

Proposition 5.9. The class Œap.r�/� in Hp;pxD .U/ does not depend on the choice of

the collection of .1; 0/-connections r� and corresponds to the Atiyah class ap.E/ bythe isomorphism of Theorem 5.6.

More generally, for a homogeneous symmetric polynomial ' of degree d, we maydefine 'A.r�/ in Ad;d .U/ and the class Œ'A.r�/� in Hd;d

xD .U/, which corresponds

to 'A.E/ by the isomorphism of Theorem 5.6.Let S , U0 and U be as in Section 5.3. If we may choose r0 so that 'A.r0/ 0,

the cocycle 'A.r�/ defines a class in the relative cohomology Hd;dxD .U; U0/. This

idea is used in the localization theory of Atiyah classes of holomorphic bundles.

6 Atiyah residues of singular distributions

We review the results in [1] from a slightly different viewpoint and also discuss thelocalization problem on singular varieties.

6.1 Actions of distributions

Let M be a complex manifold of dimension m and F a non-singular distribution ofrank r , i.e. a subbundle of TM of rank r .

Definition 6.1. A (holomorphic) action of F on a holomorphic vector bundle E overM is a C-bilinear map

˛ W A0.M;F / � A0.M;E/ �! A0.M;E/

satisfying the following conditions, forf inA0.M/,u inA0.M;F / and s inA0.M;E/:

(i) ˛.f u; s/ D f ˛.u; s/,(ii) ˛.u; f s/ D u.f /s C f ˛.u; s/, and

(iii) ˛.u; s/ is holomorphic whenever u and s are.

A vector bundle E with an action of F is called an F -bundle.


Definition 6.2. Let ˛ be an action of F on E. An F -connection (or ˛-connection, ifit is necessary to specify the action) for E is a connection for E satisfying

(i) rs.u/ D ˛.u; s/, for s 2 A0.M;E/ and u 2 A0.M;F /, and

(ii) r is of type .1; 0/.

From the fact that an action is a local operation, we see that an F -bundle alwaysadmits an F -connection.

We note that the above material can be equivalently treated in the language ofpartial holomorphic connections instead of actions (cf. [3] and [1]), the condition (iv)in Remark 6.4 below corresponding to the fact that the partial connection is flat.

We have the following Bott type vanishing theorem for F -connections. A proofof the part (1) of the following is given in [3] and the part (2) is proved in [1] in thecontext of partial connections.

Theorem 6.3. Let M be a complex manifold of dimension m and F a non-singulardistribution of rank r onM. Also letE be anF -bundle andr0; : : : ;rq F -connectionsfor E. For a homogeneous symmetric polynomial ' of degree d , we have:

(1) if d > m � r C Œ r2�, then

'.r0; : : : ;rq/ 0I(2) if d > m � r , then

'A.r0; : : : ;rq/ 0:Remark 6.4. If F is involutive and if the action satisfies

(iv) ˛.Œu; v�; s/ D ˛.u; ˛.v; s//� ˛.v; ˛.u; s//,we have

'.r/ D 0 for ' with d > m � r .

This is usually referred to as the Bott vanishing theorem.

Let M be a complex manifold of dimension m and V a complex submanifold ofdimension n of M. Let NV be a normal bundle of V in M so that we have the exactsequence

0 �! T V �! TM jV ��! NV �! 0:

Let F be a distribution of rank r . Recall that F leaves V invariant in F jV � T V(Definition 2.3).

The following is proved in [17] (see also [18]) for the case of foliations. In factthe involutivity of F is not necessary and a proof is given in [1] in terms of partialconnections. Here we reproduce the proof in our context.

Theorem 6.5. Let V be a complex submanifold ofM. Let F � TM be a distributionof rank r leaving V invariant. Then there exists a holomorphic action of F jV on thenormal bundle NV.

236 Tatsuo Suwa

Proof. Let u and � beC1 sections of F jV andNV, respectively. Take sections Qu of Fand Qv of TM such that QujV D u and �. QvjV / D �, where jV means the restriction asa section. Define

˛ W A0.V; F jV / � A0.V;NV / �! A0.V;NV /

by

˛.u; �/ D �.Œ Qu; Qv�jV /:We now show that this does not depend on the choice of Qu or Qv. First, let Qu be a

section ofF with QujV D 0. Take a frame . Qu1; : : : ; Qur/ ofF on an open setU and writeas Qu D Pr

iD1 Qai Qui with Qai (C1) functions on U. We set ui D Qui jV and ai D Qai jV,i D 1; : : : ; r . Then .u1; : : : ; ur/ is a frame of F jV on U \ V and, from the conditionQujV D 0, we have ai D 0, i D 1; : : : ; r . We compute

Œ Qu; Qv�jV DrXiD1.ai Œ Qui ; Qv�jV � Qv. Qai /jV ui / D �

rXiD1Qv. Qai /jV ui ;

which is a section of F jV � T V. Hence �.Œ Qu; Qv�jV / D 0.Second, let Qv be a section of TM with �. QvjV / D 0. Then QvjV is a section of T V

such that Œ Qu; Qv�jV D Œ QujV ; QvjV � is a section of T V, since F jV � T V. Hence we have�.Œ Qu; Qv�jV / D 0.

It is then straightforward to check that ˛ is a holomorphic action.

Corollary 6.6. In the above situation, let r be an FV -connection for NV. Then, for asymmetric homogeneous polynomial ' of degree d > n � r , we have 'A.r/ 0.

6.2 Localization and residues

LetM be a complex manifold of dimensionm and F a singular distribution of rank ron M with singular set S D S.F /. There is a rank r subbundle F0 of TM jM0

,M0 D M n S , such that F jM0

D OM0.F0/. Let U0 and U be as in Section 5.3.

Let E be a holomorphic vector bundle on M with an action of F0 on M0. Let r0 bean F0-connection for E on U0 D M0 and r˛ an arbitrary .1; 0/-connection for Efor each ˛ > 0. For a symmetric polynomial ' homogeneous of degree d , we havethe Cech–Dolbeault cocycle 'A.r�/ in Ad;d .U/. If d > m � r , by Theorem 6.3,'A.r�/ is in Ad;d .U; U0/ and defines a class in Hd;d

xD .U; U0/, which we denote by

'AU 0.E;F / and call the localization of 'A.E/ by F at U 0.Suppose S is compact. Then its image by the NA-Alexander homomorphism

Hd;dxD .U; U0/ �! H

m�d;m�dxD .U0/� ' Hm�d;m�d

N@ .U 0/�

is denoted by Res'A.F ; EIU 0/ and called the residue of F for E at U 0 with respectto 'A.


If S has a finite number of connected components .S/, we break up U0 accord-ingly ; U0 D [U0

so that we have a decomposition

Hm�d;m�dxD .U0/ D

M

Hm�d;m�dxD .U0

/

and we have the residue Res'A.F ; EIU 0/ inHm�d;m�d

xD .U0/� ' Hm�d;m�d

N@ .U 0/�

for each �. From Proposition 5.7, we have the following residue theorem.

Theorem 6.7. In the above situation, if M is compactX

.i/�Res'A.F ; EIU 0/ D KS.'A.E// in Hm�d;m�d

N@ .M/�;

where i W U 0,!M denotes the inclusion.

6.3 Atiyah classes on singular varieties

In this subsection, we discuss Atiyah classes on singular varieties, similarly as forChern classes (cf. [22], Chapter VI, 4).

Let M be a complex manifold of dimension m and V a subvariety of pure dimen-sion n of M. Let Sing.V / denote the singular set of V and V 0 D V n Sing.V / theregular part.

First we assume that V is compact and let zU be a neighborhood of V in M. Also,let U D f zU˛g˛2I be an open covering of zU and f zR˛g˛2I a system of honey-combcells adapted to U such that the regular part V 0 of V is transverse to each zR˛0��˛p

. Weset

R˛0��˛pD zR˛0��˛p

\ V:Then we may define the integrationZ

V

W Hn;nxD .U/ �! C

as in [22], Chapter IV, 2. Also the bilinear pairing

Ap;q.U/ � An�p;n�q.U/ �! An;n.U/ �! C

defined as the composition of the cup product and the integration over V induces theKodaira–Serre homomorphism on V :

KSV W Hp;qN@ . zU / ' Hp;q

xD .U/ �! Hn�p;n�qxD .U/� ' Hn�p;n�q

N@ . zU /�;

238 Tatsuo Suwa

which is not an isomorphism in general. The above homomorphism KSV sends theclass Œ�� in Hp;q

xD .U/ to the functional assigning to each class Œ � in Hn�p;n�qxD .U/

the value ZV

� Y :

Now suppose V may not be compact. Let S be a compact set in V. Let zU0 be aneighborhood of V0 D V nS inM, zU 0 a neighborhood of S inM and U0 D f zU˛g˛2I 0

an open covering of zU 0. Set I D f0g [ I 0, with the order 0 < ˛ for all ˛ 2 I 0, andconsider the open covering U D f zU˛g˛2I of zU D zU0 [ zU 0, which is a neighborhoodof V in M.

We define Ap;q.U; zU0/ and Hp;qxD .U; zU0/ as in Section 5.3. Then we have the

integration ZV

W Hn;nxD .U; zU0/ �! C:

The cup product induces the pairing

Ap;q.U; zU0/ � An�p;n�q.U0/ �! An;n.U; zU0/;which, followed by integration, gives a bilinear pairing

Ap;q.U; zU0/ � An�p;n�q.U0/ �! C:

This induces the “N@-Alexander homomorphism over V ”:

xAV W Hp;qxD .U; zU0/ �! H

n�p;n�qxD .U0/� ' Hn�p;n�q

N@ . zU 0/�;

which is not an isomorphism in general.The homomorphism xAV sends the class Œ�� inHp;q

xD .U; zU0/ to the functional which

assigns to each class Œ � in Hm�p;m�qxD .U0/ the valueZ

V

� Y :

From the above construction, we have the following:

Proposition 6.8. In the above situation, if V is compact, the following diagram iscommutative:

Hp;qxD .U; zU0/ j�

��

xAV

��

Hp;qxD .U/ ' Hp;q

N@ . zU /

KSV

��

Hp;qN@ .M/

KSM

��

��

Hn�p;n�qN@ . zU 0/�

i�

�� Hn�p;n�qN@ . zU/� �� Hn�p;n�q

N@ .M/�;

where i W zU 0 ,! zU denotes the inclusion.


Moreover, if M is compact Kähler, we have the following commutative diagram

Hp;qN@ .M/ ��

KSV

��

HpCq.M/

PV

��

HpCq.M/id��

PMo��

Hn�p;n�qN@ .M/ �� H2.n�p�q/.M/ H2.m�p�q/.M/�V

��

where V denotes the intersection product with V (cf. [25], 7.2).

Remark 6.9. Suppose S contains Sing.V / and V0 D V n S admits a neighborhoodzU0 inM with a holomorphic retraction r W zU0 ! V0 (cf. Remark 5.8). In this situation,to define �0 in a cochain � in Ap;q.U/, we only need to define it on V0, as we maytake the pull-back by r .

Again, let V be a variety of dimensionm in a complex manifoldM. First suppose Vis compact and let zU and U be as above. For a holomorphic vector bundle E overzU and a homogeneous symmetric polynomial ' of degree d, we have the character-istic class 'A.E/ in Hd;d

xD .U/ ' Hd;dN@ . zU /. We also have the class KSV .'A.E// in

Hm�d;m�dN@ . zU /�.Now let F be a singular distribution of rank r on M satisfying conditions (1)

and (2) in Section 2.3 and let FV be the singular distribution on V induced from F .Set S D .S.F /\V /[Sing.V/ and V0 D V nS , which is a submanifold ofM. Thereis a subbundle F0 of TM on M n S.F / defining F away from S.F /. We denoteby FV0

the subbundle F0jV0of T V0. Then it defines FV away from V. Let zU0, zU 0,

U, and zU D zU0 [ zU 0 be as above. We assume that zU0 is a Stein neighborhood of V0admitting a holomorphic retraction r W zU0 ! V0 (cf. Remark 6.9).

For a holomorphic vector bundle E over zU and a homogeneous symmetric poly-nomial ' of degree d, the characteristic class 'A.E/ is represented by the cocycle'A.zr�/ in Ad;d .U/, where zr� is a collection . Qr˛/ of connections, each zr˛ being aconnection forE on zU˛ . Note that it is sufficient if r0 is defined only on V0 D V nS ,since by our assumption, the bundleEj zU0

is isomorphism to r�EjV0and we may take

as zr0, the pull-back r�r0.Suppose there is an action of FV0

on EjV0and let r0 be an FV0

-connection forEjV0

. Then, if d > n � r , we have '.zr0/ D r�'.r0/ D 0, by Theorem 6.3, andthe above cocycle 'A.zr�/ is inAd;d .U; zU0/ and it defines a localization 'AzU 0

.E;FV /

in Hd;dxD .U; zU0/ of 'A.E/ in Hd;d

xD .U/. We denote the class AV .'AzU 0.E;FV // in

Hm�d;m�dxD .U0/� ' H

m�d;m�dN@ . zU 0/� by Res'A.FV ; EI zU 0/ and call it the residue

of 'A.E/ at zU 0 with respect to FV. The residue is a functional described as above. IfS has a finite number of connected components .S/, we break up U0 accordingly,

240 Tatsuo Suwa

U0 D [U0

, so that we have a decomposition

Hm�d;m�dxD .U0/ D

M

Hm�d;m�dxD .U0

/

and we have the residue Res'A.FV ; EI zU 0/ inHm�d;m�d

xD .U0/� ' Hm�d;m�d

N@ . zU 0/�

for each �. From Proposition 6.8, we have the following theorem.

Theorem 6.10. In the above situation, if V is compact, thenX

.i/�Res'A.FV ; EI zU 0/ D KSV .'

A.E// in Hm�d;m�dN@ . zU /�;

where i W zU 0,! zU denotes the inclusion.

6.4 An example

We take up again the singular distribution F on P3 considered in Example 4.3.Recall that it is an extension to P3 of the distribution defined by the 1-form

! D z dx C z dy � y dz in the affine space C3 D f.x; y; z/g. It leaves the planefz D 0g invariant. From ! ^ d! D �z dx ^ dy ^ dz, we see that ! defines a contactstructure on C3 with singular set fz D 0g (Martinet hypersurface). We will see that the(first) Atiyah class of the normal bundle of the (projectivized) Martinet hypersurfaceis localized at the singular set of the corresponding distribution.

The extended distribution F leaves the singular hypersurface V D f�0�3 D 0gin P3 invariant and we work on V. In fact F also leaves the hyperplane f�3 D 0ginvariant and this case is treated in [1].

Recall that the singular set S D S.F / of F has three irreducible components Si ,i D 1; 2; 3. Note that S2 is the singular set of V, in fact V is a union of two projectiveplanes crossing normally along S2. We also recall that P1 D .0 W 1 W 0 W 0/ is theintersection point of S1 and S2 and P2 D .0 W 0 W 1 W 0/ is the intersection point of S2and S3. There is a subbundle F0 of rank two of T P3 on P3 n S such that F D O.F0/

away from S .The normal bundle NV 0 of the regular part V 0 of V canonically extends to a line

bundle NV on V and then onto the whole P3, which is the line bundle LV associatedto the divisor V. Since P3 is compact Kähler, we know that the first Atiyah classa1.LV / inH 1;1

N@ .P3/ D H 2.P3/ ' C coincides with first Chern class c1.LV /, which

is 2c1.H3/.Let V0 D V nS . Then it is in V 0 and has two connected components V0;1 and V0;2,

given by �3 D 0 and �0 D 0, respectively. Let zU0;1 and zU0;2 be tubular neighborhoodsofV0;1 andV0;2, respectively, sufficiently small so that they are disjoint. The restrictionof the projection from the point .0 W 0 W 0 W 1/ to the plane f�3 D 0g gives aholomorphic retraction r1 W zU0;1 ! V0;1 and the restriction of the projection fromthe point .1 W 0 W 0 W 0/ to the plane f�0 D 0g gives a holomorphic retraction


r2 W zU0;2 ! V0;2. Thus if we set zU0 D zU0;1 [ zU0;2, it is a tubular neighborhoodof V0 with a holomorphic retraction r W zU0 ! V0. Let zU1, zU2 and zU3 be tubularneighborhoods of S1 n fP1g, S2 n fP1; P2g and S3 n fP2g, respectively, sufficientlysmall so that they are pairwise disjoint. Also let zU4 and zU5 be balls around P1 and P2,respectively, sufficiently small so that they are disjoint. Then U D f zU0; : : : ; zU5g is acovering of V and U0 D f zU1; : : : ; zU5g is that of S . We set zU D zU0 [ [ zU5 andzU 0 D zU1 [ [ zU5.

Let FV0D F0jV0

. We have NV0D LV jV0

and it admits an FV0-connection (The-

orem 6.5), which can be lifted to zU0. Thus we have the localization a1.LV j zU ;F / inH1;1xD .U; zU0/ of a1.LV j zU / in H 1;1

N@ . zU / and its residue in H 1;1xD .U0/�, which we will

compute. Note that the Chern class c1.LV j zU / is not localized in this context.Let r be an FV0

-connection for NV0on V0 and let r0 D r�r. Let r1; : : : ;r5

be connections for LV on zU1; : : : ; zU5. Let f zR0; : : : ; zR5g be a system of honey-combcells adapted to U whose boundaries are transverse to V, which will be given morespecifically below. We set Ri D zRi \ V.

We set � D .�i ; �ij / with �i D a1.ri / and �ij D a1.ri ;rj / (note that theform a1.ri ;rj ;rk/ is a .1;�1/-form and is zero). Let Œ � be a class in H 1;1

xD .U0/represented by D .i ; ij /. Then the residue is a functional assigning to Œ � theintegralZ

V

� ^ D5XiD1

� ZRi

�i ^ i CZR0i

�0i ^ i�

CX

1�i<j�5

� ZRij

�i ^ ij C �ij ^ j �ZR0ij

�0i ^ ij�:

(6.1)

Note that the possible combinations for .i; j / in the second term are .1; 4/, .2; 4/,.2; 5/, and .3; 5/. Since zUi , 1 � i � 5, are Stein, as in [1], Lemma 9.9, we see thateach class Œ � inH 1;1

xD .U0/ is represented by a cocycle of the form � D .0; �ij /, where

�ij D ij C �i � �j ; i D N@�i : (6.2)

With this the right hand side of (6.1) becomesX1�i<j�5

� ZRij

a1.ri / ^ �ij �ZR0ij

a1.r0;ri / ^ �ij�; (6.3)

the possible combinations of .i; j / being as above. Note that by symmetry, we needto consider only .1; 4/ and .2; 4/, the case for .3; 5/ being same as .1; 4/ and thecase for .2; 5/ as .2; 4/. Thus to express the residue, we only need to find a1.ri / anda1.r0;ri / for i D 1; 2.

242 Tatsuo Suwa

Proposition 6.11. By a suitable choice of connections,

a1.ri / D 0; 0 � i � 2;

a1.r0;r1/ D �p�12�

� dx1x1z1

� dz1z1

�;

a1.r0;r2/ D

8<ˆ:

p�12�

��1 � 1

z 1

�dx1x1C dz1

z1

�on V0;1;

p�12�

�.1 � z1/dy1

y1C dz1

�on V0;2:

Proof. Since the integrals are to be performed onRij andR0ij that are inV0, it sufficesto determine connections forNV on V0. Moreover, for .i; j / D .1; 4/ and .2; 4/, theyare in W .1/. Thus we fix r1 and r2 on the affine coordinate system .x1; y1; z1/.We have the exact sequence, for i D 1; 2,

0 �! T V0;i �! T P3jV0;i

��! NV0;i�! 0:

We may take �1 D �. @@y1/ as a frame ofNV0;1

and �2 D �. @@x1/ as a frame ofNV0;2

.

Letr1 be the connection trivial with respect to �1. Then we have a1.r1/ D 0 inW .1/.The normal bundleNV of the hypersurfaceV has a frame ��, which is the canonical

extension of the frame @@x1y1

on the regular part on W .1/. Let r2 be the connection

trivial with respect to ��. Then we have a1.r2/ D 0 in W .1/. Note that, on V0;1, wehave �� D 1

x 1�1 and on V0;2, �� D 1

y 1�2.

To compute the difference forms a1.ri ;rj /, we make the following observation(cf. Section 5.1). Let �i be the connection matrix (form, in this case) ofri with respectto some holomorphic frame � of NV0

. Then, since the �i ’s are of type .1; 0/,

a1.ri ;rj / D c1.ri ;rj / Dp�12�

.�j � �i /: (6.4)

Let r0 be an FV0-connection for NV0

and compute its connection forms withrespect to �1 and ��.

On W .1/, we may take the vector fields

v1 D x1 @

@x1C @

@z1and v2 D y1 @

@y1C z1 @

@z1

as generators of F . We set

u1;1 D v1jV0;1D x1 @

@x1C @

@z1; u2;1 D v2jV0;1

D z1 @

@z1;

u1;2 D v1jV0;2D @

@z1; u2;2 D v2jV0;2

D y1 @

@y1C z1 @

@z1:


In the above the restrictions are as sections. We find r0 on V0;1. The connectionform �0 of r0 with respect to �1 is of the form �0 D f dx1 C g dz1, as it is oftype .1; 0/. Then on the one hand we have r0.�1/.u1;1/ D .x1f C g/ �1 andr0.�1/.u2;1/ D z1g �1. On the other hand by definition,

r0.�1/.u1;1/ D ��hx1

@

@x1C @

@z1;@

@y1

ijV0

�D 0;

and

r0.�1/.u2;1/ D ��hy1

@

@y1C z1 @

@z1;@

@y1

ijV0

�D ��1:

Hence we get

�0 D dx1

x1z1� dz1z1;

which gives the expression for a1.r0;r1/ by (6.4).By similar computations we can find the connection form ��

0 of r0 with respectto ��, noting that, on V0;1, it is of the form �� D f dx1 C g dz1 and, on V0;2, it is ofthe form �� D f dy1 C g dz1.

Now we give the domains of integration more specifically. Let ı be a sufficientlysmall positive number and set

zR4 D f� 2 P3 j j�0j2 C j�2j2 C j�3j2 � ı2 j�1j2g;zR5 D f� 2 P3 j j�0j2 C j�1j2 C j�3j2 � ı2 j�2j2g;zR1 D f� 2 P3 j j�2j2 C j�3j2 � ı2 j�0j2g n Int zR4;zR2 D f� 2 P3 j j�0j2 C j�3j2 � ı2 j�1j2g \ f� 2 P3 j j�0j2 C j�3j2 � ı2j�2j2g

n .Int zR4 [ Int zR5/;zR3 D f� 2 P3 j j�0j2 C j�1j2 � ı2j�3j2g n Int zR5;

zR0 D zU0 n� 5[iD1

Int zRi [[ij

Int zRij�:

Now we express the domains R14, R014, R24 and R024 of integration explicitly.Note that the domains in question are all contained in W .1/ D f.x1; y1; z1/g. In thesequel, we let ı0 be a positive number with ı02 D ı2

1Cı2 .First,

R14 D fy1 D 0; jx1j2 C jz1j2 D ı2; jz1j � ıjx1jgD fy1 D 0; jx1j2 C jz1j2 D ı2; jz1j � ıı0g;

244 Tatsuo Suwa

which is a part of the boundary @R4 (in �3 D 0) ofR4 and has the orientation oppositeto that of @R4. We also have

R014 D f.x1; y1; z1/ j y1 D 0; jx1j D ı0; jz1j D ıı0g;which coincides with @R14 and is oriented so that dargx1 ^ dargz1 is positive.

Next, R24 is given by

R24 D fx1y1 D 0; jx1j2 C jy1j2 C jz1j2 D ı2; jx1j2 C jy1j2 � ı2jz1j2g:The set R024 has two connected components and can be expressed as a disjoint

union

R024 D fjx1j D ıı0; y1 D 0; jz1j D ı0g t fx1 D 0; jy1j D ıı0; jz1j D ı0g:The first part is oriented so that dargx1 ^ dargz1 is negative and the second part sothat dargy1 ^ dargz1 is negative. We have similar expressions for R035 and R025.

Now we consider the commutative diagram (cf. Proposition 6.8):

H1;1xD .U; zU0/ j�

��

AV

��

H1;1N@ . zU /

KSV

��

H1;1N@ .P3/ ' H 2.P3/

PV

��

�� H 2.P3/

o PP3

��

Id��

H1;1N@ . zU 0/� �� H 1;1

N@ . zU /� �� H 1;1N@ .P3/� ' H2.P3/ H4.P3/:�V

��

We denote by i� the composition of the first two maps of the second row and try tofind i�Resa1.F ; NV IS/. Recall that the map H 1;1

N@ .P3/ �! H1;1xD .U/ is induced by

7! .i ; ij / D .; 0/. Note that H 1;1N@ .P3/ ' C, which is generated by the class of

0 Dp�12�

@N@ log k�k2:On each affine coordinate system, we have

0 D 1

2�p�1

N@� Nxi dxi C Nyi dyi C Nzi dzi1C jxi j2 C jyi j2 C jzi j2

�:

Thus we may take, as �1, �2 and �4 in (6.2), the following forms.

• Since zU1 is in W .0/, we may set

�1 D 1

2�p�1

Nx dx C Ny dy C Nz dz1C jxj2 C jyj2 C jzj2 :

• Since zU2 is both in W .1/ and W .2/, for the sake of symmetry, we set

�2 D 1

2

1

2�p�1

� Nx1 dx1 C Ny1 dy1 C Nz1 dz11C jx1j2 C jy1j2 C jz1j2 C

Nx2 dx2 C Ny2 dy2 C Nz2 dz21C jx2j2 C jy2j2 C jz2j2

:


• Since zU4 is in W .1/, we may set

�4 D 1

2�p�1

Nx1 dx1 C Ny1 dy1 C Nz1 dz11C jx1j2 C jy1j2 C jz1j2 :

On V .0/1 , z D y1 D x2 D 0 and we compute

�14 D �1 � �4 Dp�12�

dx1

x1;

�24 D �2 � �4 D 1

2

p�12�

dz1

z1:

On V .0/2 , x1 D y2 D 0 and we compute

�24 D �2 � �4 D 1

2

p�12�

dz1

z1:

Using Proposition 6.11, we compute (we omit the constant�p�12�

�2)

a1.r0;r1/ ^ �14 D dz1

z1^ dx1x1

;

a1.r0;r2/ ^ �24 D

8<ˆ:1

2.1 � 1

z 1/dx1

x1^ dz1z1; on V0;1;

1

2.1 � z1/dy1

y1^ dz1z1; on V0;2:

Therefore i�Resa1.F ; NV IS/ is a functional which assigns to the canonical gen-erator Œ0� two times the value

�ZR014

a1.r0;r1/ ^ �14 �ZR024

a1.r0;r2/ ^ �24 D 1C 1

2C 1

2D 2

so that i�Resa1.F ; NV IS/.Œ0�/ D 4, as expected.The above computation appears to suggest that the residue Resa1.F ; NV IS/ is in

fact “ŒS1�C 2ŒS2�C ŒS3�”.

References

[1] M. Abate, F. Bracci, T. Suwa, and F. Tovena, Localization of Atiyah classes, to appear inRev. Math. Iberoam. 208, 228, 234, 235, 240, 241

[2] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic self-maps, Ann. ofMath. (2) 159 (2004), 819–864. 207

246 Tatsuo Suwa

[3] M. Abate, F. Bracci, and F. Tovena, Index theorems for holomorphic maps and foliations,Indiana Univ. Math. J. 57 (2008), 2999–3048. 207, 235

[4] M. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 84(1957), 181–207. 208

[5] M. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology 1 (1961),25–45. 219

[6] P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geometry 7(1972), 279–342. 208, 212, 215, 222

[7] P. Baum,W. Fulton, and R. MacPherson, Riemann–Roch for singular varieties, Inst. HautesÉtudes Sci. Publ. Math. 45 (1975), 101–145. 219

[8] R. Bott, Lectures on characteristic classes and foliations, Notes by L. Conlon, with twoappendices by J. Stasheff, in Lectures on algebraic and differential topology. Delivered atthe Second Latin American School in Mathematics, Mexico City, July 1971. Dedicated tothe memory of Heinz Hopf, ed. by S. Gitler, Lecture Notes in Mathematics 279, Springer,Berlin and New York 1972, 1–94. 212, 213

[9] R. Bott and L. Tu, Differential forms in algebraic topology, Graduate Texts in Mathemat-ics 82, Springer, New York and Berlin 1982. 215

[10] J.-P. Brasselet, Définition combinatoire des homomorphismes de Poincaré, Alexanderet Thom pour une pseudo-variété, in Caractéristique d’Euler–Poincaré. E.N.S. Seminar,1978–1979. ed. by J.-L. Verdier, Astérisque 82-83, Société Mathématique de France, Paris1981, 71–91. 220

[11] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vectorfields, Ann. of Math. (2) 115 (1982), 579–595. 208

[12] R. F. Harvey and H. B. Jr. Lawson, A theory of characteristic currents associated with asingular connection, Astérisque 213, Société Mathématique de France 1993. 219

[13] F. Hirzebruch, Topological methods in algebraic geometry, Die Grundlehren der Mathe-matischen Wissenschaften 131, Springer-Verlag New York, New York 1966. 219

[14] B. Iversen, Local Chern classes, Ann. Scient. Éc. Norm. Sup. 9 (1976), 155–169. 219

[15] T. Izawa, Residues of codimension one singular holomorphic distributions, Bull. Braz.Math. Soc. (N.S.) 39 (2008), 401–416. 208, 224

[16] D. Lehmann, Systèmes d’alvéoles et intégration sur le complexe de Cech–de Rham, Pub-lications de l’IRMA, 23, No VI, Université de Lille I, Lille 1991. 215, 231

[17] D. Lehmann, Résidus des sous-variétés invariantes d’un feuilletage singulier, Ann. Inst.Fourier (Grenoble) 41 (1991), 211–258. 208, 235

[18] D. Lehmann and T. Suwa, Residues of holomorphic vector fields relative to singularinvariant subvarieties, J. Differential Geom. 42 (1995), 165–192. 208, 235

[19] J. Milnor and J. Stasheff, Characteristic classes, Annals of Mathematics Studies 76,Princeton University Press and University of Tokyo Press, Princeton and Tokyo 1974.212

[20] R. S. Mol, Classes polaires associées aux distributions holomorphes de sous-espacestangents, Bull. Braz. Math. Soc. (N.S.) 37 (2006), 29–48. 208

[21] T. Suwa, Residues of complex analytic foliation singularities, J. Math. Soc. Japan 36(1984), 37–45. 208, 223, 224, 225


[22] T. Suwa, Indices of vector fields and residues of singular holomorphic foliations,ActualitésMathématiques, Hermann, Paris 1998. 207, 210, 212, 213, 214, 215, 231, 237

[23] T. Suwa, Characteristic classes of coherent sheaves on singular varieties, in Singularities–Sapporo 1998. Proceedings of the International Symposium on Singularities in Geometryand Topology held at Hokkaido University, Sapporo, July 6–10, 1998, ed. by J.-P. Bras-selet and T. Suwa, Advanced Studies in Pure Mathematics 29, Kinokuniya, Tokyo 2000,279–297. 208, 217, 220

[24] T. Suwa, Residues of Chern classes on singular varieties, Singularités Franco–Japonaises.Papers from the 2 nd Franco–Japanese Singularity Conference held in Marseill–Luminy,September 9–13, 2002, ed. by J.-P. Brasselet and T. Suwa, Séminaires et Congrés 10,Société Mathématique de France, Paris 2005, 265–285. 225

[25] T. Suwa, Residue theoretical approach to intersection theory, in Real and complex singu-larities. Papers from the 9 th International Workshop held at the University of São Paulo,São Carlos, July 23–28, 2006., ed. by M. J. Saia and J. Seade, Contemporary Mathemat-ics 459, American Mathematical Society, Providence, RI, 2008, 207–261. 213, 215, 220,239

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Two birational invariants in statistical learning theory

Sumio Watanabe

Precision and Intelligence Laboratory, Tokyo Institute of Technology4259 Nagatsuta, Midori-ku, Yokohama, 226-8503 Japan


Abstract. This paper introduces a recent advance in the research between algebraic geometryand statistical learning theory. A lot of statistical models used in information science containsingularities in their parameter spaces, to which the conventional theory can not be applied. Thestatistical foundation of singular models was been left unknown, because no mathematical basecould be found. However, recently new theory was constructed based on algebraic geometryand algebraic analysis. In this paper, we show that statistical estimation process is determinedby two birational invariants, the real log canonical threshold and the singular fluctuation. As aresult, a new formula is derived, which enables us to estimate the generalization error withoutany knowledge of the information source. In the discussion, a relation between mathematics andthe real world is introduced to pure mathematicians.

1 Introduction

The purpose of this paper is to introduce a singularity problem in statistics to re-searchers of mathematics. A lot of statistical models used in artificial intelligence,information science, neuroscience, natural language processing, and bioinformaticscontain singularities in their parameter spaces. A singularity is most important in suchmodels because it corresponds to the knowledge to be discovered. In other words,singularities determine the statistical estimation and statistical hypothesis test, hencewe need singular theory on which new statistical theory will be established.

Let .�;B; P / be a probability space andX W �! RN be a random variable whoseprobability distribution is represented by q.x/dx. Let X1; X2; :::; Xn be independentrandom variables which are subject to the same probability distribution as X . Theexpectation operators EX Œ � and EŒ � are defined by

EX ŒF .X/� DZF.x/q.x/dx;

EŒF .X1; X2; : : : ; Xn/� DZF.x1; x2; :::; xn/

nYiD1

q.xi /dxi ;


250 Sumio Watanabe

for functions F.x/ and F.x1; x2; :::; xn/. LetW be a subset of Rd. A statistical modelis defined by a pair .p.x j w/; '.w//, where p.x j w/ is a probability density functionof x 2 RN for a given parameter w 2 W � Rd and '.w/dw is a probability densityfunction on W. The expectation operator Ew Œ � is defined by

Ew ŒF .w/� D

ZF.w/

nYiD1

p.Xi jw/ˇ'.w/dwZ nYiD1

p.Xi jw/ˇ'.w/dw;

where 0 < ˇ <1 is a constant. Note that Ew Œ � depends on X1; X2; : : : ; Xn, henceEw ŒF .w/� is not a constant but a random variable. The distribution-valued randomvariable p�.x/ Ew Œp.x j w/� is called a predictive distribution.

Main Problem. Let G and T be real valued random variables defined by

G D �EX Œlogp�.X/�;

T D �1n

nXiD1

logp�.Xi /:

The two random variables G and T are called generalization and training error,respectively. We ask what mathematical properties of the triple

.q.x/; p.x j w/; '.w//determine the asymptotic behavior of G and T as n!1.

In this paper, we show that the above problem has a close relation to singularitytheory. On the other hand, it has direct applications in statistics. In Section 8, the reasonwhy such a problem is important in statistics is explained.

Example 1 (Regular case). Assume that N D d , W D Rd and that

p.x j w/ D 1

.2�/N=2exp

�� 12kx � wk2

�;

q.x/ D p.x j 0/;

'.w/ D 1

.2�/d=2exp

�� 12kwk2

�:

In this case, the set of parameters which make p.xjw/ be equal to q.x/ is trivial,

fw 2 W Ip.x j w/ D q.x/g D f0g:

Two birational invariants in statistical learning theory 251

Such a case can be analyzed by the conventional statistical method. The entropy Sand the empirical entropy Sn of q.x/ respectively defined by

S D �EX Œlog q.X/�;

Sn D �1n

nXiD1

log q.Xi /:

Then EŒSn� D S , and

G D S C 1

2n

1pn

nXiD1

Xi

2 COp� 1n2

�;

T D Sn � 1

2n

1pn

nXiD1

Xi

2 COp� 1

n3=2

�;

where Yn D Op.1=n˛/ if and only if n˛Yn converges in law to some random variablewhen n!1. Moreover, it is easy to show that

limn!1 EŒn.G � S/� D d

2;

limn!1 EŒn.T � Sn/� D �d

2:

By this example, we have three conjectures.

(1) The main problem of this paper has a relation to the central limit theorem.

(2) The asymptotic behaviors of G and T are determined by some invariants such asthe dimension of the triple .q.x/; p.x j w/; '.w//.

(3) There is some symmetry between EŒG� and EŒT �.

Example 2 (Singularities in statistics). Assume that N D 2, W D R4, x D .x1; x2/,and w D .a; b; c; d/ and that

p.xjw/ D 1

.2�/1=2q0.x1/ exp

�� 12kx2 � a sin.bx1/ � c sin.dx1/k2

�;

q.x/ D p.x j 0/;

'.a; b; c; d/ D 1

.2�/2exp

�� a

2 C b2 C c2 C d22

�;

where q0.x1/ is an arbitrary probability density function. In this case, the set of pa-rameters which make p.x j w/ be equal to q.x/ is a real algebraic set,

fw 2 R4Ip.x j w/ D q.x/g D fw 2 R4I ab C cd D ab3 C cd3 D 0g;

252 Sumio Watanabe

which contains a singularity at the origin. Although the same cases often occur instatistics, no method has been found by whichG and T can be analyzed. In this paper,we show that the asymptotic behaviors ofG andT are determined by the two birationalinvariants of singularities.

Acknowledgment. The author would like to thank a lot of mathematicians for theirkind teaching mathematical concepts. Without singularity theory, our research resultscould not be obtained. Modern mathematics is necessary even in information science.

2 Preparation

In this section, we prepare mathematical definitions and assumptions, and explain thesingularity problem in statistics more precisely. The function L.w/ is defined by

L.w/ D �EX Œlogp.X jw/�:By using the definition S D �EX Œlog q.X/�, it follows that L.w/ � S for arbitraryw 2 W. It is assumed that there exists w0 2 W o that minimizes L.w/, where W o isthe largest open set that is contained inW. A nonnegative functionK.w/ is defined by

K.w/ D L.w/ � L.w0/:Then the set W0 D fw 2 W IK.w/ D 0g is not the empty set. It is assumed that, forarbitrary w 2 W0, p.x j w/ is the same probability distribution, which is referred toas p0.x/. Also we use notations,

L0 D L.w0/ D �EX Œlogp0.X/�:

To study the problem, we need some assumptions.

Assumptions. LetW be a compact set and assume that the largest open set containedinW is not the empty set. It is defined by several analytic functions�1.w/; �2.w/; : : : ;�l .w/,

W D fw 2 Rd I�1.w/ � 0; �2.w/ � 0; : : : ; �l.w/ � 0g:Let W � be an open set such that W � W � � Rd . Let s � 3 be a constant and

Ls.q/ D°f I kf k

� Zjf .x/jsq.x/dx

�1=s<1

±be a Banach space. The function f .x;w/ is defined by

f .x;w/ D logp0.x/

p.xjw/:

Then it follows thatK.w/ D EX Œf .X;w/�:


It is assumed that the functionw 7! f . jw/ is an analytic function fromW � toLs.q/and that there exists � > 0 such thatZ

. supw2W

jf .x;w/j2/. supK.w/<�

p.xjw//dx <1:

Definition 1. (1) The probability density function q.x/ is said to be realizable byp.x j w/ if p0.x/ D q.x/. If otherwise, it is called unrealizable.

(2) IfW0 consists of a single element w0 and Hessian matrix r2L.w0/ is positivedefinite, then q.x/ is said to be regular for p.xjw/. If otherwise, it is called singular.

We define a function,

Kn.w/ D 1

n

nXiD1

f .Xi ; w/:

It is immediately derived that Ew Œ � is rewritten as

Ew ŒF .w/� D

ZF.w/ exp.�ˇnKn.w//'.w/dwZ

exp.�ˇnKn.w//'.w/dw;

and

G D L0 � EX Œlog Ew Œexp.�f .X;w//��;

T D Ln � 1n

PniD1 log Ew Œexp.�f .Xi ; w//�;

where Ln is defined by

Ln D �1n

nXiD1

logp0.Xi /:

Note that EŒLn� D L0. Two random variablesG �L0 and T �Ln are determined bythe function f .x;w/.

Singularity problem in statistics. In statistics, we ask the asymptotic behaviors ofG and T when W0 contains singularities. If q.x/ is singular for p.x j w/, thenW0 D fw 2 W IK.w/ D 0g is a real analytic set with singularities, henceKn.w/ cannot be approximated by any quadratic form. The function exp.�ˇnKn.w// can not bereplaced by any Gaussian function. The conventional saddle point approximation inwhichKn.w/ is approximated by a quadratic form in the neighborhood of its minimumpoint can not be applied. This is the main reason why singularity theory is necessary.

254 Sumio Watanabe

3 Real log canonical threshold

Since K.w/ � 0 is a real analytic function on an open set W � � Rd , Hironakaresolution theorem [11] and [4] ensures that there exists a set .M�; g/, where M� isa d -dimensional real analytic manifold and g is a proper analytic function

g W M� ! W �;

such that

K.g.u// D u2k dYjD1

u2kj

j ;

jg0.u/j'.g.u// D b.u/juhj b.u/dYjD1juhj

j j;

where k D .k1; k2; :::; kd / and h D .h1; h2; :::; hd / are multi-indices made of non-negative integers, jg0.u/j is the Jacobian determinant of w D g.u/, and b.u/ is afunction which satisfies b.u/ > 0. Note that a function g is said to be proper if andonly if the inverse of a compact set is also compact. We assumed thatW is a compactset, hence M g�1.W / is also compact subset of the manifold M�.

Definition 2. The real log canonical threshold � is defined by

� D minM

min1�j�d

�hj C 12kj

�;

where, if kj D 0, then .hj C 1/=kj D 1. The multiplicity m is defined as themaximum number of elements in the set made of j that attains the above minimum,in other words,

m D maxM

#nj I� D

�hj C 12kj

�o:

where # shows the number of elements of a set.

For a given analytic functionK.w/, there are infinite sets of resolution of singular-ities .M; g/. If a value which is defined using a resolution set .M; g/ does not dependon the choice of the set, then it is called a birational invariant.

Lemma 3.1. The real log canonical threshold is a birational invariant.

Short proof. This proof was originally found in [8] and [4]. A zeta function definedin Re.z/ > 0

�.z/ DZK.w/z'.w/dw


is a holomorphic function. By using resolution of singularities, it can be analyticallycontinued to a unique meromorphic function on the entire complex plane. Its poles areall real, negative, and rational numbers. The largest pole of �.z/ is equal to .��/ andits order is m. Therefore the real log canonical threshold is a birational invariant.

By using the above zeta function �.z/, we can show that there exists a SchwartzdistributionD.u/ on M whose support is contained in the set fu 2MIK.g.u// D 0g,such that the Laurent expansion of z with the topology of the Schwartz distributionholds,

u2kz juhj b.u/ D .m � 1/Š.z C �/m D.u/C :

Then, by using the inverse Mellin transform, the following convergence of the Schwartzdistribution holds when n!1

n�1

.logn/m�1 ı� tn� u2k

�juhjb.u/ �! t�1D.u/

for an arbitrary t > 0. Therefore, we obtain the asymptotic expansion [27] and [28],for n!1

� logZ

exp.�nK.w//'.w/dw D � logn � .m � 1/ log log nCO.1/;

where O.1/ is a bounded function of n.

4 Singular fluctuation

Definition 3. If there exists a constant A > 0 such that, for an arbitrary w 2 WEX Œf .X;w/� � AEX Œf .X;w/

2�;

then f .X;w/ is said to have a relatively finite variance.

Since W is compact and K.w/ D EX Œf .X;w/�, this definition is ensured if itholds in the region W�

W� D fw 2 W IK.w/ < �g;where � > 0 is sufficiently small constant. The following lemmas can be provedimmediately.

Lemma 4.1. If q.x/ is realizable by p.x j w/, then f .X;w/ has a relatively finitevariance.

Lemma 4.2. If q.x/ is regular for p.x j w/, then f .X;w/ has a relatively finitevariance.

256 Sumio Watanabe

If q.x/ is unrealizable by and singular for p.x j w/, then f .X;w/ may and maynot have a relatively finite variance. The case when f .X;w/ does not have a relativelyfinite variance is studied in Section 6. In this section, we study the case when f .X;w/has a relatively finite variance.

By using the resolution of singularities, K.g.u// is a normal crossing function,K.g.u// D u2k . By using

K.g.u// D EX Œf .X; g.u//� � AEX Œf .X; g.u//2�;

there exists an Ls.q/-valued analytic function a.x; u/ such that

f .x; g.u// D a.x; u/uk:Therefore EX Œa.X; u/� D uk . Remark that such a function a.x; u/ is well-definedon M, whereas a.x; g�1.w// is not on W in general. This is one of the reasons whyresolution of singularities is necessary in statistics. Let us define a random process�n.u/ on M by

�n.u/ D 1pn

nXiD1fa.Xi ; u/ � ukg:

Let B.M/ be a Banach space defined by

B.M/ D ff .u/ is continuous I kuk supu2M

jf .u/j <1g:

Since M is a compact set,B.M/ is a Polish space; in other words, it is a complete andseparable metric space. By using a.x; u/ is an Ls.q/-valued analytic function of u, itfollows that the set of random processes f�ngnD1;2;::: is uniformly tight inB.M/. Thenby applying Prohorov’s theorem, �n.u/ converges in law to a tight Gaussian process�.u/ on B.M/, by using the uniqueness of a Gaussian process that satisfies

E� Œ�.u/� D 0;E� Œ�.u/�.u

0/� D EX Œa.X; u/a.X; u0/� � EX Œa.X; u/�EX Œa.X; u

0/�;

where E� Œ � shows the expectation value over the random process �.u/. Such a process�n.u/ is called an empirical process or it is said that �n.u/ satisfies the central limittheorem on the Banach space.

An expectation operation h i on M is defined by

hF.u; t/i D

Z 1

0

dt

ZD.u/duF.u; t/t�1 exp.�ˇt C ˇpt�.u//Z 1

0

dt

ZD.u/du t�1 exp.�ˇt C ˇpt�.u//

:

The value hF.u; t/i is a functional of �.u/, hence it is also a random variable, but itdoes not depend on n.


Definition 4. The singular fluctuation � > 0 is defined by

� D ˇ

2E�EX Œhta.X; u/2i � h

pta.X; u/i2�:

In general, � depends on ˇ.

Lemma 4.3. The singular fluctuation is a birational invariant.

Proof. Let us define a random variable V by

V DnXiD1fEw Œ.logp.Xi jw//2� � Ew Œlogp.Xi jw/�2g:

Then we can prove that V converges in law to a random variable by the same way asthe proof of the following theorem and it satisfies

� D limn!1

ˇ

2EŒV �;

which shows that � is a birational invariant.

Then we have the following theorem.

Theorem 4.4 (Main theorem). Assume that f .X;w/ has a relatively finite variance.Then both random variables n.G � L0/ and n.T � Ln/ converge in law. Also thefollowing convergences hold:

limn!1 EŒn.G � L0/� D � � �

ˇC �;

limn!1 EŒn.T � Ln/� D � � �

ˇ� �:

Outline of proof. For the complete version of the proof, see [28] and [29], and [32].Firstly, we prove the convergences in law. We define a distribution Yn.w/dw by

Yn.w/dw exp.�nˇKn.w// '.w/ dw:Then by using the fact that

Kn.g.u// D u2k � 1pn�n.u/ u

k;

258 Sumio Watanabe

we have

Yn.w/dw D Yn.g.u//jg0.u/jduD e�nˇu2kCp

nˇuk�n.u/uhb.u/du

DZ 1

0

dt ı.t � n u2k/uh e�ˇtCptˇ �n.u/ b.u/du

Š .logn/m�1

nD.u/du

Z 1

0

dt t�1e�ˇtCptˇ�.u/;

whereŠ shows the asymptotic expansion of Schwartz distribution as n!1. There-fore the convergence in law holds,

Ew Œ.pnf .x;w//s� �! h.pt a.x; u//si

for s � 0. Here the relations of the parameters are

w D g.u/;t D nK.w/ D nu2k;

f .x; w/ D a.x; u/uk:The generalization error is

G D EXh� log Ew Œ1 � f .X;w/C 1

2f .X;w/2�

iC op

�1n

�D EXEw Œf .X;w/� � 1

2EXEw Œf .X;w/

2�C 1

2EX ŒEw Œf .X;w/�

2�C op�1n

�;

where op.1=n/ is a random variable which satisfies the convergence in probability,n op.1=n/ ! 0. Hence G converges in law. The convergence of T in law can beproved by the same way.

Secondly, let us show the convergences of expectation values. The Gaussian process�.u/ can be represented by

�.u/ D1XjD1

cj .u/gj

where fgj g are independent random variables, each of which gj is subject to thestandard normal distribution. Then

E� Œ�.u/�.u0/� D

1XjD1

cj .u/cj .u0/:

Let us introduce a generating function

Fn.˛/ D �EEXh

logZ

exp.� f .X;w/ � ˇnKn.w//'.w/dwi:


Then

EŒG� D L0 C Fn.1/ � Fn.0/;EŒT � D L0 C Fn�1.1C ˇ/ � Fn�1.ˇ/;

EŒV � D �n F 00n�1.ˇ/:

We need the asymptotic behavior of

Zn.s/ DZf .x;w/s exp.�ˇnHn.w//'.w/dw;

where s � 0 is a real value. For example,

F 0n.0/ D E

hZn.1/Zn.0/

i; (4.1)

F 00n .0/ D �E

hZn.2/Zn.0/

iC E

hZn.1/Zn.0/

i2: (4.2)

By the same method as above,

Zn.s/ Š .logn/m�1

nCs=2� Z

D.u; t/t s=2 exp.ˇpt�.u//

�;

whereR

D.u; t/ is defined by the integration over the manifold,ZD.u; t/ D

Z 1

0

dt

ZduD.u/ t�1 exp.�ˇt/:

Let us define

yZ.q; r; s/ DZ

D.u; t/ �.u/q t r=2 a.x; u/s exp.ˇpt�.u//:

Then

Zn.s/ Š .logn/m�1

nCs=2 yZ.0; s; s/: (4.3)

Firstly, since EX Œa.X; u/� D uk ,

EX Œ yZ.0; 1; 1/� D yZ.0; 2; 0/:Secondly, by using the partial integration of tZ 1

0

dt te�ˇtCˇpt�.u/ D �

ˇ

Z 1

0

dt t�1e�ˇtCˇpt�.u/

C 1

2

Z 1

0

dt t�1=2�.u/e�ˇtCˇpt�.u/;

it follows thatyZ.0; 2; 0/ D �

ˇyZ.0; 0; 0/C 1

2yZ.1; 1; 0/:

260 Sumio Watanabe

And lastly, by using the partial integration over the Gaussian process �.u/,

E�h yZ.1; 1; 0/yZ.0; 0; 0/

iD E�

"ZD.u; t/

� 1XjD1

cj .u/gj

� t1=2eˇpt�.u/Z

D.u0; t 0/eˇpt 0�.u0/

#

D E�

"ZD.u; t/

� 1XjD1

cj .u/@

@gj

� t1=2eˇpt�.u/Z

D.u0; t 0/eˇpt 0�.u0/

#

D ˇEXE�h yZ.0; 2; 2/yZ.0; 0; 0/

i� ˇEXE�

h yZ.0; 1; 1/yZ.0; 0; 0/

i2D 2�;

where we used E� Œ�.u/�.u0/� D EX Œa.X; u/a.X; u0/� on the set fuIK.g.u// D 0g.

Then by using eq. (4.3),

limn!1 sup

0�˛�1CˇjF .3/n .˛/jn D 0;

limn!1 jF

0n.0/ � F 0

n�1.0/jn D 0;

limn!1 jF

00n .0/ � F 00

n�1.0/jn D 0:At last, by (4.1) and (4.2),

EŒG� D L0 C F 0n.0/C

1

2F 00n .0/C o

�1n

�;

EŒT � D L0 C F 0n.0/C

2ˇ C 12

F 00n .0/C o

�1n

�;

EŒV � D �nF 00n .0/C o.1/;

which completes the theorem.

In the special case ˇ D 1, the predictive distribution p�.x/ is called Bayes esti-mation in statistics. The average expectation value of the generalization error is givenby

EŒG� D L0 C �

nC o

�1n

�;

where o.1=n/ is a smaller order term than 1=n. This function shows the accuracyof Bayes estimation, which is called the learning curve in statistical learning theory.The learning curve is in inverse proportion to the number of random samples, and itscoefficient is equal to the real log canonical threshold. Therefore, the learning curve isdetermined by the singularities. From Theorem 4.4, by eliminating � and �, we obtainthe following formula.


Corollary 4.5 (Equation of state in statistical estimation). Assume that f .X;w/ hasa relatively finite variance. The following formula holds.

EŒG� D EŒT �C ˇ

nEŒV �C o

�1n

�: (4.4)

This equation holds for arbitrary triple .q.x/; p.x j w/; '.w//. The method howto apply this formula to the practical problems is explained in Section 8.

5 Two birational invariants

Theorem 4.4 shows that the learning process is determined by the two birationalinvariants. In this section, we discuss the mathematical properties of them.

5.1 Real log canonical threshold

The original concept of the real log canonical threshold was found in the problem of theasymptotic expansion [8] of the singular Schwartz distribution ı.t�f .x// as t ! 0 fora polynomialf .x/ that satisfiesrf .x/ D 0. Remark that, ifrf .x/ D 0 onf .x/ D 0,then ı.f .x// is not well-defined. It was proved that resolution of singularities givesa general solution to this problem [4]. The Bernstein–Sato polynomial was found tostudy this problem from the algebraic analysis point of view; see [5], [12], [19], [17],and [20]. The log canonical threshold plays an important role in higher dimensionalalgebraic geometry [13]. Its algebraic property was studied in [14], [15], [21], and [10].Application to the oscillating integral was proposed in [26]. The relation between thereal log canonical threshold and Bayes integral was found [27], which was applied toseveral statistical models; see [6], [28], [3], [33], and [34].

It is well known that the log canonical threshold shows the relative quantity oftwo algebraic varieties W and W0, or it is defined for a pair .W;W0/. In statisticallearning theory, they correspond to the set of parametersW and the set of the optimalparametersW0. In this paper, we have shown that the learning curve is determined bythe mathematical relation between W and W0.

5.2 Singular fluctuation

From the theoretical point of view, it is still unknown what singular fluctuation is. Fora given function f .x;w/, its average and covariance are defined by

K.w/ D EX Œf .X;w/�;

�.w;w0/ D EX Œf .X;w/f .X;w0/�:

262 Sumio Watanabe

If two sets of triples .qi .x/; pi .x j w/; 'i .w// .i D 1; 2/ have the same K.w/ and�.w;w0/, then they have the same real log canonical threshold and singular fluctuation.It is well known that the central limit theorem is determined by the average andthe covariance of the original random variable. Singular fluctuation may express thevariance information on the functional space.

6 Case study

In this section, we show some examples of the real log canonical thresholds andsingular fluctuations.

6.1 Regular and realizable case

If q.x/ is regular for and realizable by p.xjw/, and if '.w/ > 0 atW0, then � D � Dd=2, where d is the dimension of the parameter space. In this case, � does not dependon ˇ. Example 1 is a special case.

6.2 Regular and unrealizable case

If q.x/ is regular for but unrealizable by p.xjw/, and if '.w/ > 0 at W0, then

� D d

2;

� D 1

2tr.IJ�1/;

can be proven [30], where I , J are d � d matrices respectively defined by

I DZrw logp.x j w0/rw logp.x j w0/q.x/dx;

J D �Zr2w logp.x j w/q.x/dx:

In this case, � does not depend on ˇ. Note that there are both cases � > � and � < �.

6.3 Singular and realizable case

In order to construct the model selection algorithm or the hypothesis testing in statistics,this case is most important.


In general, � and � depend on q.x/ and p.x j w/. Moreover, � depends on ˇ. Forexample, w D fak; bkI k D 1; 2; : : : ;H g

p.x; y j w/ D 1

.2�/1=2q0.x/ exp

�� 12

�y �

HXkD1

ak sin.bkx/�2�

and q.x/ D p.x; yj0/, then the real log canonical threshold is equal to that of

K0.w/ DHXhD1

� HXkD1

ak.bk/2h�1�2:

In this case, a resolution of singularities was found [3] and

� D ŒpH�2 C ŒpH�C 14ŒpH�C 2 ;

where Œx� shows the maximum integer that is not larger than x. Example 2 is a specialcase of this. Hence the real log canonical threshold of Example 2 is equal to 2=3.

In practical applications, statistical models have the same type function asK0.w/.In other words, its Newton diagram is degenerate and its complexity grows when thedimension of the parameter becomes large. In the above case, it follows that

� ŠpH

4;

whenH is sufficiently large. It is the future study to clarify the behavior of the real logcanonical threshold when the complexity and the dimension of the polynomial tendsto infinity. This problem might have relation to random matrix theory.

6.4 Singular and unrealizable case

If f .X;w/ does not have a relatively finite variance, then there exists an example, inwhich Theorem 4.4 does not hold. For example,

q.x; y/ D 1

2�exp

�� 12.x2 C y2/

�;

p.x; yja/ D 1

2�exp

�� 12f.x � a/2 C .y �

pa4 � a2 C 1/2g

�;

where a 2 R1 is the parameter. It is easy to check that f .X;w/ does not have arelatively finite variance. By the direct calculation, we have

EŒG� Š L0 C�12� 1ˇ

� Qn2=3

;

EŒT � Š L0 ��12C 1

ˇ

� Qn2=3

;

264 Sumio Watanabe

where Q D 27=6p2��.7

6/, therefore Theorem 4.4 does not hold [32]. However, the

equation of state in statistical estimation holds,

EŒG� D EŒT �C ˇ

nEŒV �C o

� 1

n2=3

�:

The equation of state in statistical estimation may hold more universally.

7 Problems in probability theory

In this section, we discuss the mathematical problem in probability theory. From theviewpoint of probability theory, this paper is based on two strong assumptions. Theformer is thatw 7! f .x;w/ is anLs.q/-valued analytic function, and the latter is thatthe set of parameters W is compact.

The assumption thatf .x;w/ is analytic is necessary because the resolution theoremis employed. If it is not an analytic function, it is unknown whether the theorem canbe generalized or not. It is the future study to generalize the results of this paper tonon-analytic functions.

The assumption that W is a compact set is necessary to show that B.M/ is aPolish space and that �n.u/ is a uniformly tight process. The convergence in law�n.u/ ! �.u/ was proved using Prohorov’s theorem. If W is not compact, then M

is not compact in general, hence the convergence in law of �n.u/ can not be provedin general. It is also the future study to prove the convergences in law of G and Twithout using the convergence in law of �n.u/.

8 Application to statistics

In this section, the application of singularity theory to statistics is introduced to re-searchers of mathematics. Readers who are not interested in applications can skip thissection.

8.1 Background of the problem

Firstly, we discuss the background of the problem. The probability distribution q.x/dxfrom which random variables X1; X2; :::; Xn are taken is called a true distribution ora true information source. In real world problems, the true distribution is unknownin general, and only a set of sample values of X1; X2; : : : ; Xn can be observed. Toestimate the unknown true distribution, a pair .p.x j w/; '.w// is employed, whichis called a probabilistic model, a statistical model, or a learning machine. One of the


main purposes of statistical learning theory is to establish the method how to evaluatethe probabilistic model compared to the unknown true distribution.

The predictive distribution p�.x/ Ew Œp.xjw/� is the estimated probability den-sity function for a given set of samples and a given model .p.xjw/; '.w//. In practicalapplications, although people never know the true distribution q.x/, they want to eval-uate how accurately p�.x/ approximates q.x/. It seems impossible to make such anevaluation.

The random variable G is a quantitative measure of the accuracy of p�.x/. Ifp�.x/ D q.x/, the generalization error G takes the minimum value which is equal tothe entropy of q.x/. If otherwise, G is larger than the entropy. Hence the smaller Gmeans that p�.x/ is more accurate for q.x/. However, in order to obtainG, we have tocalculate EX Œ � but this integration can not be performed without the true distributionq.x/. Instead of G, we can calculate the training error T, using only samples and themodel.

We ask whether we can estimate G from T. The equation of state in statisticalestimation, Corollary 4.5, EŒG� D EŒT �C.ˇ=n/EŒV � claims that EŒG� can be obtainedfrom EŒT � and EŒV �, where T and V can be calculated only samples and the modelwithout any knowledge of the true distribution. Since this formula holds for an arbitraryset .q.x/; p.x j w/; '.w//, one can evaluate the statistical model .p.x j w/; '.w//.

8.2 New results

Secondly, let us discuss what the original points of this paper are.If a true distribution is regular for and realizable by a statistical model, then the

equation of state in statistical estimation is equivalent to the AIC (Akaike informationcriterion); see [1]. If a true distribution is regular for and unrealizable by a statisticalmodel, then it is equivalent to TIC (Takeuchi information criterion). Therefore, if atrue distribution is regular for a statistical model, then the obtained result of this papercontains the conventional results.

If a true distribution is singular for a statistical model, there has been no formulaby which we can estimate the generalization error. Hence the equation of state is thefirst result by which we can estimate the generalization error in singular cases.

Table 1 shows the mathematical difference between the regular and singular sta-tistical theory. The regular theory studies the probability distribution on the parameterspace, whereas the singular theory does that on the functional space.

9 Conclusion

In this paper, we introduced two birational invariants by which we can estimate thegeneralization error without any knowledge about the true distribution. Singularitytheory is essential to statistical learning theory.

266 Sumio Watanabe

Table 1. Regular and singular

Regular Singularalgebra linear algebra ring and ideal

geometry differential algebraicanalysis real-valued function-valued

probability theory central limit theorem empirical processFisher inform. matrix positive definite semi-positive def.

Cramer–Rao inequality holds no meaningMaximum likelihood asymptotic normal singular

Bayes a posteriori asymptotic normal singularlog canonical threshold d=2 �

singular fluctuation d=2 �

Bayes marginal .d=2/ log n � logninformation criterion AIC, TIC equation of state

exponential mixturesexamples polynomial regression neural networks

linear prediction hidden Markov

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Frobenius morphisms of noncommutative blowups

Takehiko Yasuda�

Department of MathematicsGraduate School of Science, Osaka University

Toyonaka, Osaka 560-0043, Japanemail: [email protected]

Abstract. We define the Frobenius morphism of certain class of noncommutative blowups inpositive characteristic. Thanks to a nice property of the class, the defined morphism is flat.Therefore we say that the noncommutative blowups in this class are Kunz regular. One ofsuch blowups is the one associated to a regular Galois alteration. As a consequence of de Jong’stheorem, we see that for every variety over an algebraically closed field of positive characteristic,there exists a noncommutative blowup which is Kunz regular. We also see that a variety withF-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutativeblowup which is associated to an iteration of the Frobenius morphism of the variety.

1 Introduction

The Frobenius morphism is arguably the most important notion in the algebraic geom-etry of positive characteristic and used almost everywhere. Concerning the singularitytheory, Kunz’s theorem is classical [11]:A scheme is regular if and only if its Frobeniusmorphism is flat. The main aim of this article is to define the Frobenius morphism ofcertain class of noncommutative blowups in positive characteristic and to see that thedefined morphism is flat. Here the noncommutative blowup that we mean is basicallythe same as the noncommutative crepant resolution in [16] and the noncommuta-tive desingularization in [2] except that we remove some assumptions, especially thefiniteness of global dimension.

Let k be a field of characteristic p > 0. Recently it was found in [15] that ifX D SpecR belongs to some classes of singularities over k, then for sufficientlylarge e, the endomorphism ring EndR.R1=p

e/, whose elements are differential op-

erators on R1=pe, has finite global dimension and is regarded as a noncommutative

resolution of X. This article derives from the author’s attempt to know where the reg-ularity of EndR.R1=p

e/ comes from and to show its regularity for a broader class

of singularities. However the regularity which we will consider in this article is theflatness of Frobenius rather than the finiteness of global dimension. It is because the

�This work was supported by Grant-in-Aid for Young Scientists (20840036) from JSPS.


270 Takehiko Yasuda

former seems to the author simpler and compatible with EndR.R1=pe/. We however

consider not only noncommutative blowups of the form EndR.R1=pe/.

LetX; Y be integral normal Noetherian schemes over k with finite Frobenius mor-phisms and f W Y ! X a finite dominant morphism. We associate to f a noncommu-tative blowup NCB.Y=X/, which is the pair of the endomorphism ring EndOX

.OY /

and the left EndOX.OY /-module OY . (More generally we will consider the noncom-

mutative blowup associated to a coherent sheaf. However the examples in which weare interested are associated to finite morphisms of schemes.)Also we regard this as thecategory of left EndOX

.OY /-modules with the distinguished object OY . Let Ye ! Y

be the e-times iteration of the Frobenius morphism. We say that f is F-steady if forevery e � 0, the structure sheaves of Ye and Y locally have, as OX -modules, thesame summands (for details, see Section 4). For instance, if Y is regular, then f isF-steady. Given an F-steady morphism Y ! X, we define the Frobenius morphismof NCB.Y=X/, which is flat by construction. Hence we say that NCB.Y=X/ is Kunzregular.

If k is algebraically closed, from de Jong’s theorem [4], every k-variety X admitsa Galois alteration Y ! X with Y regular. It uniquely factors as Y ! xY ! X suchthat xY is a normal variety, Y ! xY is finite and xY ! X is a modification. Then theassociated noncommutative blowup NCB.Y=X/ D NCB.Y= xY / is Kunz regular. Thusevery variety admits a noncommutative blowup which is Kunz regular (Corollary 4.5).

Another interesting example of noncommutative blowups is the one associated toan iterated Frobenius morphism Xe ! X of a normal scheme X. In the affine case,this corresponds to the above-mentioned ring EndR.R1=p

e/. IfX has only F-pure and

FFRT (finite F-representation type) singularities, then for sufficiently large e,Xe ! X

is F-steady and the associated noncommutative blowup NCB.Xe=X/ is Kunz regular(see Section 5). The FFRT singularity was introduced in [13] and proved to have D-module theoretic nice properties; see [13], and [14]. Our result is yet another suchproperty.

1.1 Convention

Throughout the paper, we work over a fixed base field k unless otherwise noted. Wemean by a scheme a separated Noetherian scheme over k. In Sections 4, 5 and 6, weadditionally assume that k has characteristic p > 0 and that every scheme is F-finite,that is, the Frobenius morphism is finite. If f W Y ! X is an affine morphism ofschemes and M is a quasi-coherent sheaf on Y , then by abuse of notation, we denotethe push-forward f�M again by M.

Frobenius morphisms of noncommutative blowups 271

2 Noncommutative schemes

2.1 Pseudo-schemes

Following [1], p. 235, we first define the category PS.

Definition 2.1. A pseudo-scheme is the pair .A;M/ of a k-linear abelian category A

and an object M 2 A. A morphism f W .A;M/! .B; N / of pseudo-schemes is theequivalence class of pairs .f �; �/ of a k-linear functor f � W B ! A which admits aright adjoint f� W A ! B and an isomorphism � W f �M Š N . Here two such pairs.f �; �/ and ..f �/0; � 0/ are equivalent if there is an isomorphism f � Š .f �/0 whichis compatible with � and � 0. The composition of morphisms is defined in the obviousway. We denote the category of pseudo-schemes by PS. A morphism f is said to beflat if its pull-back functor f � is exact.

For a schemeX, we denote by Qcoh.X/ the category of quasi-coherent sheaves onX. We have a natural functor

.scheme/ �! PS; X 7�! Xps D .Qcoh.X/;OX /:

From a theorem of Gabriel [5], we can reconstructX fromXps (which was generalizedto the non-Noetherian case by Rosenberg [12]).

Theorem 2.2 (Reconstruction of schemes). If Xps Š Y ps, then X Š Y .

We can also reconstruct morphisms.

Proposition 2.3 (Reconstruction of morphisms). The functor X 7! Xps is faithful.

Proof. Suppose that f W Y ! X D SpecA be a morphism of schemes withX affine.Then f ps determines a k-algebra map A D End.A/ ! �.OY / D End.OY / and sodetermines f.

Next suppose thatf W Y ! X be an arbitrary morphism of schemes. Then applyingf� to the structure sheaves of integral closed subschemes ofY, we see thatf ps uniquelydetermines f as the map of sets. For each affine open subset � W U ,! X, applying f �to the sheaves ��M, M 2 Qcoh.U /, we see that f ps uniquely determines the schememorphism f jf �1.U / W f �1.U / ! U. As a consequence, f ps uniquely determines f.Hence the functor is faithful.

The above results allow us to identify a scheme X (resp. a scheme morphism f )with Xps (resp. f ps).

272 Takehiko Yasuda

2.2 Noncommutative schemes

Definition 2.4. Let Z be a scheme. A finite NC (noncommutative) scheme over Z isthe pair X D .A;M/ of a coherent sheaf A of OZ-algebras and a coherent sheaf M

of left A-module. We denote by Qcoh.X/ D Qcoh.A/ the category of quasi-coherentleft A-modules and set Xps D .Qcoh.X/;M/. Like a scheme, we often identify Xand Xps.

A morphism X D .A;M/ ! X 0 D .A0;M0/ of finite NC schemes over Z is amorphism Xps ! .X 0/ps defined by the functor

N ˝A0 �W Qcoh.X 0/! Qcoh.X/

for some coherent sheaf N of .A;A0/-bimodules and an isomorphism

N ˝A0 M0 ŠM:

Note that the functor has the right adjoint HomA.N ;�/ and indeed defines a morphismin PS.

We do not construct the correct category of finite NC schemes over differentschemes in this article. Instead we will work in the ambient category PS.

3 Alterations and noncommutative blowups

Definition 3.1. A morphism Y ! X of integral schemes is called an alteration(resp. modification) if it is generically finite, dominant and proper (resp. birational andproper). An alteration Y ! X is said to be normal (resp. regular) if Y is so. A finite-birational factorization of a normal alteration Y ! X is a factorization of Y ! X

into a finite and dominant morphism Y ! xY and a modification xY ! X with xYnormal. (This is clearly unique up to isomorphism if exist.) A normal alteration is saidto be factorizable if it admits a finite-birational factorization.An alteration f W Y ! X

is said to be Galois if there exists a finite groupG of automorphisms of Y such that f isG-equivariant under the trivial G-action on X and the field extension K.Y /G=K.X/is purely inseparable.

Lemma 3.2. Let f W Y ! X be a normal Galois alteration. Suppose that the quotientalgebraic space Y=G is a scheme. In the case where k has positive characteristic, wesuppose that Y=G is F-finite. Then f is factorizable.

Proof. In characteristic 0, the natural morphism Y=G ! X is birational, hence f isfactorizable. Let us suppose that k has characteristic p > 0. We take e 2 Z�0 suchthat .K.Y /G/p

e � K.X/. Let Y=G be the quotient variety and Y=G ! .Y=G/e themorphism corresponding to the inclusion O

pe

Y=G,! OY=G of sheaves, though this is

not a morphism ofk-schemes unlessk is perfect. Let xY be the normalization of .Y=G/e


inK.X/. Then we claim that f factorizes through xY . To see this, we take affine opencoverings Y D S

SpecSi and X D SSpecRi such that for each i , f .SpecSi / �

SpecRi and SpecSi is stable under theG-action. Then .Y=G/e DS Spec..SGi /pe/.

If we denote the normalization of .SGi /pe

in K.X/ by xSi , then xY D SSpec xSi . For

each i, we have xSi D Si \K.X/. Since Si � xSi � Ri , the claim holds.

Definition 3.3. For a torsion-free coherent sheaf M on an integral schemeX, we write

EM D EM=X D EndOX.M/:

We define the NC blowup of X associated to M, NCB.M=X/, to be the finite NCscheme .EM;M/ over X. We define the projection

NCB.M=X/ �! X

by the functor M ˝OX�; where we think of M as a .EM;OX /-bimodule.

For a dominant finite morphism f W Y ! X of integral schemes, we put

EY=X D EOY =X

and

NCB.Y=X/ D NCB.OY =X/:

Since OY is a subring of EY=X , we have the induced functor

Qcoh.EY=X / �! Qcoh.OY /;

which is identical to OYEY=X ˝EY=X

�. We call the corresponding morphism

Y �! NCB.Y=X/

the coforgetful morphism. This is obviously flat. The composition of the coforgetfulmorphism and the projection,

Y �! NCB.Y=X/! X;

is exactly the original morphism Y ! X.

Definition 3.4. For a factorizable normal alteration Y ! X, if Y ! xY ! X is thefinite-birational factorization, then we define the associated NC blowup, NCB.Y=X/,to be NCB.Y= xY /.

Remark. The normality assumption in the above definition is not really necessary,but just for simplicity.

Every factorizable normal alteration Y ! X factors also as

Y �! NCB.Y=X/ �! X:

274 Takehiko Yasuda

4 Frobenius morphisms

In this section, we shall define the Frobenius morphism for some class of noncommu-tative blowups.

From now on, we suppose that the base field k has characteristic p > 0. We alsosuppose that every scheme is F-finite.

4.1 Equivalent modules

Definition 4.1. Let R be a commutative complete local Noetherian ring. Then everyfinitely generated R-module is uniquely the direct sum of finitely many indecompos-able R-modules. We say that R-modules M and N are equivalent if

M ŠMi

L˚ai

i and N ŠMi

L˚bi

i

for some indecomposable R-modules Li and positive integers ai and bi .We say that coherent sheaves M and N on a scheme X are equivalent if for every

point x 2 X, the complete stalks yMx and yNx are equivalent yOX;x-modules.

Given coherent sheaves M and N on a schemeX. We think of M as an .EM;OX /-bimodule and similarly for N . Then Hom.M;N / D HomOX

.M;N / is an .EN ;EM/-bimodule.

Lemma 4.2. Let L, M and N be coherent sheaves on X which are mutually equiv-alent.

(i) We have a natural isomorphism of .EN ;EL/-bimodules

Hom.M;N /˝EMHom.L;M/ Š Hom.L;N /:

In particular

Hom.M;N /˝EMHom.N ;M/ Š EN :

Hence the functors

Hom.M;N /˝EM�W Qcoh.EM/! Qcoh.EN /

Hom.N ;M/˝EN�W Qcoh.EN /! Qcoh.EM/

are equivalences which are inverses to each other.

(ii) We have a natural isomorphism of .EN ;OX /-bimodules

Hom.M;N /˝EMM Š N :

Proof. These are well-known to the specialists.


(i) There exists a natural morphism

Hom.M;N /˝EMHom.L;M/! Hom.L;N /:

It is easy to see that the morphism is an isomorphism after the completion at eachpoint of X. Hence the morphism is an isomorphism.

(ii) The proof is similar to the above.

4.2 F-steady modules and Frobenius morphisms

For a scheme X, we write the e-iterated k-linear Frobenius as

F e D F eX W Xe �! X:

Sometimes we simply call this the e-th Frobenius of X. A key observation is that themorphism F e factors as Xe ! NCB.Xe=X/! X (see Definition 3.3).

Definition 4.3. LetX be an integral normal scheme and M a reflexive coherent sheaf(that is, M__ Š M). Denote by Me the sheaf on Xe corresponding to M via theobvious identification Xe D X. Then Me is identical as an OX -module to the push-forward of M by the e-iterated absolute Frobenius. We say that M is F-steady if forevery e, M and Me are equivalent OX -modules.

For a finite dominant morphism f W Y ! X of integral normal schemes, we saythat f is F-steady if OY is an F-steady OX -module.

Example. If Y is regular, then f is F-steady. Indeed being flat over OY , OYeis locally

isomorphic to O˚rY , r > 0, as an OY -module and hence also as an OX -module.

From Lemma 4.2, for an F -steady sheaf M, we have an isomorphism

NCB.Me=X/ Š NCB.Me0=X/; e; e0 � 0:We also define a morphism

NCB.Me=Xe/ �! NCB.Me=X/;

which we call the coforgetful morphism, as follows. We think of OX as a subring of

OXeD O

1=pe

X in the obvious way. Then EMe=Xeis a subring of EMe=X . Hence we

have a natural morphism

NCB.Me=Xe/ �! NCB.Me=X/

defined by EMe=XeEMe=X ˝EMe=X

�.

Definition 4.4. Let M be an F-steady sheaf on X. We define the e-th Frobenius ofNCB.M=X/ to be the composite

F e D F eNCB.M=X/ W NCB.Me=Xe/cofor.��! NCB.Me=X/

��! NCB.M=X/:

276 Takehiko Yasuda

By construction, this is flat, which we call the Kunz regularity of NCB.M=X/.(Recall that from Kunz [11], a scheme is regular if and only if its Frobenius morphismsare flat.) The morphism is also directly defined by the functor

EMe=XeHomOX

.M;Me/˝EM=X�:

Corollary 4.5. Suppose that k is algebraically closed and that X is an arbitrary k-variety. Take a regular Galois alteration Y ! X with Y quasi-projective. (Such analteration exists from de Jong’s theorem [4], 7.3). Then the associated noncommutativeblowup NCB.Y=X/ is Kunz regular.

Proof. We first note that Y ! X is factorizable. Let Y ! xY ! X be the finite-birational factorization. Since Y is regular, the morphism Y ! xY is F-steady. HenceNCB.Y= xY / D NCB.Y=X/ is Kunz regular.

Remark. Bondal and Orlov [2] conjectured the following: Let Y ! X be a finitemorphism of varieties such thatX has canonical singularities andY is regular. Then thederived category of EY=X -modules is a minimal categorical desingularization. Theirconjecture and the above corollary seem somehow related.

4.3 Compatibility of Frobenius morphisms

In this subsection, to justify our definition of the Frobenius morphism, we show somecompatibility of it (see also Section 6). We suppose that M is an F-steady reflexivecoherent sheaf on an integral normal scheme X.

Proposition 4.6. The diagram

NCB.Me=Xe/F e

��

proj.

��

NCB.M=X/

proj.

��Xe

F e�� X

is commutative.

Proof. From Lemma 4.2, we have isomorphisms of .EMe=Xe;OX /-bimodules

Me ˝OXeOXeŠMe Š HomOX

.M;Me/˝EM=XM:

The left hand side defines the composite morphism F e ı proj., while the right handside defines proj. ı F e. Hence the proposition follows.


Lemma 4.7. For e0 � e � 0, the diagram

NCB.Me0=Xe/cofor. ��

o��

NCB.Me0=X/

o��

NCB.Me=Xe/ cofor.�� NCB.Me=X/

is commutative.

Proof. There exists a natural morphism

˛ W HomOXe.Me;Me0/˝EMe=Xe

EMe=X ! HomOX.Me;Me0/;

� ˝ 7! � ı :We claim that this is an isomorphism, which proves the lemma.

Let U � X be an open subset such that X n U has codimension � 2 and MjU islocally free. Then locally onU, we have an isomorphism of OXe

-modules Me0 ŠM˚re

for some r . Hence locally on U, the source and target of ˛ are both isomorphic toE˚r

Me=X. It is now easy to see that ˛ is an isomorphism over U.

Moreover both hand sides are flat right EMe=X -modules and hence locally isomor-phic to direct summands of E˚l

Me=Xfor some l . From the normality assumption, E˚l

Me=X

is a reflexive OX -module (see [7], Proposition 1.6) and so are its direct summands. So˛ is an isomorphism all over X. We have proved the claim and the lemma.

Corollary 4.8. For e; e0 � 0, the diagram

NCB.MeCe0=Xe/F e

��

o��

NCB.Me0=X/

o��

NCB.Me=Xe/F e

�� NCB.M=X/

is commutative.

Proof. If e0 � e, then from Lemmas 4.2 and 4.7, the diagram

NCB.MeCe0=X/

��

��

NCB.MeCe0=Xe/

��

��

��

NCB.Me0=X/

��NCB.Me0=Xe/

��

��

NCB.Me=X/

��NCB.Me=Xe/

��NCB.M=X/

278 Takehiko Yasuda

is commutative. Now the corollary follows from our definition of the Frobenius mor-phism.

If e0 < e, then similarly the diagram

NCB.MeCe0=X/

��

��

NCB.MeCe0=Xe/

��

��

��

NCB.Me=X/

��NCB.Me=Xe/

��

��

NCB.Me0=X/

��NCB.Me0=Xe/

��NCB.M=X/

is commutative and the corollary follows.

Corollary 4.9. For e; e0 � 0, the diagram

NCB.MeCe0=XeCe0/F e0

��

F eCe0

��

��

��

� NCB.Me=Xe/

F e

��NCB.M=X/

is commutative. Namely we have F eCe0 D F e ıF e0. In particular, the e-th Frobenius

of NCB.M=X/ is the e-iterate of the first Frobenius.

Proof. This follows from the commutativity of the diagram

NCB.MeCe0=X/

��NCB.MeCe0=Xe/

��

��

NCB.Me=X/

��NCB.MeCe0=XeCe0/

��NCB.Me=Xe/

��NCB.M=X/:

Proposition 4.10. Let f W Y ! X be a finite dominant morphism with Y regular.Then the diagram


YeF e

��

cofor.

��

Y

cofor.

��NCB.Ye=Xe/

F e�� NCB.Y=X/

is commutative.

Proof. Because OYeis a locally free OY -module, the canonical map

OYe˝OY

EY=X ! HomOX.OY ;OYe

/:

is an isomorphism, which proves the proposition.

5 D-blowups

Among NC blowups, especially interesting are the ones associated to Frobenius mor-phisms of schemes.

Definition 5.1. For an integral scheme X, we define the e-th D-blowup of X asDBe.X/ D NCB.Xe=X/.

Remark. The D-blowup can be regarded as the noncommutative counterpart of theF-blowup (see [15]).

Definition 5.2 (Hochster–Roberts [8]). Let X D SpecR be an integral scheme. Wesay that R and X are F-pure if R ,! Re splits as an R-module map.

Definition 5.3 (Smith–Van den Bergh [13]). Suppose that R is a complete localNoetherian domain so that the Krull–Schmidt decomposition holds for finitely gen-erated R-modules. Then R and SpecR are said to be FFRT (finite F-representationtype) if there are finitely many indecomposable R-modules Mi , i D 1; : : : ; n, suchthat for any e, Re is isomorphic to

LniD1M

˚rii , ri � 0, as an R-module.

Proposition 5.4. Let R be a complete local Noetherian normal domain. Supposethat X D SpecR is F-pure and FFRT. Then for sufficiently large e, the Frobeniusmorphism F eX W Xe ! X is F-steady.

Proof. LetMi , i D 1; : : : ; n, be the irredundant set of indecomposable modules as inthe above definition. Then there exists e0 such that for every e � e0,Re is isomorphic,as an R-module, to

LniD1M

˚rii , ri > 0. Hence Xe ! X is F-steady.

As a corollary, we obtain the following.

280 Takehiko Yasuda

Corollary 5.5. Let X be an integral normal scheme with F-pure and FFRT singular-ities. Namely the completion of every local ring of X is F-pure and FFRT. Then forsufficiently large e, DBe.X/ is Kunz regular.

Example. Normal toric singularities and tame quotient singularities are F-pure andFFRT. See [14] for other examples.

6 Comparing Frobenius morphisms of commutativeand noncommutative blowups

Let X D SpecR be an integral normal affine scheme and M a finitely generatedreflexive R-module such that the associated sheaf zM is F-steady. Let g W Z ! X bea modification which is a flattening of zM , that is, M D g� zM=tors is locally free.

Lemma 6.1. We have �.M/ DM .

Proof. There exists an open subset U � X such thatX nU has codimension� 2 andzM is locally free on U. Since X is normal, from Zariski’s main theorem, .g�M/jU Dg�g�. zM jU / D zM jU. It follows that the natural morphism zM ! g�M is an injectioninto a torsion-free sheaf which is an isomorphism over U. Since zM is reflexive, this isan isomorphism. Therefore we have

�.M/ D �.g�M/ D �. zM/ DM:

SetE D EndR.M/ and E D EM=Z . Then from the preceding lemma,E D �.E/.Since M is locally free, the projection

h W NCB.M=Z/ �! Z;

which is defined by M ˝OZ�, is an isomorphism.

For F 2 Qcoh.E/ D Qcoh.NCB.M=Z//, �.F / is a left E-module. Thus wehave a left exact functor

ˆ W Qcoh.Z/ �! E-mod

F 7�! �.h�F /:

Put Ee D EndRe.Me/. Similarly we have a functor

ê W Qcoh.Ze/ �! Ee-mod:


Proposition 6.2. The diagram

Qcoh.Z/ ˆ ��

.F e/�

��

E-mod

.F e/�

��Qcoh.Ze/

ê

�� Ee-mod

is commutative up to isomorphism of functors.

Proof. We claim that for F 2 Qcoh.E/, there exists a natural isomorphism

HomR.M;Me/˝E �.F / Š �.HomOZ.M;Me/˝E F /:

Obviously there exists a natural morphism from the left-hand side to the right-handside. Since the claim is local on X, to show this, we may suppose that R is a completelocal ring. Then the claim easily follows from the definition of equivalent modules.

We have natural isomorphisms

..F e/� ıˆ/.F / D HomR.M;Me/˝E �.M ˝OZF /

Š �.HomOZ.M;Me/˝E M ˝OZ

F /

Š �.Me ˝OZF / (Lemma 4.2)

Š �.Me ˝OZeOZe˝OZ

F /

Š �.Me ˝OZe.F e/�F /

Š .ê ı .F e/�/.F /:Thus the proposition holds.

We have the right derived functor of ˆ

Rˆ W DC.Qcoh.Z//! DC.E-mod/:

Similarly for ê .

Corollary 6.3. The diagram

DC.Qcoh.Z// Rˆ ��

.F e/�

��

DC.E-mod/

.F e/�

��DC.Qcoh.Ze// Rê

�� DC.Ee-mod/

is commutative up to isomorphism of functors.

282 Takehiko Yasuda

Proof. We have

.F e/� ı Rˆ Š R..F e/� ıˆ/ Š R.ˆ ı .F e/�/ Š .Rˆ/ ı .F e/�:

The functor RˆmapsDb.Coh.Z// intoDb.E-modfg/. Here Coh.Z/ denotes thecategory of coherent sheaves and E-modfg that of finitely generated left E-modules.As shown in [6], [10], [3], and [16], in some situations, the functor

Rˆ W Db.Coh.Z// �! Db.E-modfg/

is an equivalence, a kind of Fourier–Mukai transform. Then through this equivalence,the Frobenius morphisms on both hand sides correspond to each other at the level ofderived category.

Example. Let G � SLd .k/ be a small finite subgroup of order prime to p withd D 2; 3. Set R D kŒx1; : : : ; xd �G and X D SpecR. Let Y be either RAd

kor Xe for

e � 0, and let Z be the universal flattening of Y ! X, which is isomorphic to theG-Hilbert scheme of Ito–Nakamura [9] (for the case Y D Xe , see [15], and [17]). Ifwe putM to be the coordinate ring of Y , then the above functor is an equivalence (forinstance, see [15], and [16]).

References

[1] M. Artin and J. J. Zhang, Noncommutative projective schemes. Adv. Math. 109 (1994),228–287. 271

[2] A. Bondal and D. Orlov. Derived categories of coherent sheaves, in Proceedings of theInternational Congress of Mathematicians. Invited lectures. Held in Beijing, August 20–28,2002, Vol. II, ed. by T. Li, Higher Education Press, Beijing 2002, 47–56. 269, 276

[3] T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence ofderived categories. J. Amer. Math. Soc. 14 (2001), 535–554 (electronic). 282

[4] A. J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ.Math. (83) (1996), 51–93. 270, 276

[5] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. 271

[6] G. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondancede McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983). 282

[7] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176. 277

[8] M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Ad-vances in Math. 21 (1976), 117–172. 279

[9] Y. Ito and I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad.Ser. A Math. Sci. 72 (1996), 135–138. 282

[10] M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras,Math. Ann. 316 (2000), 565–576. 282


[11] E. Kunz, Characterizations of regular local rings for characteristic p, Amer. J. Math. 91(1969), 772–784. 269, 276

[12] A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112 (1998), 93–125. 271

[13] K. E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in primecharacteristic, Proc. London Math. Soc. (3) 75 (1997), 32–62. 270, 279

[14] S. Takagi and R. Takahashi,D-modules over rings with finiteF -representation type, Math.Res. Lett. 15 (2008), 563–581. 270, 280

[15] Y. Toda and T. Yasuda, Noncommutative resolution, F -blowups and D-modules, Adv.Math. 222 (2009), 318–330. 269, 279, 282

[16] M. Van den Bergh, Non-commutative crepant resolutions, in The legacy of Niels HenrikAbel. Papers from the Abel Bicentennial Conference held at the University of Oslo, Oslo,June 3–8, 2002, ed. by O. A. Laudal and R. Piene, Springer , Berlin 2004, 749–770. 269,282

[17] T. Yasuda, Universal flattening of Frobenius, Amer. J. Math. 134 (2012), 349–378. 282

Bivariant motivic Hirzebruch classand a zeta function of motivic Hirzebruch class

Shoji Yokura�

Department of Mathematics and Computer ScienceFaculty of Science, Kagoshima University

1-21-35 Korimoto, Kagoshima 890-0065, Japanemail: [email protected]

Abstract. The Euler–Poincaré characteristic is a generalization of the cardinality (or counting)and its higher homological extension for singular varieties as a natural transformation (whatcould be put in as its “categorification”) is MacPherson’s Chern class transformation. Thistransformation furthermore has two main developments: a bivariant-theoretic analogue and agenerating series of it, i.e. a zeta function. The motivic Hirzebruch class is a unified theoryof the three well-known characteristic classes of singular varieties, i.e. the above MacPher-son’s Chern class transformation, Baum–Fulton–MacPherson’s Riemann–Roch and Cappell–Shaneson’sL-class transformation, which extends Goresky–MacPherson’sL-class. In this paperwe discuss a bivariant-theoretic analogue and a zeta function of the motivic Hirzebruch class.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

2. Counting: from cardinality to Hodge–Deligne polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 288

3. A “categorification” of an additive homology class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

4. Motivic characteristic classes: the most sophisticated categorificationof additive-multiplicative homology classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304

5. A bivariant-theoretic analogue of the motivic Hirzebruch class Ty� . . . . . . . . . . . . . . . . . . 306

6. A zeta function of the motivic Hirzebruch class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

1 Introduction

It is safe to say that theories of characteristic classes are super generalizations of“counting points of a finite set” (see e.g. [62]), in other words the notion of cardinality

�Partially supported by Grant-in-Aid for Scientific Research (No. 21540088), the Ministry of Education,Culture, Sports, Science and Technology (MEXT), and JSPS Core-to-Core Program 18005, Japan.


286 Shoji Yokura

has been generalized to theories of characteristic classes of singular spaces. On theother hand it has another development into the so-called “generating function” or “zetafunction” in a much fancier way. Namely we have:

cardinality

��

A �� zeta functions

��characteristics

��

B �� zeta functions

��characteristic classes

C �� zeta functions

where A, B, and C are described below.

A: Hasse–Weil zeta functions, for which the Weil conjecture was solved affirmativelyby Deligne in [16] and [17] (see [54]).

In the Hasse–Weil zeta function just the cardinality of a finite set is used. Nowthe cardinality is generalized or extended as the notion of “characteristic” and wehave

B: • the zeta function for the Euler–Poincaré characteristic was studied by Mac-donald [35];

• the zeta function for the arithmetic genus and more generally for the Hirze-bruch �y-characteristic was studied by Moonen [41];

• the zeta function for the signature was studied by Zagier [67].

These three important characteristics, i.e. Euler–Poincaré characteristic, arithmeticgenus and signature have class versions, which are all described as a natural trans-formation from its corresponding covariant functor to the homology functor. Theyare respectively the Chern–MacPherson class [36], Baum–Fulton–MacPherson’sRiemann–Roch or Todd class [4] and Cappell–Shaneson’s L-class [12] (also see[60]) which extends Goresky–MacPherson’s homologyL-class as a natural trans-formation (see [8] and also [60]). The zeta functions of these three characteristicclasses are the following:

C: • the zeta function for the Chern–MacPherson class was studied by Ohmoto[43] (see also [42]);

• the zeta function for Baum–Fulton–MacPherson’s Riemann–Roch or Toddclass was studied by Moonen [41];

• the zeta function for Thom–Hirzebruch L-class for smooth manifolds wasstudied by Zagier [67].

Bivariant motivic Hirzebruch class and a zeta function of motivic Hirzebruch class 287

As the Hirzebruch’s �y-characteristic unifies the above three characteristics (��1,the Euler–Poincaré characteristic, �0, the arithmetic genus, and �1 the signature),the motivic Hirzebruch class Ty�, see [8] and [9], in a sense unifies the above threecharacteristic classes. Thus it is quite natural to consider the zeta function of the motivicHirzebruch class as a class version of Moonen’s zeta function of the Hirzebruch �y-characteristic.

On the other hand, Fulton and MacPherson introduced the theory of BivariantTheory [21] (also see [19]) and they showed the existence of a bivariant-theoreticanalogue of Baum–Fulton–MacPheron’s Riemann–Roch W G0.X/! H�.X IQ/:

td W K0.X ! Y / �! H.X ! Y /:

Furthermore they conjectured (or posed as a problem) the existence of a bivariant-theoretic analogue of the Chern–MacPherson class and J.-P. Brasselet [7] proved itaffirmatively:

c W F .X ! Y / �! H.X ! Y /:

Thus it is also quite natural to think of the existence of a bivariant-theoretic analogueof the above motivic Hirzebruch class, since it unifies the Chern–MacPherson classand Baum–Fulton–MacPheron’s Riemann–Roch. If we get a bivariant-theoretic ana-logue of the motivic Hirzebruch class, then we could speculate a reasonable bivariant-theoretic analogue of the Cappell–Shaneson’s L-homology class, which is not yetavailable, as far as the author knows.

Thus the above flow of diagrams extends as follows:

characteristic classes

��

C �� zeta functions

��motivic characteristic classes

��

�� motivic zeta functions

��bivariant motivic characteristic classes �� ?

As we will see, when it comes to thinking of zeta functions of the motivic character-istic classes, it seems quite natural to consider generalizing the above story or thoughtsto an arbitrary natural transformation of two covariant functors on a reasonably nicecategory so that one can consider such a generalized zeta function associated to a givencovariant functor or a given natural transformation. In such a more general category itis not necessarily guaranteed that the symmetric productX .n/ and hence the projection�n W Xn ! X .n/ exist. We formulate a general and formal zeta function of naturaltransformation associated to a covariant functor and a natural transformation in sucha way that it specializes to usual zeta functions.

288 Shoji Yokura

In the present paper we first deal with bivariant-theoretic analogues of the motivicHirzebruch classes and then the zeta functions of them. As indicated in the abovediagram, however, we do not know what is the implication of “a bivariant zeta function”corresponding to the bivariant motivic Hirzebruch class.

Acknowledgements

The author would like to thank the organizers of The 5th Franco–Japanese Sympo-sium on Singularities (FJ2009) for such a wonderful organization. He would alsolike to thank P. Aluffi, M. Banagl, J.-P. Brasselet, D. Eisenbud, S. Ishii, S.-I. Kimura,L. Maxim, T. Ohmoto, Y. B. Rudyak, J. Schürmann, and T. Yasuda for useful discus-sions and comments, and the referee for pointing out typos and so on.

2 Counting: from cardinality to Hodge–Deligne polynomials

For a finite field k, counting k-points of varieties gives rise to the following homo-morphism from the Grothendieck ring K0.V.k// of varieties:

] W K0.V.k// �! Z

defined by ].ŒX�/ D ].X.k//:This homomorphism is extended to the following formalpower series via the symmetric product X .n/ D Xn=Sn, where the symmetric groupSn acts on the Cartesian product Xn of n-copies of X as permutations of factors:

�HW .X; t/ D1XnD0

].X .n/.k//tn 2 ZŒŒt ��:

This function is called the Hasse–Weil zeta function of X and it is a rational functiondue to Dwork.

M. Kapranov [29] modified the Hasse–Weil zeta function just a bit to define thefollowing motivic zeta function:

�.X; t/ D1XnD0

ŒX .n/.k/�tn 2 K0.V.k//ŒŒt ��;

which shall be denoted by �KapŒX�.t/, called the Kapranov motivic zeta function. More

generally, for any ring homomorphism

� W K0.V.k// �! R

he considered the following zeta function

��.X; t/ D1XnD0

�.ŒX .n/.k/�/tn 2 RŒŒt ��:


This zeta function is called the Kapranov zeta function associated to �.

The above ring homomorphism � is a “motivic” generalization of the counting orcardinality. Indeed, certainly the counting ].A/ of points of a finite set A satisfies thefollowing basic properties:

• A Š A0 (bijection or equipotent) H) ].A/ D ].A0/,

• ].A/ D ].A n B/C ].B/ for B � A,

• ].A � B/ D ].A/ ].B/,• ].pt/ D 1. (Here pt denotes one point.)

If we consider the following “topological counting” ]top on the category Top oftopological spaces such that ]top.X/ 2 Z and it satisfies the following four properties:

• X Š X 0 (homeomorphism = Top-isomorphism) H) ]top.X/ D ]top.X0/,

• ]top.X/ D ]top.X n Y /C ]top.Y / for Y � X,

• ]top.X � Y / D ]top.X/ ]top.Y /,

• ]top.pt/ D 1;then one can show that if such a ]top exists, then we must have that

]top.R1/ D �1; hence ]top.R

n/ D .�1/n:Hence ifX is a finite CW -complex with �n.X/ denoting the number of open n-cells,then

]top.X/ DXn

.�1/n�n.X/ D �.X/

is the Euler–Poincaré characteristic of X. Namely, sloppily speaking, the topologicalcounting ]top is uniquely determined and it is the topological Euler–Poincaré charac-teristic.

Now, let us consider such a counting on the category V of algebraic varieties:

• X Š X 0 (V -isomorphism) H) ]alg.X/ D ]alg.X0/,

• ]alg.X/ D ]alg.X n Y /C ]alg.Y / for a closed subvariety Y � X,

• ]alg.X � Y / D ]alg.X/ ]alg.Y /,

• ]alg.pt/ D 1:If such an “algebraic” counting ]alg exists, then it follows from the decomposition

of the n-dimensional complex projective space

Pn D C0 tC1 t tCn�1 tCn

that we must have

]alg.Pn/ D 1 � y C y2 � y3 C C .�y/n

290 Shoji Yokura

where y D �]alg.C1/ 2 Z. In fact, it follows from Deligne’s theory of mixed Hodgestructures that the following Hodge–Deligne polynomial (cf. [15])

�u;v.X/ DX

i;p;q�0.�1/i .�1/pCq dimC.GrpFGrWpCqH i

c .X;C//upvq

satisfies the above four properties, namely any Hodge–Deligne polynomial �u;v withuv D �y is such a ]alg. The Hirzebruch �y-characteristic is nothing but �y;�1 and themost important and interesting ones are the following:

y D �1: ��1 D �, the topological Euler–Poincaré characteristic,

y D 0: �0 D �a, the arithmetic genus,

y D 1: �1 D � , the signature.

These three important characteristics are extended as higher class analogues, i.e.“categorified” or natural transformations of additive-multiplicative homology classes,as explained in the following section. In fact they are most sophisticated ones andwell-known characteristic classes:

y D �1: Chern–MacPherson class

c� W F.X/ �! H�.X IZ/:

y D 0: Baum–Fulton–MacPheron’s Riemann–Roch

W G0.X/ �! H�.X IQ/

y D 1: Cappell–Shaneson’s L-homology class

! W �.X/ �! H�.X IQ/:

It turns out (see [8] and [62]) that the Hodge–Deligne polynomial

�u;v W K0.V/ �! ZŒu; v�

can be extended as a class version only when u D y; v D �1, just like Hirzebruch–Riemann–Roch was extended by A. Grothendieck as a natural transformation fromthe covariant functor of coherent sheaves to the rational homology theory, which iscalled Grothendieck–Riemann–Roch. Namely only the Hirzebruch �y-characteristic

�y W K0.V/ �! ZŒy�

can be extended as a class version

Ty� W K0.V=�/ �! H�.�IQŒy�/:This is called the motivic Hirzebruch class and it “unifies” the above three characteristicclasses (see later sections).


3 A “categorification” of an additive homology class

3.1 An additive homology class

Let Top be the category of topological spaces and continuous maps. Let C be anothercategory of topological spaces equipped with possibly extra more geometric structures,e.g. such as differentiable manifolds, almost complex manifolds, complex algebraicvarieties, real algebraic varieties, complex analytic varieties, etc. Let f W C ! Top bethe forgetful functor, i.e. the functor forgetting those extra more geometric structures.

Definition 3.1 (An additive homology class on the category C ). Let AB be thecategory of abelian groups and let C be a possibly more geometric category as above.LetH� W C ! AB be the homology covariant functor. If an element ˛.X/ 2 H�.X/is defined uniquely (up to isomorphism) for an object X, and it satisfies that

(i) for an isomorphism f W X Š��! X 0, f�˛.X/ D ˛.X 0/

(ii) ˛.X t Y / D ˛.X/C ˛.Y /;then the element ˛.X/ is called an additive homology class of X. Furthermore, if theadditive homology class ˛ is non-trivial and satisfies the following “cross product”formula

(iii) ˛.X � Y / D ˛.X/ � ˛.Y /;then it is called an additive-multiplicative homology class of X.

Remark 3.2. (i) It is easy to see that the non-triviality of ˛ means that

˛.pt/ D 1;namely, the non-trivial additive-multiplicative homology class is a normalized one.

(ii) In fact, the homology functor H� can be replaced by any covariant functorH W C ! AB equipped with the cross product structure such that the coefficient ringH .pt/ is a domain. In this case we call such an additive class an additive H -class.However, we stick to the homology theory for the sake of simplicity.

(iii) Almost all topological invariants are additive classes: the Euler–Poincaré char-acteristic on Top, all the characteristic classes and the characteristic numbers on thecategory C1 of differentiable manifolds and the category AC of almost complexmanifolds, all the characteristic classes and characteristic numbers of singular vari-eties, etc. They are all in fact additive-multiplicative classes. Fulton’s canonical class,Fulton–Johnson’s Chern class and Milnor class are additive but not multiplicative.

(iv) The Euler–Poinaré characteristic is an integer, but we still call it a class. Notethat any additive-multiplicative homology class of a compact 0-dimensional manifold,i.e. a finite set, is nothing but the cardinality of the finite set. In this sense, an additive-multiplicative homology class is a generalization of counting points.

292 Shoji Yokura

3.2 A natural transformation associated to an additive homology class

Given an additive homology class on a category C with the forgetful functor

f W C �! Top;

the correspondence˛ W M 7�! ˛.M/ 2 H�.M/

is not “categorical” as it is, although the receiver of the correspondence ˛ is thecovariant homology functor, but the source of the correspondence ˛ is not a functor.We can make it a natural transformation from a certain functor � to the covarianthomology functor

˛ W �.X/ �! H�.X/:

This will be called a categorification of the correspondence ˛ .

Here is one simple answer for the functor � .

Definition 3.3. Let X be a topological space and let C be a category of topologicalspaces equipped with possibly extra more geometric structures and let f W C ! Topbe the forgetful functor. Let

M prop.Cf��! Top=X/

be the monoid consisting of isomorphism classes ŒVp�! X� of proper morphisms

p W V ! X (more precisely,p W f.V /! X 2 HomTop.f.V /; X/), whereV 2 Obj.C/.Hereh W V ! X andh0 W V 0 ! X (more precisely,h W f.V /! X andh0 W f.V 0/! X )are called isomorphic over X 2 Obj.Top/

• if there is an isomorphism

� W V �! V 0 2 HomC .V; V0/

and

• if h0 ı f.�/ D h in HomTop.f.V /; X/.

The addition and zero are defined by

• ŒVh�! X�C ŒV 0 h0

�! X� D ŒV t V 0 hCh0

��! X�,

• 0 D Œ� ! X�:

Then we define

Kprop.Cf��! Top=X/

to be the Grothendieck group of the monoid M prop.Cf��! Top=X/, which shall be

provisionally called the Grothendieck–Thom relative group over X.


Proposition 3.4. (i) Kprop.Cf��! Top=X/ is a covariant functor on the category

Top with pushforwards for proper morphisms, i.e. for a proper morphism f W X ! Y

the pushforward

f� W Kprop.Cf��! Top=X/ �! Kprop.C

f��! Top=Y /

defined by

f�.ŒVp��! X�/ D ŒV f ıp��! Y �

is covariantly functorial.

(ii) Kprop.Cf��! Top=X/ has a cross product structure on the category Top:

Kprop.Cf��! Top=X/ �Kprop.C

f��! Top=Y /��! Kprop.C

f��! Top=X � Y /is defined by

ŒVp��! X� � ŒW k��! X� D ŒV �W p�k��! X � Y �:

(iii) Let ˛ be an additive homology class defined on the category C . Then thereexists a unique natural transformation on the category Top of topological spaces:

˛ W Kprop.Cf��! Top=�/ �! H�.�/

satisfying that for X 2 Obj.C/

˛.ŒXidX��! X�/ D ˛.X/:

(iv) Let ˛ be an additive-multiplicative homology class defined on the category C .Then the above natural transformation

˛ W Kprop.Cf��! Top=�/ �! H�.�/

commutes with the cross product, i.e. the following diagram commutes:

Kprop.Cf��! Top=X/ �Kprop.C

f��! Top=Y /�˛��˛ ��

��

H�.X/ �H�.Y /

�

��Kprop.C

f��! Top=X � Y /�˛

�� H�.X � Y /:

Remark 3.5. Depending on the additive homology class ˛, the above Grothendieck–

Thom covariant functor Kprop.Cf��! Top=�/ can be made into a much finer one.

Here are typical examples.

(i) Suppose that the additive homology class ˛ on C is bordism invariant, i.e. ifX and Y are bordant, i.e. if there exists W such that @W D X t Y then we

294 Shoji Yokura

have ˛.X/ D ˛.Y /. (Here we are allowed to be sloppy and we do not considerorientation; if we consider orientation, we should put the sign.) In this case we

can divide Kprop.Cf��! Top=�/ out by the bordism relation, i.e.

�prop.Cf��! Top=X/ D Kprop.C

f��! Top=X/= �;where the equivalence relation � is defined by

ŒVh��! X� � ŒV 0 h0

��! X� () there exists H W W �! X such that

(a) @W D V t V 0;(b) H j@W D h t h0:

(ii) If it satisfies the “Grothendieck additivity”

˛.V / D ˛.V n S/C ˛.S/ for S � V;then we can devide Kprop.C

f��! Top=�/ out by this strong additivity relation:

Kprop0 .C

f��! Top=X/ D Kprop.Cf��! Top=X/= �;

where the equivalence relation � is defined by

ŒVh��! X� � ŒV n S hjV nS��! X�C ŒS hjS��! X� for S � V:

Remark 3.6. It should be emphasized that even though we consider such a finer cat-egory C for a source space V the map h W V ! X of course has to be considered

in the crude category Top. The above �prop.Cf�! Top=X/ (with C being the cate-

gory of closed oriented manifolds) is the so-called bordism group ��.�/, which is ageneralized homology theory, in particular ��.�/ is a covariant functor

�� W Top �! AB;

where AB is the category of abelian groups. Clearly we can consider this covariantfunctor on a different category finer than the category Top of topological spaces, e.g.consider the category VC of complex algebraic varieties. Namely we consider con-tinuous maps h W M ! V from closed oriented manifolds M to a complex algebraicvariety V . We still get a covariant functor

�� W VC �! AB:

In this set-up the following three different categories are involved:

• coC1 of closed oriented manifolds,

• Top of topological spaces,

• VC of complex algebraic varieties.


More precisely, we have the following forgetful functors fs W coC1 ! Top andft W VC ! Top (here “s” and “t” mean “source object” and “target object”):

coC1 fs��! Topft �� VC:

And the commutative triangle

M

h

��

��

��

� �� M 0

h0

��

V

really means

fs.M/

h

��

��

��

�fs.�/ �� fs.M 0/

h0

��

ft .V /:

As suggested by this situation in [65] we deal with a more general situation of cospanof categories Cs;Ct ;B:

CsS��! B

T �� Ct :

From this cospan CsS��! B

T �� Ct we get the canonical generalized .S; T /-re-

lative Grothendieck groups K.CsS��! B=T .�// and also from the following com-

mutative diagrams of categories and functors

Cs

S 0

��

��

��

S �� B

ˆ

��

Ct

T 0

��

T��

B 0

we obtain a categorification of an additive function ˛.X/ on objects Obj.Cs/ withvalues ˛.X/ 2 T 0.X/:

˛ W K.Cs S��! B=T .�// �! T 0.�/:In particular, for the following commutative diagram

Cs

S 0

��

��

��

S �� B

ˆ

��

Cs

S 0

��

S��

B 0

296 Shoji Yokura

with S W C ! B being a full functor, then the natural transformation

˛ W K.Cs S��! B=T .�// �! S 0.�/satisfying the condition that

˛.Œ.V; V; idV /�/ D ˛.V / 2 S 0.V /

for V 2 Obj.Cs/ is unique. For more details, see [65].

Remark 3.7. (i) If ˛.M/ D cl�.M/ D cl.M/ \ ŒM � is the Poincaré dual of anycharacteristic cohomology class cl, then the above natural transformation ˛ is denoted

by cl W Kprop.Cf�! Top=�/ ! H�.�/ and this can be considered as a very general

theory of characteristic homology classes of topological spaces.

(ii) If we consider cl W Kprop.Cf��! Top=�/ ! H�.�/ on a category B geo-

metrically finer than the category Top of topological spaces with the forgetful functorf W C ! B (e.g. an inclusion functor), then we get a general theory of characteristichomology classes on the subcategory B; for example, a general theory of character-istic homology classes on pseudo-manifolds, on complex algebraic varieties, on realalgebraic varieties, etc.

The Grothendieck–Thom covariant functor Kprop.Cf�! Top=�/ cannot in gen-

eral become a contravariant functor with a reasonable pullback. It is because in thefollowing fiber square

M 0 f 0

��

h0

��

M

h

��X

f�� Y:

with M 2 Obj.C/ the fiber product M 0 D X �Y M does not necessarily belong tothe category C . If it does, we can define the pullback homomorphism

f � W Kprop.Cf��! Top=Y / �! Kprop.C

f��! Top=X/

by

f �.ŒM h�! Y �/ D ŒM 0 h0

��! X�

and with this Kprop.Cf��! Top=�/ becomes a contravariant functor.

Lemma 3.8. Let us consider the category VC of complex algebraic varieties, in-stead of the category Top, and consider the subcategory �VC of smooth varieties asthe source category C , thus f W �VC ! VC is the inclusion functor. Then the functor


Kprop.�VCf��! VC=�/ also becomes a contravariant functor for smooth morphisms

on the category VC, where for a smooth morphism f W X ! Y the pullback homo-morphism

f � W Kprop.�VCf��! VC=Y / �! Kprop.�VC

f��! VC=X/

is defined by

f �.ŒM h��! Y �/ D ŒM 0 h0

��! X�:

Theorem 3.9 (Verdier-type Riemann–Roch, see [65]). Let the situation be as de-scribed in Lemma 3.8. Let cl be any multiplicative characteristic cohomology class of

complex vector bundles. Then the above natural transformation cl W Kprop.�VCf��!

VC=�/! H�.�/ on the category VC satisfies the following Verdier-type Riemann–Roch formula: For a smooth morphism f W X ! Y the following diagram commutes:

Kprop.�VCf��! VC=Y /

f �

��

�cl �� H�.Y /

cl.Tf /\f �

��Kprop.�VC

f��! VC=X/ �cl�� H�.X/:

In the case when we consider the identity functor i W C ! C for an object

X 2 Obj.C/, the functor Kprop.Ci��! C=X/ is simply denoted by Kprop.C=X/.

Theorem 3.10 (SGA-6-type Riemann–Roch [65]). Consider the category �VC ofsmooth complex varieties. Let Kprop.sm.�VC=X/ be the Grothendieck group of themonoid of the isomorphism classes of proper smooth morphisms h W V ! X. On thecategory �VC let us define

Tcl W Kprop.sm.�VC=X/ �! H�.X/

by

Tcl.ŒVh�! X�/ D PDX

�1�h�.cl.Th/ \ ŒV �/�:

Here PDX W H�.X/ ! H�.X/ is the Poincaré duality isomorphism given by takingthe cup product with the fundamental class. Then the following diagram is commutativefor a proper smooth morphism f W X ! Y :

298 Shoji Yokura

Kprop.sm.�VC=X/Tcl ��

f�

��

H�.X/

fŠ.cl.Tf /[ /

��Kprop.sm.�VC=Y /

Tcl

�� H�.Y /:

Here the Gysin homomorphism fŠ W H�.X/! H�.Y / is defined by

fŠ D PDY�1 ı f� ı PDX :

Remark 3.11. One would not be able to expect such a SGA-6-type Riemann–Rochtheorem on the category Top of topological spaces.

3.3 Examples

Example 3.12 (The case of fundamental class). Let us consider taking the fundamentalclass Œ�� on the category C1 of differentiable manifolds. The fundamental class Œ��is certainly an additive-multiplicative homology class and we have the unique naturaltransformation on the category Top of topological spaces:

Œ � W Kprop.C1 f��! Top=�/ �! H�.�/:Then the classical Steenrod’s realization problem can be interpreted as the problem ofasking for the surjectivity of the homomorphism

Œ � W Kprop.C1 f��! Top=X/! H�.X/

for a topological space.The following results are known (see [44]).

• ([51] and [44], Chapter IV, Theorem 7.37)

Œ � W Kprop.C1 f��! Top=X/ �!M0�i�6

Hi .X/

is surjective.

• ([33]) Let CPoincaré be the category of Poincaré complexes, i.e. topological spaceswhich satisfies the Poincaré duality. Then the following is surjective:

Œ � W Kprop.CPoincaré f��! Top=X/ �!Mi 6D3

Hi .X/:


• ([49] and [44], Chapter VIII, Example 1.25(a)) Let Cpseudo be the category ofpseudo-manifolds. Then the following is surjective:

Œ � W Kprop.Cpseudo f��! Top=X/ �! H�.X/:

Example 3.13 (The case of Stiefel–Whitney class). Let C D C1 the category ofC1 manifolds. Let V be a differentiable manifold and let P.w/�.V / 2 H�.V;Z2/be the Poincaré dual P.w/\ ŒV � of a polynomial P.w/ D P.w1; w2; : : : / of Stiefel–Whitney classes w�.T V / 2 H�.V;Z2/. P.w/�.V / is clearly an additive homologyclass. Then we have a unique natural transformation on the category Top of topologicalspaces

P.w/� W Kprop.C1 f��! Top=�/ �! H�.�;Z2/

such that for a differentiable manifold X we have

P.w/�ŒXidX��! X�/ D P.w/�.X/:

In particular the Stiefel–Whitney class w� is a typical one.If we restrict ourselves to the subcategory VR of real algebraic varieties, instead

of Top and we let �VR be its subcategory of smooth real algebraic varieties, then wehave a finer natural transformation on the category VR

P.w/� W Kprop.�VRf�! VR=�/ �! H�.�;Z2/:

In the case when P.w/ D w, we have the following more geometric “realization”on the category VR through constructible functions:

Kprop.�VRf�! VR=X/

w�

��

��

��

�const �� F.X/

w�

��

H�.X;Z2/ :

Example 3.14 (The case of Pontryagin class). Let C D C1 the category of C1manifolds. Let V be a differentiable manifold and let P.p/�.V / 2 H�.V;Z/ bethe Poincaré dual of a rational-coefficient polynomial P.p/ D P.p1; p2; : : : / ofPontryagin classes p�.T V / 2 H�.V;Q/. Then P.p/�.V / is clearly an additivehomology class with Q-coefficients. Then we have a unique natural transformationon the category Top

P.p/� W Kprop.C1 f��! Top=�/ �! H�.�;Q/

300 Shoji Yokura

such that for a differentiable manifold X we have

P.p/�ŒXidX��! X�/ D P.p/�.X/:

Here of course we can consider a Z-coefficient polynomial.If we restrict ourselves to the subcategory VR instead of Top and we let �VR

be its subcategory of smooth real algebraic varieties, then we have a finer naturaltransformation on the category VR

P.p/� W Kprop.�VRf��! VR=�/ �! H�.�;Q/:

If we further restrict ourselves to the subcategory VC instead of Top and we let�VC be its subcategory of smooth complex algebraic varieties, then we have a finernatural transformation on the category VC

P.p/� W Kprop.�VCf��! VC=�/ �! H�.�;Q/:

In the case when P.p/ D L is a Hirzebruch’s L-class, then we have the follow-ing more geometric “realization” on the category VC through Cappell–Shaneson–Youssin–Balmer’s cobordism groups:

Kprop.�VCf��! VC=X/

L�

��

��

��

�cobordism �� .X/

L�

��

H�.X;Q/ :

It follows from [8] that we can replaceKprop.�VCf�! VC=X/ by the finer relative

Grothendieck group K0.VC=X/:

K0.VC=X/

L�

��

��

��cobordism �� .X/

L�

��

H�.X;Q/ :

This will be recalled in the following section on motivic characteristic classes.We have the following natural transformation

P.p/� W �prop.C1 f��! Top=�/ �! H�.�;Q/:Which is due to the fact that the Pontryagin classes are bordism invariant. We wouldlike to speculate that for any multiplicative sequence P.p/ of Pontryagin classes we


have the following commutative diagram

�prop.�VCf�! Top=X/

P.p/�

��

��

��

cobordism �� .X/

P.p/�

��

H�.X;Q/ :

In our paper [8] (see [9] also) we could deal with the only L-class, but not withany other multiplicative sequence, because we require a stronger additivity on a givenH -class ˛.X/, i.e. the following additivity (provisionally called “Grothendieck addi-tivity”):

˛.X/ D ˛.X n Y /C ˛.Y /for a closed subvariety Y � X.

Example 3.15 (The case of Chern classes). Let AC be the category of almost complexmanifolds. LetV be an almost complex manifold and letP.c/�.V / 2 H�.V;Z/ be thePoincaré dual of a Z-coefficient polynomial P.c/ D P.c1; c2; : : : / of Chern classesc�.T V / 2 H�.V;Z/. P.c/�.V / is clearly an additive H -class with H D H�.�;Z/.Then we have a unique natural transformation on the category Top

P.c/� W Kprop.ACf��! Top=�/ �! H�.�;Z/

such that for an almost complex manifold X we have

P.c/�ŒXidX��! X�/ D P.c/�.X/:

If we further restrict ourselves to the subcategory VC instead of Top and we let�VC be its subcategory of smooth complex algebraic varieties, then we have a finernatural transformation on the category VC

P.c/� W Kprop.�VCf��! VC=�/ �! H�.�;Z/:

In the case whenP.c/ D c is the Chern class, then we have the following more geo-metric “realization” on the category VC through constructible functions via MacPher-son’s theorem:

Kprop.�VC=X/

c�

��

��

��

const �� F.X/

cMac�

��

H�.X;Z/ :

302 Shoji Yokura

It follows from [8] that we can replace Kprop.�VCf��! VC=X/ by the finer

relative Grothendieck group K0.VC=X/:

K0.VC=X/

c�

��

��

��const �� F.X/

cMac�

��

H�.X;Z/ :

More on this will be discussed in the following section.

Example 3.16 (The case of Chern classes of other types). Let EMVC be the sub-category of complex algebraic varieties embeddable into smooth varieties and let cFJ�(resp., cFJ� ) be Fulton–Johnson’s Chern class (resp., Fulton’s canonical class) definedon a scheme embeddable into a nonsingular scheme. Namely, let X be a scheme em-beddable into a nonsingular scheme M. cFJ� .X/ ([19], Example 4.2.6(c)) is definedby

cFJ� .X/ D c.TM jX / \ s.NXM/;

where TM is the tangent bundle ofM and s.NXM/ is the Segre class of the conormalsheaf NXM of X in M [19], §4.2. Fulton’s canonical class cF�.X/ ([19], Exam-ple 4.2.6(a)) is defined by

cF�.X/ D c.TM jX / \ s.X;M/;

where s.X;M/ is the relative Segre class [19], §4.2. The local complete intersectionvariety X defines a normal bundle NX in M , from which we can define the virtualtangent bundle TX of X by

TX D TM jX �NXMwhich is a well-defined element of the Grothendieck groupK0.X/. As shown in [19],Example 4.2.6, for a local complete intersection variety X in a non-singular varietyM these two classes are both equal to

cFJ� .X/ D cF�.X/ D c.TX / \ ŒX�:Then there exists the following unique natural transformations on the category Top

cF�W Kprop.EMVC

f��! Top=�/ �! H�.�;Z/

satisfying that cF�.ŒX

idX��! X�/ D cF�.X/ for X 2 Obj.EMVC/, and

cFJ�W Kprop.EMVC

f��! Top=�/ �! H�.�/


satisfying that cFJ�.ŒX

idX��! X �/ D cFJ� .X/, and

cM�W Kprop.EMVC

f��! Top=�/ �! H�.�/

satisfying that cM�.ŒX

idX��! X�/ D cM� .X/, the Chern–Mather class.Since we have the canonical map

ˆ W Kprop.EMVCf��! VC=�/ �! K0.VC=�/;

we can consider the natural transformation

Ty� ıˆ W Kprop.EMVCf��! VC=�/ �! H�.�/˝QŒy�;

which is simply denoted by

Ty� W Kprop.EMVCf��! VC=�/ �! H�.�/˝QŒy�:

The natural transformation

Ty� W K0.VC=�/ �! H�.�/˝QŒy�

will be described in the following §4.Then we can consider the following three natural transformations:

MF DcF�� T�1� W Kprop.EMVC

f�! VC=�/ �! H�.�/;

MFJ DcFJ�� T�1� W Kprop.EMVC

f�! VC=�/ �! H�.�/;

MM DcM�� T�1� W Kprop.EMVC

f�! VC=�/ �! H�.�/:A geometric realization problem for these natural transformations is, e.g. if or how

one can factorize them through the covariant functor of constructible functions (whichis supposed to be adapted or admissible with these distinguished Chern homologyclasses, i.e. cl� D c the Chern class):

Kprop.EMVCf�! VC=�/

const

��

��

��

MF; MFJ; MM

��

��

��

�

F.X/‹

�� H�.X/˝Q :

Here const W Kprop.EMVCf�! VC=�/! F.X/ is defined by

const.ŒVp�! X�/ D p�11V :

304 Shoji Yokura

4 Motivic characteristic classes:the most sophisticated categorification

of additive-multiplicative homology classes

As mentioned in the previous section, in this section we recall the three well-knowntheories of characteristic classes of singular varieties, which are the most sophisticatedcategorification of additive-multiplicative homology classes:

• MacPherson’s Chern class transformation [36]:

cMac� W F.X/ �! H�.X/;

which is a unique natural transformation satisfying the normalization conditionthat for a smooth varietyX the value of the characteristic function is the Poincarédual of the total Chern cohomology class: cMac� .11X / D c.TX/ \ ŒX�,

• Baum–Fulton–MacPherson’s Todd class or Riemann–Roch [4]:

tdBMF� W G0.X/ �! H�.X/˝Q;

which is a unique natural transformation satisfying the normalization conditionthat for a smooth variety X the value of the structure sheaf OX is the Poincarédual of the total Todd cohomology class: tdBMF� .OX / D td.TX/ \ ŒX�,

• Goresky–MacPherson’s homology L-class [22], which is extended as a naturaltransformation by Sylvain Cappell and Julius Shaneson [12] (also see [60]):

LCS� W �.X/ �! H�.X/˝Q;

which is a unique natural transformation satisfying the normalization conditionthat for a smooth variety X the value of the constant sheaf QX is the Poincarédual of the total Hirzebruch–Thom L-class: LCS� .QX / D L.TX/ \ ŒX�.

Here the homology theory H�.X/ is the Borel–Moore homology theory.

From now on the category VC shall be simply denoted by V without the suffix C.

In [8] (cf. [9], [48], and [62]) we introduced the motivic Hirzebruch class

Ty� W K0.V=X/ �! H�.X/˝QŒy�;

which is a unique natural transformation satisfying the normalization condition that

for a smooth varietyX the value of the isomorphism class ŒXidX��! X� of the identity

map idX is the Poincaré dual of the total Hirzebruch cohomology class:

Ty�.ŒXidX��! X�/ D td.y/.TX/ \ ŒX�:


Here the Hirzebruch class td.y/.E/ of the complex vector bundleE (see [26] and [27])is defined to be

td.y/.E/ DrankEYiD1

� ˛i .1C y/1 � e�˛i .1Cy/ � ˛iy

�2 H�.X/˝QŒy�;

where ˛i is the Chern root of E, i.e. c.E/ DrankEYiD1

.1C ˛i /: Note that

• td.�1/.E/ D c.E/ the Chern class,

• td.0/.E/ D td.E/ the Todd class and

• td.1/.E/ D L.E/ the Thom–Hirzebruch L-class.

The motivic Hirzeruch class Ty� W K0.V=X/ ! H�.X/ ˝ QŒy� “unifies” theabove three characteristic classes cMac� ; tdBMF� ; LCS� (also see §3) in the sense that thefollowing diagrams commute:

K0.V=X/

�

��

T�1�

��

��

��

F.X/cMac

�

�� H�.X/˝Q;

K0.V=X/

�

��

T0�

��

��

��

G0.X/tdBMF

�

�� H�.X/˝Q;

and

K0.V=X/

!

��

T1�

��

��

��

�.X/LCS

�

�� H�.X/˝Q:

306 Shoji Yokura

This “unification” could be considered as a positive answer to the following remarkin MacPherson’s survey article [37]:

“It remains to be seen whether there is a unified theory of characteristicclasses of singular varieties like the classical one outlined above.”1

In the rest of the paper we consider the following two problems on the motivicHirzebruch class Ty�:

(i) A bivariant-theoretic analogue of the motivic Hirzebruch class Ty�;

(ii) A zeta function of the motivic Hirzebruch class Ty�.

5 A bivariant-theoretic analogueof the motivic Hirzebruch class Ty�

In early 1980’s Fulton and MacPherson introduced Bivariant Theory as a categoricalframework for the study of singular spaces [21] (see also Fulton’s book [19]): A bi-variant theory is defined on morphisms, instead of objects, and unifies both a covariantfunctor and a contravariant functor. Important objects to be investigated in bivarianttheories are Grothendieck transformations between given two bivariant theories; aGrothendieck transformation is a bivariant version of a natural transformation.

5.1 Fulton–MacPherson’s bivariant theory

We quickly recall some basic ingredients of Fulton–MacPherson’s bivariant theory;see [21].

Let V be a category which has a final object pt and on which the fiber product orfiber square is well-defined. Let us suppose that V has a class of maps, called confinedmaps (e.g. proper maps), which are closed under composition and base change andcontain all the identity maps, and a class of fiber squares, called independent squares(e.g. Tor-independent), which satisfy the following conditions.

(i) If the two inside squares in

X 00 h0��

f 00

��

X 0

f 0

��

g0

�� X

f

��Y 00

h�� Y 0

g�� X

1At that time Goresky–MacPherson’s homology L-class was not available yet and it was defined only afterthe theory of Intersection Homology was invented by Mark Goresky and Robert MacPherson in 1980.


or

X 0

f 0

��

h00�� X

f

��Y 0

g0

��

h0�� X 0

g

��Z0 h �� Z

are independent, then the outside square is also independent,

(ii) Any square of the following forms is independent:

X

f

��

idX �� X

f

��Y

idX

�� X

Xf ��

idX

��

Y

idY

��X

f�� Y

where f W X ! Y is any morphism.

A bivariant theory B on a category V with values in the category of graded abeliangroups is an assignment to each morphism

Xf��! Y

in the category V a graded abelian group

B.Xf��! Y /

which is equipped with the following three basic operations. The i-th component of

B.Xf��! Y /, i 2 Z, is denoted by Bi .X

f��! Y /.

Product operations. For morphisms f W X ! Y and g W Y ! Z, the productoperation

W Bi .X f��! Y /˝ Bj .Yg��! Z/! BiCj .X

gf��! Z/

is defined.

308 Shoji Yokura

Pushforward operations. For morphisms f W X ! Y and g W Y ! Z with fconfined, the pushforward operation

f� W Bi .X gf��! Z/ �! Bi .Yg��! Z/

is defined.

Pullback operations. For an independent square

X 0 g0

��

f 0

��

X

f

��Y 0

g�� Y;

the pullback operation

g� W Bi .X f��! Y / �! Bi .X 0 f 0

��! Y 0/

is defined.

And these three operations are required to satisfy the seven compatibility axioms(see [21], Part I, §2.2, for details):

(B-1) product is associative,

(B-2) pushforward is functorial,

(B-3) pullback is functorial,

(B-4) product and pushforward commute,

(B-5) product and pullback commute,

(B-6) pushforward and pullback commute, and

(B-7) projection formula.

We also assume that B has units, i.e. there is an element 1X 2 B0.XidX��! X/ such

that ˛ 1X D ˛ for all morphismsW ! X, all ˛ 2 B.W ! X/; such that 1X ˇ D ˇfor all morphisms X ! Y , all ˇ 2 B.X ! Y /; and such that g�1X D 1X 0 for allg W X 0 ! X.

Let B;B0 be two bivariant theories on a category V . Then a Grothendieck transfor-mation from B to B0

W B �! B0

is a collection of homomorphisms

B.X ! Y / �! B0.X ! Y /


for a morphism X ! Y in the category V , which preserves the above three basicoperations:

(i) .˛ B ˇ/ D .˛/ B0 .ˇ/,

(ii) .f�˛/ D f�.˛/, and

(iii) .g�˛/ D g�.˛/.A bivariant theory B is called commutative (see [21], §2.2), if whenever both

Wg0

��

f 0

��

X

f

��Y

g�� Z

Wf 0

��

g0

��

Y

g

��X

f�� Z

are independent squares, then for ˛ 2 B.Xf��! Z/ and ˇ 2 B.Y

g��! Z/

g�˛ ˇ D f �ˇ ˛:If

g�˛ ˇ D .�1/deg.˛/ deg.ˇ/f �ˇ ˛;then it is called skew-commutative.

B�.X/ D B.X ! pt/ becomes a covariant functor for confined morphisms and

B�.X/ D B.Xid��! X/ becomes a contravariant functor for any morphisms. As to the

grading, Bi .X/ D B�i .X ! pt/ and Bj .X/ D Bj .Xid��! X/.

Definition 5.1 ([21], Definition 2.6.2, Part I). Let � be a class of maps in V , whichis closed under compositions and containing all identity maps. Suppose that to each

f W X ! Y in � there is assigned an element �.f / 2 B.Xf�! Y / satisfying that

(i) �.g ı f / D �.f / �.g/ for all f W X ! Y , g W Y ! Z 2 � and

(ii) �.idX / D 1X for all X with 1X 2 B�.X/ D B.X idX��! X/ the unit element.

Then �.f / is called a canonical orientation of f. If we need to refer to which bivarianttheory we consider, we denote the bivariant theory B by �B.f /.

A canonical orientation makes B� a contravariant functor and B� a covariant functorwith the corresponding Gysin homomorphisms:

Proposition 5.2. For the composite Xf�! Y

g�! Z, if f 2 � has a canonical orien-tation �B.f /, then we have the Gysin homomorphism defined by f Š.˛/ D �.f / ˛:

f Š W B.Y g�! Z/ �! B.Xgf��! Z/;

310 Shoji Yokura

which is functorial, i.e. .gf /Š D f ŠgŠ: In particular, whenZ D pt, we have the Gysinhomomorphism:

f Š W B�.Y / �! B�.X/:

Proposition 5.3. For an independent square

X 0 g0

��

f 0

��

X

f

��Y 0

g�� Y:

if g 2 C \ � and g has a canonical orientation �B.g/, then we have the Gysinhomomorphism defined by gŠ.˛/ D g0�.˛ �.g//:

gŠ W B.X 0 f 0

��! Y 0/ �! B.Xf��! Y /;

which is functorial, i.e. .gf /Š D gŠfŠ: In particular, for an independent square

Xf��! Y

idX

??y ??yidY

X ��!f

Y;

with f 2 C \ � , we have the Gysin homomorphism:

fŠ W B�.X/ �! B�.Y /:

The symbols f Š and gŠ should carry the information of � and the canonical orien-tation � , but it will be usually omitted unless some confusion is possible.

Let W B ! B0 be a Grothendieck transformation of two bivariant theories B and

B0 and let us assume that there is a bivariant element uf 2 B�.X/ D B0.XidX��! X/

such that

.�B.f // D uf �B0.f /:

This formula is called Riemann–Roch formula (see [21]).The Grothendieck transformation W B ! B0 induces natural transformations

� W B� ! B0� and � W B� ! B0�.


For any morphism f W X ! Y we have the commutative diagram

B�.Y / ��

��

f �

��

B0�.Y /

f �

��B�.X/

�� B0�.X/:

For a confined morphism f W X ! Y we have the commutative diagram

B�.X/� ��

f�

��

B0�.X/

f�

��B�.Y / ��

�� B0�.Y /:

Furthermore the above Riemann–Roch formula .�B.f // D uf �B0.f / gives riseto the following commutative diagrams for the above Gysin homomorphisms fŠ; f Š :

B�.X/ ��

��

fŠ

��

B0�.X/

fŠ. uf /

��B�.Y /

�� B0�.Y /

B�.Y /��

f Š

��

B0�.Y /

uf f Š

��B�.X/ ��

�� B0�.X/:

5.2 A pre-motivic bivariant theory on the categoryof complex algebraic varieties

Let V be the category of complex algebraic varieties, let Prop be the class of propermorphisms, Sm be the class of smooth morphisms, and let any fiber square be anindependent square.

Theorem 5.4. We define

MPropSm .V=X

f�! Y /

to be the free abelian group generated by the set of isomorphism classes of propermorphisms h W W ! X such that the composite of h and f is a smooth morphism:

h 2 Prop and f ı h W W ! Y 2 Sm;

312 Shoji Yokura

in other words, h W V ! X is “a left quotient” of a smooth morphism s W V ! Y

divided by the given morphism f :

f ı h D s or h D s

f;

V

s

��

��

��

h �� X

f��

Y:

Then the association MPropSm is a bivariant theory if the three operations are defined as

follows.

• Product operations. For morphisms f W X ! Y and g W Y ! Z, the productoperation

W MPropSm .V=X

f��! Y /˝MPropSm .V=Y

g��! Z/ �!MPropSm .V=X

gf��! Z/

is defined for ŒVp��! X� 2 MProp

Sm .V=Xf��! Y / and ŒW

k��! Y � 2MProp

Sm .V=Yg��! Z/, by

ŒVp��! X� ŒW k��! Y � D ŒV 0 pık00

��! X�;

and bilinearly extended. Here we consider the following fiber squares

V 0 p0

��

k00

��

X 0 f 0

��

k0

��

W

k

��V

p�� X

f�� Y

g�� Z:

• Pushforward operations. For morphisms f W X ! Y and g W Y ! Z withf 2 Prop, the pushforward operation

f� W MPropSm .V=X

gf��! Z/ �!MPropSm .V=Y

g��! Z/

is defined by

f�.ŒVp��! X�/ D ŒV f ıp��! Y �:

and linearly extended.


• Pullback operations. For an independent square

X 0 g0

��

f 0

��

X

f

��Y 0

g�� Y;


g� W MPropSm .V=X

f��! Y / �!MPropSm .V=X 0 f 0

��! Y 0/

is defined by

g�.ŒVp��! X�/ D ŒV 0 p0

��! X 0�;

and linearly extended. Here we consider the following fiber squares:

V 0 g00

��

p0

��

V

p

��X 0 g0

��

f 0

��

X

f

��Y

g �� Y:

The bivariant theory MPropSm shall be called a pre-motivic bivariant relative Grothen-

dieck group on the category of complex algebraic varieties.

Theorem 5.5. Let cl be a multiplicative characteristic cohomology class of complexvector bundles, i.e. for a complex vector bundleE overX we have cl.E/ 2 H�.X/˝Rwith a certain commutative ringR and we have cl.E˚F / D cl.E/cl.F /. Then thereexists a unique Grothendieck transformation from the pre-motivic bivariant relativeGrothendieck group MProp

Sm to the Fulton–MacPherson bivariant homology theory H

cl W MPropSm .V=X

f��! Y / �! H.Xf��! Y /˝R

satisfying the normalization condition that for a smooth morphism f W X ! Y

cl.ŒXidX��! X�/ D cl.Tf / Uf 2 H.X

f��! Y /˝R:

Here Uf 2 H.Xf��! Y / is the canonical orientation of the smooth morphism f.

314 Shoji Yokura

Remark 5.6. (i) cl W MPropSm .V=X

f��! Y / ! H.Xf��! Y /˝ R can be called a

general bivariant theory of pre-motivic characteristic classes. When Y is a point pt,

cl� W MPropSm .V=X ! pt/ �! H.X ! pt/ D HBM� .X/˝R

is a unique natural transformation satisfying the normalization condition that for asmooth variety

cl�.ŒXidX��! X�/ D cl.TX/ \ ŒX�:

In other words, this gives rise to a pre-motivic characteristic classes for singularvarieties. In a sense, this could be also a very general solution or answer to the afore-mentioned MacPherson’s question about the existence of a unified theory of character-istic classes for singular varieties. We emphasize that when it comes to the pre-motiviccharacteristic class for singular varieties we do not have to require the characteristicclass cl to be multiplicative, but it can be any characteristic class.

(ii) In particular, we have the following commutative diagrams:Here we set MProp

Sm �.V=X/ DMPropSm .V=X ! pt/.

MPropSm �.V=X/

�

��

�c�

��

F.X/cMac

�

�� H�.X/:

Here �.ŒVh��! X�/ D h�11V .

MPropSm �.V=X/

�

��!!!!!!!!!!!!!

�td�

��

��

��

G0.X/tdBMF

�

�� H�.X/˝Q:

Here �.ŒVh��! X�/ D h�OV .

MPropSm �.V=X/

!

��

�L�

��

��

��

�.X/LCS

�

�� H�.X/˝Q:

Here !.ŒVh�! X�/ D h�QV .


(iii) In fact it follows from Hironaka’s resolution of singularities that there exists asurjection

MPropSm �.V=X/ �! K0.V=X/:

And it turns out that if we require the normalization condition and another extracondition that the degree of the0-dimensional component of the classcl�.CPn/ equals1�yCy2C: : : .�y/n, the natural transformation cl� W MProp

Sm �.V=X/! H�.X/˝Rcan be pushed down to the relative Grothendieck group K0.V=X/, the multiplicativecharacteristic class has to be the Hirzebruch class and that is the only case; i.e. thefollowing diagram commutes:

MPropSm �.X/

q

��!!!!!!!!!!!!!

�td.y/ �

��"""""

""""""

""""

K0.V=X/Ty �

�� H�.X/˝QŒy�:

And one of the main results of our previous paper [8] claims that the above three dia-grams also commute with MProp

�m �.V=X/ being replaced by the finer groupK0.V=X/.

Thus we are led to the following natural problem.

Problem 5.7. Formulate a reasonable bivariant-theoretic analogue

K0.V=Xf��! Y /

of the relative Grothendieck group K0.V=X/ so that the following hold.

(i) K0.V=X ! pt/ D K0.V=X/.(ii) Bq W MProp

Sm .V=Xf��! Y /! K0.V=X

f�! Y / is a certain quotient map whichspecializes to the quotient map q W MProp

Sm �.V=X/ ! K0.V=X/ when Y is apoint.

(iii) Ty W K0.V=X f��! Y /! H.Xf��! Y /˝QŒy� is a bivariant-theoretic ana-

logue so that when Y is a point it specializes to the original motivic Hirzebruchclass transformation Ty� W K0.V=X/! H�.X/˝QŒy�, and

(iv) the following diagram commutes:

MPropSm .V=X

f��! Y /

Bq

!!######

######

####

�td.y/

��

��

��

K0.V=Xf��! Y /

Ty

�� H.Xf��! Y /˝QŒy�:

316 Shoji Yokura

Remark 5.8. If we can get a reasonable one K0.V=Xf��! Y /, then its associated

contravariant functor K0.V=X/ D K0.V=XidX��! X/ can be considered as the

contravariant counterpart of the relative Grothendieck group K0.V=X/.

5.3 A bivariant-theoretic relative Grothendieck group

K0.V=Xf��! Y /

First we recall the following result of Franziska Bittner [6].

Theorem 5.9. The groupK0.V=X/ is isomorphic to MPropSm �.V=X/modulo the blow-

up relation

Œ; ! X� D 0 and ŒBlYX0 ! X� � ŒE ! X� D ŒX 0 ! X� � ŒY ! X�; (bl)

for any following Cartesian diagram (which shall be called the blow-up diagram fromhere on)

Ei 0 ��

q0

��

BlYX 0

q

��Y

i�� X 0

f�� X:

with i a closed embedding of smooth (pure dimensional) spaces and f W X 0 ! X

proper. Here BlYX 0 ! X 0 is the blow-up of X 0 along Y with exceptional divisor E.Note that all these spaces over X are also smooth (and pure dimensional and/orquasi-projective).

We want a bivariant-theoretic analogue of the above “blow-up relation”.

Lemma 5.10. Let h W X 0 ! X be a smooth morphism, i W S ! X 0 be a closedembedding such that the composite h ı i W Z ! X is also a smooth morphism.

Ei 0 ��

q0

��

BlSX 0

q

��S

i�� X 0

h�� X:

In the above diagram, let q W BlSX 0 ! X 0 be the blow-up of X 0 along S and letq0 W E ! S be the exceptional divisor map of the blow up q W BlSX 0 ! X 0. Then itfollows that h ı q W BlSX 0 ! X is a smooth morphism and h ı q ı i 0 W E ! X is asmooth morphism.


Definition 5.11. For a morphism f W X ! Y we consider the above diagram

Ei ��

q0

��

BlSX 0

q

��S

i�� X 0

h�� X

f�� X:

such that

(i) h W X 0 ! X is proper,

(ii) f ı h W X 0 ! Y is smooth,

(iii) f ı h ı i W Z ! Y is smooth,

(iv) q W BlSX 0 ! X 0 is the blow-up ofX 0 along S and q0 W E ! S is the exceptionaldivisor map of the blow up q W BlSX 0 ! X 0.

Let BL.V=Xf�! Y / be the free abelian subgroup of MProp

Sm .V=Xf�! Y / generated

byŒBlSX

0 ! X� � ŒE ! X� � ŒX 0 ! X�C ŒS ! X�

and we set

K0.V=Xf��! Y / D MProp

Sm .V=Xf��! Y /

BL.V=Xf��! Y /

:

The class ŒVp��! X� C BL.V=X

f��! Y / represented by ŒVp��! X� shall be

denoted by ŒVp��! X�.

Remark 5.12. When Y D pt, the blowup diagram defining BL.V=Xf��! pt/ is

nothing but the following:

Ei 0 ��

q0

��

BlSX 0

q

��S

i�� X 0

h�� X:

such that

(i) S and X 0 are nonsingular,

(ii) h W X 0 ! X is proper,

(iii) q W BlSX 0 ! X 0 is the blow-up of X 0 along S , q0 W E ! S is the exceptionaldivisor map of the blow up q W BlSX 0 ! X 0:

318 Shoji Yokura

Hence BL.V=Xf�! pt/ is nothing but BL.V=X/, i.e. we have

K0.V=X �! pt/ D K0.V=X/:

Theorem 5.13. K0.V=Xf��! Y / becomes a bivariant theory with the following

three operations induced by the corresponding ones defined in Theorem 5.4.

• Product operations. For morphisms f W X ! Y and g W Y ! Z, the productoperation

? W K0.V=X f��! Y /˝ K0.V=Yg��! Z/ �! K0.V=X

gf��! Z/

is defined by

ŒVp��! X�?ŒW

k��! Y � D ŒV h��! X� ŒW k��! Y �:

and bilinearly extended.

• Pushforward operations. For morphisms f W X ! Y and g W Y ! Z withf 2 Prop, the pushforward operation

f� W K0.V=X gf��! Z/ �! K0.V=Yg��! Z/

is defined by

f��ŒV

p��! X�

�D f�.ŒV

p��! X�/


Pullback operations. For an independent square

X 0 g0

��

f 0

��

X

f

��Y 0

g�� Y;


g� W K0.V=X f��! Y / �! K0.V=X0 f 0

��! Y 0/

is defined by

g��ŒV

p��! X�

�D g�.ŒV

p��! X�/



5.4 A bivariant-theoretic motivic Hirzebruch class

Theorem 5.14 (A bivariant-theoretic motivic Hirzebruch class). The transformation

Ty W K0.V= / �! H. /˝QŒy�

defined by

Ty

�ŒV

p��! X�

�D Ty.ŒV p��! X�/

is a unique natural transformation satisfying the normalization condition that for asmooth morphism f W X ! Y

Ty

�ŒX

idX��! X�

�D td.y/.Tf / Uf :

Corollary 5.15. For a morphism X ! pt to a point, the above Grothendieck trans-formation td.y/ � W K0.V=X ! pt/ ! H.X ! pt/ ˝QŒy� is equal to the motivicHirzebruch class Ty� W K0.V=X/! H�.X/˝QŒy�.

Let Kalg.Xf��! Y / be Fulton–MacPherson’s bivariant K-theory, introduced in

[21]. In the case when y D 0, we have the following corollary.

Corollary 5.16. The following diagram commutes

K0.V=Xf�! Y /

�Kalg

��$$$$$$$$$$

$$$$

�td

��%%%%%

%%%%%%

%%%%

Kalg.Xf�! Y /

�BMFtd

�� H.Xf�! Y /˝Q:

Let

Brc W F .X

f��! Y / �! H.Xf��! Y /

be a bivariant Chern class constructed by J.-P. Brasselet [7], i.e. a bivariant-theoreticversion of the MacPherson’s Chern class transformation c� W F.X/! H�.X/. This isa Grothendieck transformation satisfying the normalization condition (called “weak”normalization condition) that for a smooth variety X

Brc .11X / D c.TX/ \ ŒX�

where 11X 2 F .X ! pt/, c.TX/ 2 H.XidX��! X/, and ŒX� 2 H.X ! pt/:

320 Shoji Yokura

Conjecture 5.17. The Brasselet bivariant Chern class Brc satisfies the normaliza-

tion condition (called “strong” normalization condition) that for a smooth mophismf W X ! Y

Brc .11f / D c.Tf / Uf

where 11f D 11X 2 F .Xf��! Y /, c.Tf / 2 H.X

idX��! X/ and Uf 2 H.Xf��! Y /:

Remark 5.18. The “strong” normalization condition implies the “weak” normaliza-tion condition.

Corollary 5.19. If Conjecture 5.17 is correct, then the following diagram commutes:

K0.V=Xf��! Y /

�F

��

��

��

�c

��%%%%%

%%%%%%

%%%%

F .Xf��! Y /

�Brc

�� H.Xf��! Y /:

Problem 5.20. Define a bivariant-theoretic analogue B�.Xf��! Y / of the cobor-

dism group �.X/ in such a way that the following diagram commutes:

K0.V=Xf�! Y /

�B

��$$$$$$

$$$$$$

$$

�L

��"""""

""""""

"""

B�.Xf�! Y /

BL�

�� H.Xf�! Y /:

6 A zeta function of the motivic Hirzebruch class

Now in the rest of the paper we will discuss a generating function of the motivic Hirze-bruch class. We simply call it the “zeta function” of the motivic Hirzebruch class. Wetake a general look at the zeta functions of covariant functors and natural transforma-tions.


6.1 Some zeta functions

Given a sequence fcngnD1nD0 , the formal power series

1XnD0

cntn

is called an ordinary generating function (of the sequence), or simply a generatingfunction without qualifier ordinary. There are other types of generating functions suchas 1X

nD0cntn

nŠ;

called exponential generating function and1XnD0

cne�n tn

nŠ;

called Poisson generating function (e.g. see [55]).If each cn is some invariant � .X .n// of then-th symmetric productX .n/ D Xn=Sn

of some geometric object X, then the generating function1XnD0

cntn

is called a zeta function of X , and denoted by ��.X/.t/:

Example 6.1 (The case when X is a finite set). Let X be a finite set of cardinalitym,i.e. ].X/ D m. Let cn be the cardinality of the n-th symmetric product of X, i.e.

cn D ].X .n// D mHn D m�1CnCn:

Then we have the following

�].X/.t/ D1XnD0

cntn

D1XnD0

m � 1C n

n

!tn

D 1

.1 � t/mD 1

.1 � t/].X/Thus the zeta function �].X/.t/ of a finite set X is a rational function. Here we notethat for a given sequence fcng the generating function

P1nD0 cntn is expressed as a

rational function if and only if the sequence is a linear recursive sequence or LRS .

322 Shoji Yokura

Example 6.2 (The case whenX is a compact topological space). LetX be a compacttopological space and let the invariant � be the Euler–Poincaré characteristic �. Thenwe have, as mentioned in Introduction, the following formula due to I. Macdonald:

��.X/.t/ D 1

.1 � t/�.X/ :

This is clearly a generalization of the above formula.

Example 6.3 (Hasse–Weil zeta function and Weil conjecture). Let X be an alge-braic variety defined over the finite field Fp . Consider the following Hasse–Weil zetafunction:

�].X.Fp//.t/ D1XnD0

]�X.Fp/

.n/�tn:

The celebrated Weil conjecture is about the rationality of this zeta function of a non-singular projective algebraic variety; to be more precise, it is

• �].X.Fp//.t/ D P1.t/P3.t/ : : : P2N�1.t/P2.t/P4.t/ : : : P2N .t/

, where N D dimX and Pi .t/ is an

integral polynomial whose degree degPi .t/ D dimH i .X/ for the cohomologyH�.X/ and

• the absolute value of the roots of Pi .t/ is equal to pi2 .

This conjecture was solved by Pierre Deligne in [16] and [17].

Example 6.4 (The Kapranov motivic zeta function). Kapranov introduced the fol-lowing motivic zeta function:

�Kap.X/.t/ D1XnD0

ŒX .n/�tn 2 K0.V/ŒŒt ��:

Kapranov [29] showed that for a non-singular projective curve the motivic zeta function�Kap.X/.t/ is a rational function. But Larsen and Lunts [32] showed that for a surfaceXthe motivic zeta function �Kap.X/.t/ is not necessarily a rational function, in fact theyshowed that it is a rational function if and only if the Kodaira dimension of it isnegative. However, when it comes to the Grothendieck ring of Chow motives, whichis finer that the Grothendeick ring of algebraic varieties, Y. André [2] showed that ifthe Chow motive ofX is Kimura-finite (see [30]) then the Chow motivic zeta function

�Chow.X/.t/ D1XnD0

ŒCh.X .n//�tn 2 K0.CM/ŒŒt ��

is a rational function. Here CM denotes the category of Chow motives.


6.2 Covariant functors and their zeta functions

Let C be a category satisfying that terminal objects and the limit exist, hence, inparticular, for any object X 2 C the product Xn D X �X � �X exists in C . Letpt denote a terminal object.

Let A be a category of Q-vector spaces, or one can think of a category of abeliangroups and then tensor each object with Q. Or more generally one could think of atensor category with coefficients in a field of characteristic zero. Let F W C ! A bea covariant functor which carries the structure of cross products

�F W F .X/ � F .Y / �! F .X � Y /:In the rest of the paper, for any functor F W C ! A and for any object X the

abelian group F .X/ is considered as an F .pt/-module by the cross product

�F W F .pt/ � F .X/ �! F .pt �X/ D F .X/:

From now on, unless some confusion is possible, we omit the subscript F from thecross product �F .

Let F ;H W C �! A be two covariant functors carrying cross products and letT W F ! H be a natural transformation which is compatible with the structures ofcross products:

F .X/ � F .Y /� ��

T

��

F .X � Y /

T

��H .X/ �H .Y / � �� H .X � Y /:

For a covariant functor K W C ! A, the Cartesian product T n W C ! A is simplydefined by

Kn.X/ DK.Xn/:

And for a morphism f W X ! Y , f n� W Kn.X/!Kn.Y / is of course defined by

f n� D .f � f � � f /�:Then it is obvious that Kn is a covariant functor. When n D 0, we understand that forany object X

K0.X/ D K.pt/

and for any morphism f W X ! Y

K0.f / D idK.pt/ W K.pt/ �!K.pt/:

The formal power series of the covariant functor is naturally defined:1XnD0

Kntn W C �! AŒŒt �� D1XnD0

Atn

324 Shoji Yokura

is defined by: for any object X

� 1XnD0

Kntn�.X/ D

1XnD0

Kn.X/tn:

A natural transformation T W F ! H gives rise to the obvious natural transfor-mation:

T W1XnD0

F ntn �!1XnD0

Hntn:

For ˛ 2K.X/ we define

�n W K.X/ �!Kn.X/ DK.Xn/I �n.˛/ D ˛n Dn‚ …„ ƒ

˛ �K � �K ˛ :

Obviously �n W K !Kn is a natural transformation and in the case when n D 0�0 W K �!K0 DK.pt/

is nothing but the pushforward to a terminal object pt. For any power series f .t/ DP1nD0 antn with an 2 Q we define

f .�/ W K �!1XnD0

Kntn

by, for an element ˛ 2K.X/,

f .�/.˛/ D1XnD0

an˛ntn 2

1XnD0

Kn.X/tn:

For a natural transformation T W F ! H the composite

T ı �n W F �! F n �! Hn

is denoted by T n W F ! Hn: because the diagram

F .X/ � � F .X/��

T ��T

��

F .Xn/

T

��H .X/ � �H .X/ �� H .Xn/

commutes we have

T n.˛/ D T .˛n/:


Then the composite

T ı f .�/ W F �!1XnD0

F n.X/tn �!1XnD0

Hn.X/tn

is simply denoted by f .T /.

Remark 6.5. Since the Cartesian product T n is defined by cross products, the abovesimple Cartesian product Kn can be replaced by the image of the homomorphism ofcross products

�K �K W K.X/ � �K.X/ �!K.Xn/:

If we use this finer one, we denote the finer covariant functor by�Kn.X/ D Image .�K �K W K.X/ � �K.X/ �!K.Xn// :

For just now let us assume furthermore that the category C satisfies the followingconditions:

(A) for any object X the symmetric product X .n/ D Xn=Sn exists in C ;

(B) for any object X the projection �n W Xn ! X .n/ exists in C .

Then similarly we can define a finer version of the above. For a covariant functorK W C ! A, the symmetric product K.n/ W C ! A is simply defined by

K.n/.X/ DK.X .n//:

For a natural transformation T W F ! H , we define the symmetric product T .n/

T .n/ W F �! H .n/

by, for any object X and for any ˛ 2 F .X/

T .n/.˛/ D �n� .T .˛/ � � T .˛// 2 H .n/.X/ D H .X .n//:

Here �n W Xn ! X .n/ is the canonical projection to the quotient space. Then T .n/ isa natural transformation.

Since the diagram

F .X/ � � F .X/

T ��T

��

�� F .Xn/

T

��

�n� �� F .X .n//

T

��H .X/ � �H .X/ �� H .Xn/

�n�

�� H .X .n//

commutes, we haveT .n/.˛/ D T .˛.n//;

where ˛.n/ D �n�.˛n/:

326 Shoji Yokura

In this paper we deal with a more general category for which the above addi-tional conditions (A) and (B) might not be satisfied. So we consider the followingSn-invariant one. For a covariant functor K W C ! A, the “Sn-invariant product”KSn W C ! A is defined as the invariant subgroup under the action of Sn:

KSn.X/ DK.Xn/Sn D fx 2K.Xn/ j g�x D x; far all g 2 Sng:

Remark 6.6. As above, we can consider the following finer one:

�KSn.X/ D ��Kn.X/�Sn D fx 2 �Kn.X/ j g�x D x; for all g 2 Sng:

For a partition … D .k1; k2; : : : ; kn/ of n, i.e. a collection of non-negative in-tegers ki such that

PniD1 iki D n, the number ].…/ of permutation � 2 Sn of

cycle-type … is given by

].…/ D nŠ

k1Šk2Š : : : knŠ1k12k2 : : : nkn:

For ˛i 2 H ri .X/ D H .X ri / with r1C r2C C rm D n, the symmetrized crossproduct of ˛i ’s , denoted by ˛1 � ˛2 � � ˛m, is defined by

˛1 � ˛2 � � ˛m D 1

nŠ

X2Sn

��.˛1 � ˛2 � � ˛m/:

Note that the operator

�n D 1

nŠ

X2Sn

��

is usually called symmetrizer in representation theory. In passing, the following oper-ator

An D 1

nŠ

X2Sn

sign.�/��

is called alternizer.

Definition 6.7. Let … D .k1; k2; : : : ; kn/ be a partition and let ˛ 2 F .X/, and let‰ D f n j n 2 F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/.

(i) The ‰-weighted …-cross products of ˛, denoted by ˛�…‰ , is defined by

˛�…‰ D . 1˛/k1 � . 2�2�.˛//k2 � � . n�n�.˛//kn 2 F n.X/:

Here �m W X ! Xm is the diagonal morphism and

�m� W F .X/ �! F m.X/ D F .Xm/:


(ii) The ‰-weighted symmetrized …-cross products of ˛, is defined by

˛�…‰ D �n.˛

�…‰ / 2 F Sn.X/ D F .Xn/Sn ;

that is to say

˛�…‰ D . 1˛/k1 � . 2�

2�.˛//k2 � � . n�n�.˛//kn :

(iii) The average of all‰-weighted symmetrized…-cross products of ˛ with all cycletypes …’s, denoted by ˛�Sn

‰ , is defined by

˛�Sn

‰ D 1

nŠ

X…

].…/˛�…‰ 2 F Sn.X/;

which is called the total ‰-weighted symmetrized cross products of ˛.

(iv) The following formal power series of ˛ with ˛�Sn

‰ as coefficients

Z�S‰ .˛/.t/ D 1C

1XnD1

˛�Sn

‰ tn 2 1C1XnD1

F Sn.X/tn

shall be called the (pre- or stacky) ‰-weighted zeta function of ˛. If we need torefer to which covariant functor we treat, we put the suffix F as Z�S

F ;‰.˛/.t/,

otherwise we omit the suffix F to avoid messy symbols.

When the sequence ‰ is unital, i.e. when n D 1 for all n, then Z�S‰ .˛/.t/ is

simply denoted by Z�S.˛/.t/ and called the zeta function of ˛.

Proposition 6.8. The ‰-weighted zeta function Z�S‰ .˛/.t/ of ˛ is exponential:

Z�S‰ .˛/.t/ D exp

� 1XrD1

t r

r r�

r�.˛/�:

and we have

Z�S‰ .˛ C ˇ/.t/ D Z�S

‰ .˛/.t/� Z�S‰ .ˇ/.t/:

Proof. The proof is standard:

Z�S‰ .˛/.t/ D 1C

1XnD1

˛�Sn

‰ tn

328 Shoji Yokura

D1XnD0

tnX

Pi iki Dn

].…/

nŠ. 1˛/

k1 � . 2�2�.˛//k2 � � . n�

n�.˛//kn

D1XnD0

XP

i iki Dn

tk1C2k2C:::nkn

k1Š : : : knŠ1k1 : : : nkn

. 1˛/k1 � . 2�

2�.˛//k2 � � . n�n�.˛//kn

D1XnD0

XP

i iki Dn

nYrD1

t rkr

kr Šrkr. r�

r�.˛//kr (Q

with respect to �)

D1XnD0

XP

i iki Dn

nYrD1

1

kr Š

� t rr r�

r�.˛/�kr

D1YrD1

� 1Xkr D1

1

kr Š

� t rr r�

r�.˛/�kr

�

D1YrD1

exp� t rr r�

r�.˛/�

D exp� 1XrD1

t r

r r�

r�.˛/�:

Remark 6.9. Let T be an indeterminate. Since we have

log.1 � T /�˛ D �˛ log.1 � T / D ˛1XrD1

T r

r;

we get the following equality:

exp�˛

1XrD1

T r

r

�D exp.log.1 � T /�˛/ D .1 � T /�˛:

So, in the case when the sequence ‰ is unital, if we understand that

.t��/r D t r�r� and� 1XrD1

t r

r�r�

�˛ D

1XrD1

t r

r�r�˛

then we can express the above formula

Z�S.˛/.t/ D exp� 1XrD1

t r

r�r�˛

�


symbolically as follows:

Z�S.˛/.t/ D .1 � t��/�˛:

Definition 6.10. Suppose that conditions (A) and (B) are satisfied. The pushforward ofthe total symmetrized cross products ˛�Sn of ˛ under the projection �n W Xn ! X .n/

is denoted by˛

�Sn

‰ D �n�˛�Sn

‰ 2 F .X .n//:

We set

Z�S‰ .˛/.t/ D 1C

1XnD1

˛�Sn

‰ tn 2 1C1XnD1

F .X .n//tn:

Corollary 6.11. If conditions (A) and (B) are satisfied, the ‰-weighted zeta functionZ�S‰ .˛/.t/ of ˛ is exponential exactly the same as above and we have

Z�S‰ .˛ C ˇ/.t/ D Z�S

‰ .˛/.t/� Z�S‰ .ˇ/.t/:

Definition 6.12. Suppose that the conditions (A) and (B) are satisfied. If there is anassignment fX 2 F .X/ for each object X such that

fX.n/ D .fX /�Sn 2 F .X .n//

then the assignment f is called a symmetrically distinguished assignment and theelement fX is called a symmetrically distinguished element of F .X/.

Let V be the category of complex algebraic varieties. Then conditions (A) and (B)are satisfied and we have the following proposition, the proof of which is given inT. Ohmoto [43] and thus is omitted.

Proposition 6.13. The characteristic function 11X 2 F.X/ is a symmetrically distin-guished element, namely we have

11X.n/ D .11X /�Sn 2 F.X .n//:

Remark 6.14. Note that for any integerd such thatd 6D 0; 1 the constructible functiond 11X 2 F.X/ is not a symmetrically distinguished element.

Corollary 6.15. We have

Z�S.11X /.t/ D 1C1XnD1

11X.n/ tn 2 1C1XnD1

F.X .n//tn:

For the constructible function covariant functor F (note that we considered theQ-tensored one F.X/˝Q), the pushforward for a mapping �X W X ! pt to a point

�X� W F.X/ �! F.pt/ D Q

330 Shoji Yokura

is nothing but taking the Euler characteristic �, in particular �X�.11W / D �.W / for asubvariety W � X.

So, suggested by this, for a covariant functor F and a morphism �X W X ! pt,where pt is a terminal object, the pushforward

�X� W F .X/ �! F .pt/

is called the F -characteristic and denoted by �F

, mimicking the Euler characteris-tic �. If F is the usual Z-homology theory H�, then �

H�W H�.X/ ! H�.pt/ D Z

is the integrationRX

or taking the degree of the 0-dimensional component of any ho-mology class.

For a symmetrically distinguished element fX of F .X/,

�F.X/ D �

F.fX /

shall be called the F -characteristic of X. The function

��F.˛/.t/ D �

F.Z�S.˛/.t//

is called the zeta function of the F -characteristic of ˛ and in particular, for a symmet-rically distinguished element fX of F .X/,

��F .X/.t/ D ��F.fX /.t/

is called the zeta function of the F -characteristic of X.We set

F S.X/ŒŒt �� D 1C1XnD1

F Sn.X/tn;

which is called the symmetrized F -valued formal power series of X.

Proposition 6.16. (i) Let ‰ D f nj n 2 F .pt/ .n D 1; 2; 3; : : : /g be a sequence.For a covariant functor F W C ! A, the correspondence

Z�SF ;‰.t/ W F ! F S.X/ŒŒt ��

defined by

Z�SF ;‰.t/.˛/ D Z�S

F ;‰.˛/.t/


is a natural transformation and the following diagram commutes:

F S.X/ŒŒt ��

ddt

jtD0

��

�F �� F .pt/ŒŒt ��

ddt

jtD0

��F .X/ � 1

��

Z�SF ;‰

.t/

��&&&&&&&&&&&&&&&&&&&&&&&&&&F .X/

�F

�� F .pt/:

(ii) �F.Z�S.˛/.t// D exp

� 1XrD1

t r

r�

F.˛/

�D .1 � t/��F .˛/:

(iii) Suppose that the above conditions (A) and (B) are satisfied and there existsymmetrically distinguished elements fX in F .X/ for each object X. Then we have

��F .X/.t/ D .1 � t/��F .X/:

The above natural transformation Z�SF ;‰

.t/ W F ! F S.�/ŒŒt �� shall be called the‰-weighted zeta function of natural transformation, abusing words.

Remark 6.17. Of course we defined the above‰-weighted zeta function Z�S‰ .˛/.t/

backward from the expected formula:

Z�S‰ .˛/.t/ D exp

� 1XrD1

t r

r r�

r�.˛/�:

A typical model of this is of course the constructible function functor F.X/ and theEuler characteristic. In this sense the‰-weighted zeta natural transformation Z�S

‰ .t/

could be called a ‰-weighted zeta function of natural transformation of Euler-type.

Theorem 6.18. Let ‰ D f nj n 2 K0.V=pt/ D K0.V/ .n D 1; 2; 3; : : : /g be asequence of “motivic” classes of complex algebraic varieties. Then the ‰-weightedzeta function of natural transformation

Z�S‰ .t/ W K0.V=�/ �! K0.V=�/SŒŒt ��

is a generalization of the Kapranov motivic zeta function �KapŒM�.t/ in the following

sense:

332 Shoji Yokura

�K0

�Z�S.t/.ŒM

k�! X�/.t/�D exp

� 1XrD1

t r

r�K0

.ŒMk�! X�/

�

D exp� 1XrD1

t r

rŒM �

�

D .1 � t/�ŒM�

D �KapŒM�.t/:

Remark 6.19. (i) In the case of the constructible function functorF.X/, our‰-weigh-ted zeta function of natural transformation Z�S

‰ .t/ W F.X/! FS.X/ŒŒt �� is describedas follows, using the expression of Ohmoto’s paper [43], Remark 3.8. If we let

'.t/ D 1C1XnD1

n�n�.˛/ 2 F.X/˝QŒŒt ��;

where the diagonal of Xn is identified with X, Z�S‰ .˛/.t/ D T .'.t// with the corre-

sponding operator T DP1nD0 Tn W F.X/˝QŒŒt ��! FX;symŒŒt ��:

(ii) In the case of K0.V=X/, �K0.Z�S

‰ .ŒM ! X�/.t/ is almost the same as

the “motivic power series” A.t/ŒM� introduced by Gusein-Zade, Luengo and Melli-Hernández [23], where A.t/ D 1CP1

nD1 ntn 2 K0.V/ŒŒt ��. More precisely, if nis the first zero term, then

�K0.Z�S

‰ .ŒM ! X�/.t/ A.t/ŒM�.mod tn/:

In particular, if the sequence ‰ is unital, then we have

�K0.Z�S.ŒM ! X�/.t/ D

� 1XnD0

tn�ŒM� D .1 � t/�ŒM�:

6.3 Natural transformations and their zeta functions

Definition 6.20. Let F ;H W C ! A be covariant functors. Let ‰ D f n j n 2F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/. For a natural transfor-mation T W F ! H , we define the Sn-invariant product T Sn :

T Sn

‰ W F �! HSn

by, for any object X and for any ˛ 2 F .X/

T Sn

‰ .˛/ D T .˛�Sn

‰ /:

Lemma 6.21. T Sn

‰ is a natural transformation.


Proposition 6.22. Suppose that the above conditions (A) and (B) are satisfied. Thenfor any natural transformation T W F ! H , we have the canonical natural transfor-mation

�n� W HSn �! H .n/

and the following diagram commutes:

HSn � Hn

�n�

��

F

T.n/

‰��'''

''''''''

''''''''

''''''

TSn

‰

��

H .n/:

Definition 6.23. If the above conditions (A) and (B) are satisfied, then we set

H .1/.X/ŒŒt �� D1XnD0

H .n/.X/tn:

Here we set that H .0/.X/ D Q.

Definition 6.24. Let T W F ! H be a natural transformation and let ‰ D f nj n 2F .pt/ .n D 1; 2; 3; : : : /g be a sequence of elements of F .pt/.

(i) The ‰-weighted zeta function of the natural transformation T is defined by

Z�ST ;‰.t/ D

1XnD0

T Sn

‰ tn W F .�/ �! HS.�/ŒŒt ��:

Here we set that TS0

‰ .X/ D 1 for any object X.

(ii) If the above conditions (A) and (B) are satisfied, then the‰-weighted zeta functionof the natural transformation T is defined by

ZT ;‰.t/ D1XnD0

�n�T Sn

‰ tn W F .�/ �! H .1/.�/ŒŒt ��:

Here we set that T.0/‰ .X/ D 1 for any object X.

334 Shoji Yokura

In other words, since the natural transformation T W F ! H induces the canonicalnatural transformation

T W F S.�/ŒŒt �� ! HS.�/ŒŒt ��;

the above ‰-weighted zeta function of the natural transformation T is nothing but

Z�ST ;‰.t/ D T ıZ�S

F ;‰.t/:

Corollary 6.25. (1) The ‰-weighted zeta functions Z�ST ;‰

.t/ and ZT ;‰.t/ are bothnatural transformations.

(2) The‰-weighted zeta functions Z�ST ;‰

.t/ and ZT ;‰.t/ are both exponential, i.e.we have

Z�ST ;‰.˛ C ˇ/.t/ D Z�S

T ;‰.˛/.t/� Z�ST ;‰.ˇ/.t/;

ZT ;‰.˛ C ˇ/.t/ D ZT ;‰.˛/.t/� ZT ;‰.ˇ/.t/

and explicitly we have

Z�ST ;‰.t/ D ZT ;‰.t/ D exp

� 1XrD1

t r

r r�

r�T�:

In particular, in the case when the sequence ‰ is unital, then we have

Z�ST .t/ D ZT .t/ D exp

� 1XrD1

t r

r�r�T

�D .1 � t��/�T :

(3) The following diagram commutes:

HS.�/ŒŒt �� .or H .1/.�ŒŒt �� /

ddt

jtD0

��

F .�/

T

""(((((((

((((((((

((((((((

(((

Z�ST

.t/ .or ZT .t//

��

H .�/:


For a natural transformation T W F ! H , we let us suppose that F .pt/ D H .pt/and T .pt/ W F .pt/! H .pt/ is the identity. Then we have the following commutativediagram:

HS.X/ŒŒt ��

ddt

jtD0

��

�H �� H .pt/ŒŒt ��

ddt

jtD0

��F .X/

�F

##)))

))))

))))

))))

))))

)))

T ��

Z�ST

.t/

$$**********************H .X/

�H

��

�H �� H .pt/

F .pt/ D H .pt/:

The above naïve zeta function of natural transformation Z�ST.t/ W F ! HS.�/ŒŒt ��

of a given natural transformation T W F ! H could be called a zeta function ofnatural transformation of Chern class-type in the sense that if T W F ! H is theChern–MacPherson class transformation c� W F ! H� then ZT .t/ is nothing but thegenerating series of the Chern–MacPherson class constructed by T. Ohmoto [43].

6.4 A zeta function of the motivic Hirzebruch class

For other well-studied characteristic classes such as Baum–Fulton–MacPherson’sTodd class or Riemann–Roch [4] and Goresky–MacPherson’s homology L-class [22]and Cappell–Shaneson’s homology L-class [12], the canonical generating series ofthem, i.e. the zeta function cannot be described as above. In a sense the above‰-weighted zeta function of natural transformation is more or less a correct one,but we need to interpret the “multiplication by r” differently and also we need tomodify the symmetrized product T .˛�Sn

‰ /, as done in the following section. For thatpurpose the description of the motivic characteristic class or the motivic Hirzebruchcharacteristic class Ty� W K0.V=X/ ! H�.X/ ˝ QŒy� is essential and crucial. Inother words, for a general category or in a general set-up, there is no reason to thinkof such a modification or trick.

For the sake of presentation given below, from here we consider T�y� insteadof Ty�, in other words we change our parameter y by �y. Thus we have that set-

336 Shoji Yokura

ting y D 1, y D 0, and y D �1 produce the Chern class, Todd class and L-class,respectively.

In this section we consider another kind of‰-weighted zeta natural transformationfor the above motivic Hirzebruch characteristic class T�y�, but with a completelydifferent meaning of the weight, i.e. the multiplication by each weight r is understoodas an Adams operator together with another extra “deformation”.


‰Z�ST�y �

.t/ D1XnD0

‰T�ySn� tn W K0.V=�/! HS.�IQŒy�/ŒŒt ��;

with ‰T�yS0� .X/ D 1 for any object X, and for any ˛ 2 K0.V=X/,

‰T�ySn� .˛/ D 1

nŠ

X…D.k1;:::;kn/

].…/‰T�y�.˛�…/ 2 HSn.X/˝QŒy�

and

‰T�y�.˛�…/ D 1T�yk1� .˛/� : : : rT�yr

kr� .�r�.˛//� � nT�yn

kn� .�n�.˛//;

thus by a linear extension we have

‰T�ySn� .˛/ D ‰T�y�.˛�Sn/;

where the operator r is defined by

rT�yrkr� .˛/ D

1XmD0

1

rm

�T�yr

kr� .˛/�2m;

where A2m denotes the degree-2m part of the total homology class A. (In this caseonly even-degree homology classes appear.)

To emphasize that “‰” functions as an operator, not simply as a multiplication, weshall call ‰T�ySn� .˛/ a ‰-operated symmetrized product of T�y�.˛/.

Remark 6.27. (i) Note that the above ‰-operated symmetrized product‰T�ySn� .˛/ has a “deformation” in each “r-th component” unlike the previous formal‰-weighted symmetrized product, which would be simply

1T�yk1� .˛/� : : : rT�ykr� .�r�.˛//� � nT�ykn� .�

n�.˛//:

Namely, in each “r-th component” rT�yrkr� .�r�.˛//, the original natural transfor-

mation T�y� is replaced by the deformed one T�yr � with the different parameter�yr .

(ii) By the definition T�yrkr� .�r�.˛// D

�T�yr �.�r�.˛//

�kr . Thus it is clear thatby the definition of the operator r we have

rT�yrkr� .�

r�.˛// D� rT�yr �.�

r�.˛//�kr

:


(iii) As to the above ‰-operated symmetrized product, in fact one can also definesuch a ‰-operated symmetrized product (but without the above deformation) for anynatural transformation T W F ! H as long as the Q-vector space H .X/ has a Z-grading. Namely, for a natural transformation T W F ! H , we can define the “simple”‰-operated symmetrized product of T .˛/ as follows:

1Tk1.˛/� : : : rT

kr .�r�.˛//� � nT kn.�n�.˛//;

where the operator r is defined by

rT .˛/ D1XmD0

1

.pr/m

.T .˛//m :

We make such a definition so that it is compatible with the above special case. Other-wise, if we do not take care of such a compatibility, putting aside the issue of geometricmeaning, we could simply define it as follows:

rT .˛/ D1XmD0

1

rm.T .˛//m :

From here on we simply denote Z�ST�y �

.t/ without ‰, since the operator “‰00 isused in a unique way.

Theorem 6.28. (1) The zeta function Z�ST�y �

.t/ is a natural transformation.

(2) The zeta function Z�ST�y �

.t/ is exponential, i.e. for any object X and for any

element ˛; ˇ 2 K0.V=X/, we have

Z�ST�y �

.˛ C ˇ/.t/ D Z�ST�y �

.˛/.t/� Z�ST�y �

.ˇ/.t/

and explicitly we have

Z�ST�y �

.t/ D exp� 1XrD1

t r

r r�

r�T�yr ��:

More in detail we have

Z�ST�y �

.t/ D

8ˆˆ<ˆˆ:

exp� 1XrD1

t r

r�r�T�1�

�D .1 � t��/�T�1� ; y D 1;

exp� 1XrD1

t r

r r�

r�T0��; y D 0;

exp� 1XrDeven

t r

r�r�T�1�

�exp

� 1XrDodd

t r

r r�

r�T1��

D .1 � t2�2�/�T�1�

2 exp� 1XrDodd

t r

r r�

r�T1��; y D �1:

338 Shoji Yokura

(3) The following diagram commutes:

HS� .X IQŒy�/ŒŒt ��ddt

jtD0

��

�H� �� QŒy�ŒŒt ��

ddt

jtD0

��K0.V=X/

K0

��

T�y �

��

Z�ST�y �

.t/

%%################H�.X I QŒy�/

�H�

��

H�

��

QŒy�

K0.V/ y�� QŒy�:

The diagram

K0.V=Xn/Sn

�n�

��K0.V=X/ �Sn

��

�Sn

&&��K0.V=X

.n//

commutes and we have the following result.

Proposition 6.29. ŒXidX��! X� is a symmetrically distinguished element ofK0.V=X/.

Corollary 6.30. For any complex quasi-projective variety X the following holds:

�H�.Z�S

T�y �.ŒX

idX��! X�/.t//: D 1C1XnD1

��y.X .n//tn:

In particular,

(i) when y D 1, we have

�H�.Z�S

T�1�.ŒX

idX��! X�/.t// D ��.X/.t/ D .1 � t/��.X/I

(ii) when y D 0, we have

�H�.Z�S

T0�.ŒX

idX��! X�/.t// D ��a.X/.t/ D .1 � t/��a.X/:

Here �a.X/ is the arithmetic genus, i.e. the degree of the 0-dimensional compo-nent of the Todd class T0�.X/.


(iii) when y D �1, we have

�H�.Z�S

T1�.ŒX

idX��! X�/.t// D �.X/.t/

D .1 � t2/� .X/2

�1C t1 � t

��.X/2

D .1C t/�.X/�.X/2

.1 � t/�.X/C.X/2

:

Here �.X/ is the Goresky–MacPherson’s signature, i.e. the degree of the 0-di-mensional component of the Goresky–MacPherson’s homologyL-class T1�.X/.

Corollary 6.31. Theorem 6.28 and Corollary 6.30 also hold with � and HS� .�/ŒŒt ��being replaced by � and H .1/� .�/ŒŒt ��, respectively.

Corollary 6.32. For any complex projective variety X the following holds:

Z�ST�y �

.ŒXidX��! X�/.t// D exp

� 1XrD1

t r

r r�

r�T�yr �.X/�

D 1C1XnD1

T�y�.X.n//tn:

More details and more related things will be treated in different papers.

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Minimality of hyperplane arrangementsand basis of local system cohomology

Masahiko Yoshinaga�

Department of Mathematics, Kyoto UniversitySakyo-ku, Kyoto, 606-8502, Japan


Abstract. The purpose of this paper is applying minimality of hyperplane arrangements to localsystem cohomology groups. It is well known that twisted cohomology groups with coefficientsin a generic rank one local system vanish except in the top degree, and bounded chambers forma basis of the remaining cohomology group. We determine precisely when this phenomenonhappens for two-dimensional arrangements.

1 Introduction

The purpose of this paper is applying minimality of hyperplane arrangements to localsystem cohomology groups. In §1.1 and §1.2, we will recall basic notions and resultson these topics. In §1.3, we will give the plan of the paper.

1.1 Minimality of hyperplane arrangements

Let A D fH1; : : : ;Hng be a hyperplane arrangement in C`. Namely a finite set ofaffine hyperplanes. We assume each hyperplane Hi D f˛i D 0g � C` is defined byan affine linear equation ˛i . We denote the complement of hyperplanes by

M.A/ D C` nn[iD1

Hi :

After the discovery of a combinatorial description of the ringH�.M.A/;Z/ in [15]and of the K.�; 1/-property for simplicial arrangements in [4], it has been found thatthe complement M.A/ of a hyperplane arrangement A has a very special homotopytype among other complex affine varieties. In particular, the following minimalityproperty seems one of the most peculiar characteristics of M.A/; see [6], [18], [17],and [10].

�Work partially supported by JSPS Grant-in-Aid for Young Scientists (B) 20740038.


346 Masahiko Yoshinaga

Theorem 1.1 (Minimality of arrangements). The complement M.A/ is homotopyequivalent to a finite minimal CW-complex X . Namely, X satisfies the following min-imality property: the number of k-dimensional cells ]fk- dim cellsg is equal to thek-th Betti number bk.X/.

The minimality is expected to be useful for computations of local system cohomol-ogy groups. An immediate corollary is the following upper bounds for dimensions ofrank one local system cohomology groups, which were conjectured by Aomoto andfirst proved in [2] by using another method.

Corollary 1.2. Let L be a complex rank one local system on M.A/. Then the dimensionof the L-coefficients cohomology group is bounded by the Betti number:

dimH k.M.A/;L/ � bk.M.A//;for k D 0; 1; : : : ; `.

For further applications of the minimality to computations of local system co-homology groups, the description of the minimal CW-complex X, in particular theattaching map of each cell, is needed. However Theorem 1.1 does not tell it. It shouldbe noted that the proof of Theorem 1.1 is based on Morse theoretic arguments. Theconstructions of cells are relying on a transcendental method, namely using gradientflows of a Morse function.

Both of the two recent approaches to the problem of describing attaching maps ofminimal cells are:

• assuming A is defined over the real numbers R, and

• describing attaching maps by using combinatorial structure of chambers.

However they used different methods.

• In [23], we studied Lefschetz’s hyperplane section theorem for M.A/, and de-scribed the attaching maps of the top cells.

• In [20], Salvetti and Settepanella developed discrete Morse theory on the Salvetticomplex, and then described the minimal cell complex by using discrete Morseflows.

See [5] and [7] for subsequent developments. Furthermore, in [11], a 2-dimensionalalgebraic minimal chain complex is described. The present article can be consideredas a counterpart of [11].

1.2 Non-resonant local systems

A nonempty intersection of elements of A is called an edge. We denote by L.A/ theset of edges. An edgeX 2 L.A/ is called a dense edge if the localization AX D fH 2A j H � Xg is indecomposable. We denote by D.A/ � L.A/ the set of dense edges.

Minimality of hyperplane arrangements and basis of local system cohomology 347

Let � D .�1; : : : ; �n/ 2 Cn. Then � determines a rank one representation of�1.M.A// by

�1.M.A// 3 7�! exp� Z

�

nXiD1

�id log˛i�2 C�

and the associated local system L D L. In other words, L is determined by thelocal monodromy qi D e2�

p�1i 2 C� around each hyperplane Hi . For an edgeX 2 L.A/, denote

qX DYX�Hi

qi :

We also denote the half twist by q1=2i D e�p�1i .

We can embed the affine space C` in CP` as C` D CP` nH1. We call

A1 D f xH j H 2 Ag [ fH1gthe projective closure of A. The monodromy of L around the hyperplane at infinityH1 is

QniD1 q�1

i . It is natural to define

q1 DnYiD1

q�1i :

The structure of the cohomology group H k.M.A/;L/ with local system coeffi-cients has been well studied; see [1], [9], [13], and [21]. In particular, it is known thatif L is generic, then the cohomology vanishes except in k D `. Among others, let usrecall two results in this direction; see [8], [14], and [3].

Theorem 1.3 ([8]). Suppose that A is defined over R and the local system L satisfies

qX ¤ 1; for X 2 D.A1/: (1.1)

Then

H k.M.A/;L/ D

8<:0 for k ¤ `;LC2bch.A/

C ŒC � for k D `; (1.2)

where bch.A/ stands for the set of all bounded chambers. A chamber ŒC � can be con-sidered as a locally finite cycle, in other words, an element of Borel–Moore homologyŒC � 2 HBM

`.M.A//. In (1.2) we identify the chamber C with cohomology via the

canonical isomorphism HBM`.M.A// ' H `.M.A//.


D1.A1/ D fX 2 D.A1/ j X � H1g:


Theorem 1.5 ([3]). Suppose that the local system L satisfies

qX ¤ 1, for 8X 2 D1.A1/: (1.3)

Then

H k.M.A/;L/ '8<:0 for k ¤ `;

Cj�.M.A//j for k D `;where �.M.A// is the Euler characteristic of M.A/.

1.3 Plan of the paper

The purpose of this paper is to refine vanishing results of Theorem 1.3 and Theorem 1.5for ` D 2 by using minimal complex arising from minimal CW-decomposition ofM.A/. We will prove that the assertion (1.2) of Theorem 1.3 is true under the weakerassumption (1.3). Furthermore, if A is indecomposable, we also prove that the as-sumption can not be weakened any more. Our main result asserts that (1.3) and (1.2)are equivalent (for ` D 2).

In §2, we treat combinatorial structures of chambers, which will play a crucial rolein the study of minimal complex.

In §3, we will describe the minimal cochain complex arising from Lefschetz’shyperplane section theorem. Particularly, we treat the case ` D 2 in details.

In §4, we prove the main result, that is, that for an indecomposable two dimensionalarrangement A, conditions (1.3) and (1.2) are equivalent.

Acknowledgements. This is an expanded version of the author’s talk at “Fifth Franco-Japanese Symposium on Singularities.” He is grateful to the organizers.

2 Chambers and flags

2.1 Involution on unbounded chambers

Let A be a hyperplane arrangement in R`. We denote the set of chambers, boundedchambers, unbounded chambers by ch.A/; bch.A/; uch.A/, respectively. Note thatch.A/ D bch.A/ t uch.A/.

LetC 2 uch.A/be an unbounded chamber. Then the closure cl.C / in the projectivespace RP` intersects the hyperplane H1 at infinity.

Definition 2.1. Let C 2 uch.A/. (i) Define X.C/ to be the smallest subspace ofH1which contains cl.C /\H1. (ii) There exists a unique chamber which is the oppositewith respect to cl.C / \H1. We denote the opposite chamber by C_ (see Figure 1).Obviously we have C__ D C .


See Figure 1 for an example. In this figure,X.C1/ D X.C4/ D H1, andX.C2/ DX.C3/ D cl.C2/ \H1.

C1 C2 C3 C4

C_4

C_4

C_2

C_2

C_3

C_3 C_

1

cl.C2/ \H1

cl.C4/ \H1

H1

Figure 1. C and C_.

Definition 2.2. Define the involution � by

� W uch.A/ �! uch.A/;

C 7�! C_:

We now characterize dense edges contained inH1 by usingX.C/. First we provean easy lemma.

Lemma 2.3. Let A be an essential central arrangement in R`. Then the followingproperties are equivalent.

(1) A is indecomposable.

(2) There exist H 2 A and C 2 ch.A/ such that cl.C / \H D f0g.(3) For any H 2 A, there exists C 2 ch.A/ such that cl.C / \H D f0g.Proof. LetH 2 A and consider the deconing dHA with respect toH. Note that dHA

is an affine arrangement of rank .`� 1/. Using [16], §3.3, A is indecomposable if andonly if the ˇ-invariant of dHA is nonzero. By the famous result of Zaslavsky [25], itis equivalent to the existence of bounded chambers of the deconing dHA. Choose abounded chamber of dHA, and let C be its cone. Then cl.C /\H D f0g. This proves(1) H) (3). The other implications can also be similarly proved.


Using the above lemma, we obtain the following result.

Proposition 2.4. Let A be an affine arrangement in R`. An edgeX 2 L.A1/ satisfiesX 2 D1.A1/ if and only if X D X.C/ for some C 2 uch.A/.

2.2 Generic flags

Let F be a generic flag in R`

F W ; D F �1 � F 0 � F 1 � � F ` D R`;

where each F q is a generic q-dimensional affine subspace, that is, dim F q \ X DqC dimX � ` for X 2 L.A1/. Let fh1; : : : h`g be a system of defining equations ofF, that is,

F q D fhqC1 D D h` D 0g; for q D 0; 1; : : : ; ` � 1;where each hi is an affine linear form on R`. Using the flag F, we decompose the setof chambers into several subsets.

Definition 2.5. Define

chq.A/ D fC 2 ch.A/ j C \ F q 6D ; and C \ F q�1 D ;g;for q D 0; 1; : : : ; `.Proposition 2.6 ([23]). ]chq.A/ D bq.M.A//.Remark 2.7. The above proposition gives a refinement of Zaslavsky’s formulaX

iD0bi .M.A// D ]ch.A/I

see [25].

We assume that F satisfies the following conditions. For q D 0; : : : ; `, set

Fq>0 D fhqC1 D hqC2 D D h` D 0; hq > 0g:

(i) For an arbitrary chamber C , if belonging to chq.A/, then C \ F q � Fq>0.

(ii) For any two X , X 0 2 L.A/ with dimX D dimX 0 D ` � q (i.e. satisfyingX \ F q D fptg and X 0 \ F q D fptg), if X 6D X 0,

hq.X \ F q/ 6D hq.X 0 \ F q/:

In the remainder of the paper we fix a generic flag F satisfying the above conditions.And also fix the orientation of F q by the oriented basis .@h1

; : : : ; @hq/ of the tangent

space TxF q .Next we further decompose chq.A/ into two subsets.


Definition 2.8. Define subsets bchq.A/ and uchq.A/ of chq.A/ by

bchq.A/ D fC 2 chq.A/ j C \ F q is boundedg;uchq.A/ D fC 2 chq.A/ j C \ F q is unboundedg:

We note that bch`.A/ D bch.A/.

Example 2.9. Let us consider the arrangement of four lines A D fH1;H2;H3;H4gwith a generic flag F as in Figure 2.

C0

C1

C2

C3

C4 C_4

C_3 C_

1

C_2

H1 H2 H3 H4

F 0 F 1

F 2 D R2

Figure 2. bchq.A/ and uchq.A/.

Then we have by definition

ch0.A/ D fC0g; ch1.A/ D fC1; C2; C3; C_0 g; ch2.A/ D fC_

1 ; C_2 ; C

_3 ; C4g;

bch0.A/ D fC0g; bch1.A/ D fC1; C2; C3g; bch2.A/ D fC4g;uch0.A/ D ;; uch1.A/ D fC_

0 g; uch2.A/ D fC_1 ; C

_2 ; C

_3 g:

Theorem 2.10. The involution � induces a bijection

� W bchq�1.A/ �! uchq.A/:

Proof. Suppose C 2 bchq�1.A/, that is, C \ F q�1 is bounded. Then we haveC_ \ F q�1 D ;. By the assumption on the flag, C \F q is unbounded. Since F q isgeneric, cl.F q/ intersects cl.C / \H1 transversally. Hence C_ \ F q ¤ ; and it isunbounded. We have C_ 2 uchq.A/.

Conversely ifC_ 2 uchq.A/, thenC__ D C intersects F q�1. SupposeC\F q�1is unbounded. In this case, C_ also intersects F q�1. This contradicts the hypothesisC_ 2 uchq.A/ � chq.A/.

Corollary 2.11. ]bchq�1.A/ D ]uchq.A/.


Remark 2.12. (1) Corollary 2.11 together with Proposition 2.6 and ]chq.A/ D]bchq.A/C ]uchq.A/, gives a “bijective proof” for Zaslavsky’s formula ]bch.A/ DPìD0.�1/`�ibi .M.A//.(2) The bijective correspondence (Theorem 2.10) plays a crucial role in §4.

3 Minimal complexes

Let A be an essentially real arrangement and F be a generic flag as in the previoussection. SetF D F `�1˝C, the complexification of F `�1. Compare the complexifiedcomplement M.A/ with the generic hyperplane section M.A/ \ F. Lefschetz’s hy-perplane section theorem [12] tells us that M.A/ is homotopy equivalent to the spaceobtained from M.A/\F by attaching some `-dimensional cells. Namely we have thefollowing homotopy equivalence:

M.A/ � .M.A/ \ F / ['i

[i

D`;

where 'i W @D` ! M.A/\F is the attaching map. In [23], we described the homotopytype of the attaching maps. The `-dimensional cells are naturally encoded by the setch`.A/ of chambers which do not intersect F `�1. By using the description of attachingmaps, we constructed a cochain complex

.CŒchq.A/�; dL/`qD0 W : : : �! CŒchq.A/�

dL�! CŒchqC1.A/� �! : : :

which computes local system cohomology groups for arbitrary rank one local sys-tem L. Namely, we have H�.CŒch.A/�; dL/ ' H�.M.A/;L/. In §3.1, we shalldescribe the cochain complex .CŒch.A/�; dL/ based on [23], and in §3.2 we inves-tigate the case ` D 2 closely.

3.1 Minimal complex arising from Lefschetz’s Theorem

Definition 3.1 (Separating hyperplanes). Let C1; C2 2 ch.A/ be chambers. Set

Sep.C1; C2/ D fH 2 A j H separates C1 and C2gand

q1=2

Sep.C1;C2/D

YHi 2Sep.C1;C2/

q1=2i :

To describe the coboundary map dL W CŒchq.A/� ! CŒchqC1.A/�, we need thenotion of degree map

deg W chq.A/ � chqC1.A/ �! Z;


which we will define below.Suppose C 2 chq.A/ and C 0 2 chqC1.A/ are given. Let D D Dq � F q

be a q-dimensional ball with sufficiently large radius so that every 0-dimensionaledge x 2 L.A \ F q/ is in the interior of Dq . There exists a tangent vector fieldU.x/ 2 TxF q for x 2 D which satisfies the following properties:

• if x 2 @D, then U.x/ … Tx.@D/, and U.x/ directs inside of D;smallskip

• if x 2 H with H 2 A, then U.x/ … Tx.H \ F q/ � TxF q and U.x/ directsthe side in which C 0 is contained.

From the properties above, we have U.x/ ¤ 0 for x 2 @.cl.C /\D/, where cl.C / isthe closure of C in F q . Roughly speaking, the degree deg.C; C 0/ is defined to be thedegree of the Gauss map

U

jU j W @.cl.C / \D/ �! Sq�1:

Definition 3.2. Let C 2 chq.A/ and C 0 2 chqC1.A/. Fix U as above. Then definedeg.C; C 0/ as follows.

(1) When q D 0, then deg.C; C 0/ D 1.

(1) When q D 1, then cl.C / \D ' Œ�1; 1�. In this case S0 ' f˙1g. The degree ofthe Gauss maps

g D U

jU j W f˙1g �! f˙1g

is defined by

deg.g/ D

8<:0; if g.¹˙1º/ D ¹C1º or g.¹˙1º/ D ¹�1º;1; if g.˙1/ D ˙1;�1; if g.˙1/ D �1:

(2) When q � 2,

deg.C; C 0/ D deg� UjU j W @.cl.C / \D/! Sq�1�:

(It is easily seen that deg.C; C 0/ does not depend on U.)

Now let us define the map

dL W CŒchq.A/� �! CŒchqC1.A/�

by

chq.A/ 3 ŒC � 7�!X

C 02chqC1.A/

deg.C; C 0/ .q1=2Sep.C;C 0/� q�1=2

Sep.C;C 0// ŒC 0�: (3.1)


Theorem 3.3 ([23], §6.4.1). With notation as above, .CŒch.A/�; dL/ is a cochaincomplex. Furthermore,

H�.CŒch.A/�; dL/ ' H�.M.A/;L/:

In the above formula (3.1), the degree deg.C; C 0/ 2 Z is difficult to determine.The author wonders how to compute deg.C; C 0/. Let us pose a problem which mightbe interesting from the view point of combinatorics of polytopes.

Problem 3.4. LetP � Rd be a bounded d -dimensional convex polytope. Let fFege2Ebe the set of facets (i.e. .d � 1/-dimensional faces). Let U.x/ 2 TxRd be a vectorfield on Rd . Suppose that U satisfies U.x/ ¤ 0 when x 2 @P and, furthermore,U.x/ … TxFe for any point x 2 Fe in a facet. We can associate a sign vectorX 2 fC1;�1gE by

X.e/ D8<:C1 if Udirects outside of P on Fe;

�1 if Udirects inside of P on Fe:

Then how to compute the degree deg�U

jU j W @P ! Sd�1� of the Gauss map from the

sign vector X 2 f˙1gE?

3.2 The case ` D 2

In this section, we look at the minimal complex .CŒch.A/�; dL/ for ` D 2 moreclosely.

First note that ch0.A/ D fC0g consists of a chamber. The map

dL W CŒch0.A/� �! CŒch1.A/�

is determined by

dL.ŒC0�/ DX

C2ch1.A/

.q1=2

Sep.C0;C/� q�1=2

Sep.C0;C// ŒC �:

As in §2.1, we decompose ch1.A/ D bch1.A/ t uch1.A/. Note that by Theorem2.10, uch1.A/ D fC_

0 g consists of a chamber which is the opposite one of C0. Thesecond coboundary map

dL W CŒch1.A/� �! CŒch2.A/�

is given by (3.1).The degree deg.C; C 0/behaves differently according asC 2 bch1.A/or C 2 uch1.A/.


(i) Suppose C 2 bch1.A/. Then C \ F 1 is a closed interval, the boundary (twopoints) can be expressed as .H \F 1/[ .H 0\F 1/ forH;H 0 2 A. deg.C; C 0/can be computed as

deg.C; C 0/ D

8<:1 if H;H 0 2 Sep.C; C 0/;�1 if H;H 0 … Sep.C; C 0/;0 otherwise.

(ii) Suppose C 2 uch1.A/. Then C \ F 1 is an unbounded interval, the boundary(a point) can be expressed asH \F 1. The degree deg.C; C 0/ can be computedas

deg.C; C 0/ D´�1 if H … Sep.C; C 0/;0 if H 2 Sep.C; C 0/:

In particular, we have the following result.

Lemma 3.5. LetC 2 bch1.A/. The boundary ofC \F 1 is expressed as .H \F 1/[.H 0 \ F 1/. Then

deg.C; C_/ D´1 if H and H 0 are not parallel,

�1 if H and H 0 are parallel.

Example 3.6. Consider the arrangement of four lines A D fH1;H2;H3;H4g in R2

and a generic flag F as in Figure 3.

C0

C1 C2 C3

D C_0

C_3 C_

2 C_1

H1 H2 H3 H4

F 0 F 1

F 2 D R2

Figure 3. Example 3.6.

We have

bch0.A/ D fC0g; bch1.A/ D fC1; C2; C3g; bch2.A/ D fDg;uch0.A/ D ;; uch1.A/ D fC_

0 g; uch2.A/ D fC_1 ; C

_2 ; C

_3 g:


The coboundary mapdL W CŒch0� �! CŒch1�

is determined by

dL.ŒC0�/ D .q12

1 � q� 12

1 /ŒC1�C .q12

12 � q� 12

12 /ŒC2�

C .q 12

123 � q� 12

123/ŒC3�C .q12

1234 � q� 12

1234/ŒC_0 �;

anddL W CŒch1� �! CŒch2�

is given as follows:

dL.ŒC1�/ D .q12

1234 � q� 12

1234/ŒC_1 �C .q

12

124 � q� 12

124/ŒC_2 �C .q

12

12 � q� 12

12 /ŒD�;

dL.ŒC2�/ D �.q12

14 � q� 12

14 /ŒC_2 � � .q

12

1 � q� 12

1 /ŒD�;

dL.ŒC3�/ D C.q12

134 � q� 12

134/ŒC_2 �C .q

12

1234 � q� 12

1234/ŒC_3 �;

dL.ŒC_0 �/ D �.q

12

1 � q� 12

1 /ŒC_1 � � .q

12

13 � q� 12

13 /ŒC_2 � � .q

12

123 � q� 12

123/ŒC_3 �:

The coefficients of the diagonals have another expressions. Observe that we haveX.C1/ D X.C3/ D H1 and X.C2/ D SHN2 \ SHN3 \ H1. Since q1 D q�1

1234 andqX.C2/ D q2q3q1 D q�1

14 , we have

.q12

1234 � q� 12

1234/ D �.q12

X.C1/� q� 1

2

X.C1// D �.q 1

2

X.C3/� q� 1

2

X.C3//;

�.q 12

14 � q� 12

14 / D q12

X.C2/� q� 1

2

X.C2/:

In general, we have the following result.

Proposition 3.7. Let C 2 bch1.A/. Then the coefficient of ŒC_� in dL.ŒC �/ is givenby ˙.q1=2

X.C/� q�1=2

X.C//.

Proof. Let H 2 A. Then H separates C and C_ if and only if xH does go throughX.C/ 2 H1. Using q1q2; : : : qn; q1 D 1, we have

qSep.C;C_/ D q�1X.C/:

Hence˙.q1=2Sep.C;C_/� q�1=2

Sep.C;C_// D �.q1=2

X.C/� q�1=2

X.C//.

For use in the next section, we analyze the induced map

CŒbch1.A/� ,�! CŒch1.A/�dL��! CŒch2.A/� �!! CŒuch2.A/�:


Consider the composed map SdL W CŒbch1.A/� �! CŒuch2.A/�. By Theorem 2.10,the bases of the source and the target of SdL are naturally identified by the involution �.Thus the determinant det. SdL/ 2 C makes sense. The matrix SdL is expressed by anupper triangular matrix, and the determinant can be computed.

Theorem 3.8. The determinant det. SdL/ can be expressed as

det. SdL/ D ˙Y

X2D1.A1/

.q1=2X � q�1=2

X /nX ; (3.2)

where nX is a positive integer.

Proof. First note that, forC 2 uch.A/,X.C/ is either 0-dimensional or equal toH1.We call an unbounded chamber C 2 uch.A/ narrow (resp. wide) if X.C/ � H1is 0-dimensional (resp. X.C/ D H1). We decompose CŒbch1.A/� and CŒuch2.A/�into direct sum of subspaces. Set

N 1 D CŒfC 2 bch1.A/ j C W narrowg�; W 1 D CŒfC 2 bch1.A/ j C W wideg�;N 2 D CŒfC 2 uch2.A/ j C W narrowg�; W 2 D CŒfC 2 uch2.A/ j C W wideg�:

Then clearly CŒbch1.A/� D W 1 ˚ N 1 and CŒuch2.A/� D W 2 ˚ N 2. The mapSdL preserves N i . Furthermore, the matrix presentation of SdLjN1 W N 1 ! N 2 is

diagonal. Indeed suppose that C 2 bch1.A/ is a narrow chamber with wallsH \F 1

and H 0 \ F 1. Then H and H 0 are parallel. By definition of degree map, dL.ŒC �/

is a linear combination of chambers which are put between H and H 0. The oppositechamber C_ is the unique such element in uch2.A/. By Proposition 3.7, we obtainthe explicit formula

SdL.ŒC �/ D .q1=2X.C/� q�1=2

X.C//ŒC_�

for a narrow chamber C 2 bch1.A/. Next we consider W 1 and W 2. Since we haveCŒbch1�=N 1 ' W 1 and CŒuch2�=N 2 ' W 2, we have the induced map�dL W W 1 �! W 2:

This map is again expressed by a diagonal matrix. Indeed, for a wide chamber C 2bch1.A/, we have�dL.ŒC �/ D �.q1=21 � q�1=21 /ŒC_� D �.q1=2

X.C/� q�1=2

X.C//ŒC_�:

Thus

det� SdL W CŒbch1�! CŒuch2�

�D det. SdLjN1/ det. �dL/

D ˙Y

C2bch1.A/

.q1=2

X.C/� q�1=2

X.C//:

By Proposition 2.4,X.C/ in the above formulas runs all dense edges contained inH1.Hence we obtain (3.2).


Corollary 3.9. The map SdL W CŒbch1.A/� �! CŒuch2.A/� is nondegenerate if andonly if qX ¤ 1 for any dense edge X 2 D1.A1/ in H1.

The decomposability of A is related to W 2 as follows. We omit the proof (cf.Figure 2 and Figure 3).

Proposition 3.10. For ` D 2, A is decomposable if and only if dimW 2 D 1.

4 An application

As we saw in the previous sections, the basis of our cochain complex is encoded bythe set of chambers. There is also an involution � among unbounded chambers. In thissection, we prove that if the monodromies around dense edges at infinity are nontrivial,then the bases corresponding to unbounded chambers C and C_ D �.C / are canceledeach other, and finally, only bounded chambers survive. This leads to a proof of therefined version of vanishing theorem.

Our main result is the following.

Theorem 4.1. Let A be an indecomposable line arrangement in R2. Let L be a rankone local system. Then the following statements are equivalent:

(i) qX ¤ 1 for any dense edge X 2 D1.A1/ contained in H1;

(ii) we have

H k.M.A/;L/ D

8<:0; for k D 0; 1;LC2bch.A/

C ŒC �; for k D 2;

(iii) H 2.A;L/ is generated by fŒC � j C 2 bch.A/g.Remark 4.2. (i) H) (ii) H) (iii) holds for any arrangement A (without indecompos-ability). However (iii) H) (i) requires the indecomposability of A. (See Remark 4.3.)For comments to the higher dimensional cases (` � 3) see the next §5.

Proof of Theorem 4.1. (i) H) (ii). Let C0 2 ch0.A/. Since

dL.ŒC0�/ D �.q1=21 � q�1=21 /ŒC_0 �C : : : ;

and q1 ¤ 1, we have rank.dL W CŒch0.A/� ! CŒch1.A/�/ D 1 (and, in particular,H 0.CŒch.A/�; dL/ D Ker.dL W CŒch0�! CŒch1�/ D 0).

To show that

• H 1.CŒch.A/�; dL/ D 0 and

• H 2.CŒch.A/�; dL/ D Coker.dL W CŒch1� ! CŒch2�/ has fŒC �gC2bch.A/ as abasis,


it suffices to prove that the induced map

SdL W CŒbch1.A/� �! CŒuch2.A/�

is surjective (hence bijective). However this easily follows from Corollary 3.9.

(ii) H) (iii). Trivial.

(iii) H) (i). Let us assume (iii). Since H 2 D Coker.dL W CŒch1�! CŒch2�/, theassumption implies that the induced map

CŒch1.A/� �! CŒuch2.A/� (4.1)

is surjective. As in the proof of Theorem 3.8, dL maps N 1 to N 2. Thus the inducedmap

�dL W W 1 ˚C ŒC_0 �

��

o��

W 2

o��

CŒch1�=N 1 CŒuch2�=N 2

is surjective. Now if q1 D 1, then �dL is the zero map on W 1, and hence W 2 is atmost one dimensional. This is a contradiction to the assumption A is indecomposable(see Proposition 3.10). Thus we have q1 ¤ 1.

Set

bch1.A/ D fC1; : : : ; Ckg:Then ch1.A/ D fC_

0 ; C1; : : : ; Ckg. Now dL.ŒC0�/ is expressed as

dL.ŒC0�/ D a0ŒC_0 �C

kXiD1

ai ŒCi �:

Note that a0 D �.q1=21 � q�1=21 / ¤ 0. Since d2LD 0, dL.ŒC

_0 �/ 2 CŒch2� can be

expressed as a linear combination of dL.ŒC1�/; : : : ; dL.ŒCk�/. The surjectivity of (4.1)implies that (recall that CŒch1.A/� D CŒbch1.A/�˚C ŒC_

0 �)

CŒbch1.A/� �! CŒuch2.A/�

is surjective. Again by Theorem 3.8, we conclude that qX ¤ 1 for any dense edgeX 2 D1.A1/ in H1.

Remark 4.3. The assumption “A is indecomposable” is necessary to prove the part(iii) H) (i) in Theorem 4.1. Indeed, consider the arrangement in Figure 2, which isdecomposable. Let L be a rank one local system such that q1; q2; q3 2 C� are genericand q4 D q�1

1 q�12 q�1

3 . Then q1 D 1. The map �dL W CŒch1�! CŒuch2� is computed


as

�dL.ŒC1�/ D �.q12

34 � q� 12

34 /ŒC_1 �;

�dL.ŒC2�/ D .q12

234 � q� 12

234/ŒC_1 � � .q

12

12 � q� 12

12 /ŒC_3 �;

�dL.ŒC3�/ D �.q12

123 � q� 12

123/ŒC_3 �;

�dL.ŒC_0 �/ D �.q

12

2 � q� 12

2 /ŒC_1 � � .q

12

12 � q� 12

12 /ŒC_2 �:

Hence the map �dL W CŒch1� ! CŒuch2� has rank three. Thus H 2.CŒch�; dL/ isgenerated by bch2 D fC4g. Thus (iii) holds true, while (i) is false because q1 D 1.

5 Remarks and conjectures

We conclude this paper with some remarks on higher dimensional cases ` � 3. As inthe case ` D 2, it seems natural to focus on the induced map

SdL W CŒbchq�1� �! CŒuchq�

defined by the composition CŒbchq�1� ,! CŒchq�1�dL�! CŒchq� � CŒuchq�. Since

the bases of two spaces CŒbchq�1� and CŒuchq� are naturally identified by the invo-lution �, it makes sense to consider the determinant of SdL.

Conjecture 5.1. The determinant det. SdL W CŒbchq�1�! CŒuchq�/ can be expressedin the form

det. SdL/ D ˙YX

.q1=2X � q�1=2

X /nX ;

where X runs all dense edge X 2 D1.A1/ with dimX � ` � q and nX > 0.

Once the above conjecture is established, we deduce the following one.

Conjecture 5.2. Let A be an essential affine arrangement in R`. If the rank 1 localsystem L satisfies (1.3), then (1.2) holds.

“Proof of (5.1) H) (5.2).” Since the composition

SdL W CŒbchq�1� ,�! CŒchq�1�dL��! CŒchq� �!! CŒuchq�


is bijective, the rank of the map CŒchq�1�dL�! CŒchq� is at least ]bchq�1 D ]uchq .

Hence,

dim Im.dL W CŒchq�1�! CŒchq�/ � ]uchq D ]bchq�1;

dim Ker.dL W CŒchq�! CŒchqC1�/ � ]chq � ]bchq D ]uchq;

for q � ` � 1. This implies H k.CŒch�; dL/ D 0 for k � ` � 1. Also from this wededuce that H `.CŒch�; dL/ is generated by bch`.A/ D bch.A/.

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