C. R. Acad. Sci. Paris, t. 332, Série I, p. 209–214, 2001Analyse complexe/Complex Analysis

Nonuniformizable skew cylinders. A counterexampleto the simultaneous uniformization problemAlexey GLUTSYUK a,b

a Institut des hautes études scientifiques (IHES), 35, route de Chartres,91440 Bures-sur-Yvette, Franceb Unité de mathématiques pures et appliquées, M.R., École normale supérieure de Lyon, 46 allée d’Italie, 69364

Lyon cedex 07, FranceE-mail: aglutsyu@umpa.ens-lyon.fr

(Reçu le 19 décembre 2000, accepté le 10 janvier 2001)

Abstract. We disprove (Theorem 2) Ilyashenko’s simultaneous uniformization conjecture (dated late1960s): we construct a two-dimensional Stein manifold that admits a topologically trivialholomorphic fibration over the unit discD by simply connected holomorphic curves, thathas a holomorphic section and is not biholomorphically fiberwise equivalent to a subdomainof C×D. This together with a previous result of Ilyashenko (Lemma 1) implies that Bers’simultaneous uniformization theorem [3] for topologically trivial holomorphic fibrations bycompact Riemann surfaces does not extend to general holomorphic fibrations by compactRiemann surfaces with singularities (Theorem 1). 2001 Académie des sciences/Éditionsscientifiques et médicales Elsevier SAS

Cylindres tordus non uniformisables. Un contre-exemple à la conjectured’uniformisation simultanée

Résumé. On apporte une réponse négative à la conjecture d’uniformisation simultanée émise parYu.S.Ilyachenko à la fin des années 1960(théorème2) : on construit une variété deStein de dimension deux qui admet une fibration holomorphe topologiquement trivialeau-dessus du disque unitéD par des courbes holomorphes simplement connexes, qui aune section holomorphe, mais qui n’est pas biholomorphiquement équivalente(commefibration au-dessus deD) à un domaine deC×D. Cela implique, par un résultat précédentd’Ilyachenko (lemme1), que le théorème de Bers d’uniformisation simultanée[3] desfibrations holomorphes topologiquement triviales par surfaces de Riemann compactes nes’étend pas à des fibrations holomorphes générales par surfaces de Riemann compactessingulières(théorème1). 2001 Académie des sciences/Éditions scientifiques et médicalesElsevier SAS

Version française abrégée

En 1960, L. Bers [3] a démontré le théorème classique d’uniformisation simultanée, qui dit que les fibresde la fibration tautologique au-dessus d’un espace de TeichmüllerTg, g � 2, admettent une uniformisation

Note présentée par Étienne GHYS.

S0764-4442(00)01843-7/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 209

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holomorphe en le paramètre deTg. Autrement dit, la réunion des revêtements universels des fibres estbiholomorphiquement équivalente (comme fibration au-dessus deTg) à un ouvert dans le produitC × Tg.Ce théorème implique l’affirmation analogue pour toute fibration rationnelle topologiquement triviale parcourbes algébriques.

Il se trouve que la généralisation de cette dernière affirmation à des fibrations rationnellestopologique-ment non triviales et singulièrespar courbes algébriques est fausse en général.

THÉORÈME. – Il existe une surfaceS (de dimension2) algébrique lisse dansCn (Pn) fibrée par descourbes de niveaux d’une fonction rationnelle(avec singularités) telle que la famille de surfaces deRiemann correspondante n’admette pas d’uniformisation simultanée par une famille d’ouverts simplementconnexes deC avec la fonction uniformisante holomorphe en le paramètre transversal. Le nombren peutêtre choisi égal à5. Plus précisément, il existe une fonction rationnelle non constanteR : S→ C (définiepartout, un polynôme enn variables, siS est affine) et une application holomorphe inversei :D→ S dudisque unitéD (contenu dans l’image deR), R ◦ i= Id, avec les propriétés suivantes:

1) L’image i(D) ne contient pas de points critiques deR. Le feuilletageF par composantes connexesde courbes de niveau deR en dehors des points critiques deR est transverse ài(D).

2) SoitM la réunion de revêtements universels de feuilles intersectanti(D) du feuilletageF munie dela structure complexe naturelle. La variétéM n’admet pas d’application biholomorphe sur un domaine deC×D qui forme un diagramme commutatif avec les projections naturelles surD.

1. Skew cylinders and simultaneous uniformization

By Tg, g � 2, we denote the Teichmüller space of closed Riemann surfaces of genusg. The tautologicalfibration overTg is the union of all the Riemann surfaces: the fiber over each point ofTg is the correspondingRiemann surface. The tautological fibration admits a natural complex structure, and so does the union ofthe universal coverings of its fibers. The latter is fibered overTg with fibers conformally equivalent to adisc.

In 1960 L. Bers [3] proved the classical simultaneous uniformization theorem saying that fibers of thetautological fibration admit a uniformization holomorphic in the Teichmüller space parameter. Namely, heproved that the union of the universal coverings of the fibers is biholomorphically fiberwise equivalent to asubdomain in the direct productC×Tg. This theorem implies the analogous statement for any topologicallytrivial rational fibration by algebraic curves.

It appears that the generalization of the last statement to topologically nontrivial rational fibrations byalgebraic curveswith singularitiesis wrong in general.

THEOREM 1. –There exists a smooth algebraic surfaceS (of dimension2) in Cn (Pn) (the numbernmay be chosen to be equal to5) fibered by algebraic curves with singularities such that the familyof Riemann surfaces thus obtained does not admit simultaneous uniformization by a family of simplyconnected domains inC such that the uniformizing function is holomorphic in the transversal parameter.More precisely, there exists a nonconstant rational functionR : S → C (well-defined everywhere, apolynomial inn variables ifS is affine) and an inverse holomorphic mappingi :D→ S of the unit discD(in the image ofR), R ◦ i= Id, with the following properties:

1) The imagei(D) contains no critical point ofR. The foliationF by connected components of levelcurves ofR outside its critical points is transversal toi(D).

2) Let M be the union of the universal coverings of the leaves intersectingi(D) of F equipped withthe natural complex structure. There is no biholomorphism ofM onto a domain inC × D forming acommutative diagram with the natural projections toD.

Theorem 1 is proved at the end of the section.

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Counterexample to the simultaneous uniformization conjecture

DEFINITION 1. – Let D be a simply-connected domain inC, M be a two-dimensional complexmanifold, p : M → D be a holomorphic surjection having nonzero derivative. We say that the triple(M,p,D) is askew cylinderwith the baseD and the total spaceM , if:

1) all the level sets of the mapp are connected and simply connected holomorphic curves;2) the ‘fibration’M admits a holomorphic section, i.e., there exists a holomorphic mappingi :D→M

such thatp ◦ i= Id.

Example1. – LetS be a smooth affine (projective) two-dimensional complex surface,R : S → C be anonconstant rational function (well-defined everywhere),D be the unit disc in the image ofR, i :D→ Sbe an inverse holomorphic mapping,R ◦ i= Id (then it is transversal to the level curves ofR). Considerthe foliationF by connected components of level curves of the functionR outside its critical points. LetMbe the union of the universal coverings of its leaves intersectingi(D) equipped with the natural complexstructure. Then the manifoldM equipped with the standard projection toD is a skew cylinder. A skewcylinder thus constructed is calledaffine(projective) algebraic.

DEFINITION 2. – A skew cylinder(M,p,D) is said to beuniformizable, if there exists a biholomorphicmappingg :M → C ×D (not necessarily ‘onto’) that forms a commutative diagram with the projectionstoD (this mapping is called a uniformization). It is calledlocally uniformizableat a pointz ∈D, if z admitsa neighborhoodV such that the cylinder(p−1(V ), p, V ) is uniformizable.

DEFINITION 3. – A skew cylinder is said to be Stein, if its total space is Stein.

The main result of the paper is the following

THEOREM 2. –There exists a nonuniformizable Stein skew cylinder. It can be chosen either locallyuniformizable everywhere, or not, e.g., locally nonuniformizable at a single point.

Theorem 2 is proved in Subsection 2.3.

DEFINITION 4. – Let (M,p,D) be a skew cylinder, letB ⊂M be its subdomain. ThenB is called asubcylinder, if the triple(B,p, p(B)) is a skew cylinder.

Remark1. – Any compact subcylinder of a Stein skew cylinder is locally uniformizable [6,8].A projective algebraic cylinder (seeExample 1) is locally uniformizable provided that the closure1 ofthe union of the leaves intersectingi(D) of F contains at most Morse critical points ofR. This followsfrom results of Yu.S. Ilyashenko, see Subsection 2.1.

DEFINITION 5. – Two skew cylinders are said to beequivalent, if there exist biholomorphisms of theirtotal spaces and bases that form a commutative diagram with the projections.

LEMMA 1 2 (Yu. S. Ilyashenko). –Any compact subcylinder of a Stein skew cylinder is equivalent to asubcylinder of an affine(projective) algebraic skew cylinder(seeExample1). The dimension of the ambientaffine(projective) space of the corresponding surface may be chosen to be equal to5. The affine cylindermay be chosen so that the corresponding rational functionR is the restriction toS of a polynomial in theambient affine space(C5).

A sketch of the proof of Lemma 1 is presented in Subsection 2.4.

PROPOSITION 1 2 (Yu. S. Ilyashenko,see[8]). – Let a Stein skew cylinder be exhausted by an increasingsequence of uniformizable subcylinders. Then it is uniformizable.

Proof of Theorem1. – By Theorem 2, there exists a nonuniformizable Stein skew cylinder. ByProposition 1, it has a nonuniformizable compact subcylinder. By Lemma 1, the latter is equivalent to asubcylinder of an affine (projective) algebraic cylinder. The last subcylinder is nonuniformizable as well, asis the ambient algebraic cylinder. This proves Theorem 1.✷

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2. Nonuniformizability

In this section we prove Theorem 2 (Subsection 2.3) and Lemma 1 (Subsection 2.4).In Subsections 2.1–2.2 we give a survey of previous results.

2.1. Previous results for algebraic skew cylinders

In 1973 Yu. S. Ilyashenko [5] extended Bers’ simultaneous uniformization theorem to the tautologicalfibration with a unique singular fiber over a disc embedded in the Deligne–Mumford compactificationof the moduli space of Riemann surfaces. This implies that in the notations of Example 1, a projectiveskew cylinder is uniformizable, if the closure (seeendnote 1) of the union of the leaves of the foliationFintersectingi(D) contains at most Morse critical points of the functionR and their critical values coincide(cf. the previous remark).

2.2. Previous results for general skew cylinders

It is easy to construct non-Stein nonuniformizable skew cylinders.

Example2. – LetD be the unit disc,ψ :D→ C be a nonholomorphic function (say,ψ(z) = z). LetMbe the universal covering over the complement to the graph ofψ in the productC ×D. The manifoldMadmits a natural structure of skew cylinder, and the latter is not uniformizable.

Other examples of non-Stein nonuniformizable skew cylinders with fibersC may be found in [6]. Yu.S.Ilyashenko [6] proposed the conjecture that any Stein skew cylinder is uniformizable. In 1969 T. Nishino [7]independently proved the positive answer for skew cylinders with fibersC. A.A. Shcherbakov [8] provedthat any Stein cylinder can be exhausted by a growing sequence of compact subcylinders with smoothstrictly pseudoconvex boundaries. His result together with Proposition 1 and Theorem 2 implies thefollowing:

COROLLARY 1. –There exists a compact skew cylinder with a smooth strictly pseudoconvex boundarythat is nonuniformizable.

2.3. Proof of Theorem 2

LetD be the unit disc. For the proof of Theorem 2 we show that there exists a closed subsetK ⊂ C×Dfibered overD by discs (an infinite number of them degenerate into single points) such that the universalcovering over its complementC×D\K (equipped with the canonic projection toD) is a nonuniformizableStein skew cylinder. We prove this statement for the following setK constructed by Bo Berndtsson andT.J. Ransford in their joint paper [2].

THEOREM 3 ([2]). – Let D be the unit disc in complex line with the coordinatez, p be the standardprojection toD of the direct productC ×D. LetE+ = { 1

2 ,n

2n+1}n∈N, E− = −E+ ⊂D. There exists a

closed subsetK ⊂ C×D such that:1) the complementM ′ = C×D \K is pseudoconvex;2) for anyz /∈E+ ∪E− the fiberK ∩ p−1(z) is a disc;3) for anyz ∈E+ K ∩ p−1(z) = 0× z;4) for anyz ∈E− K ∩ p−1(z) = 1× z.

For the completeness of presentation, we recall the construction of the setK from [2]. Let w be thecoordinate in the fiberC on the direct productC×D. Let

u(z) = ln

∣∣∣∣z − 1

2

∣∣∣∣+ ln

∣∣∣∣z +1

2

∣∣∣∣++∞∑n=1

2−n

(ln

∣∣∣∣z − n

2n+ 1

∣∣∣∣+ ln

∣∣∣∣z +n

2n+ 1

∣∣∣∣), A ∈ R

+.

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Counterexample to the simultaneous uniformization conjecture

The functionu is harmonic,u(E±) =−∞. Letψ :D→ C be aC∞ function with bounded derivatives (upto the second order) that is constant in a neighborhood of each setE± so thatψ|E+ = 0, ψ|E− = 1. Define:

K ={∣∣w− ψ(z)

∣∣ � eu(z)+|z|2+A}. (1)

The fibers ofK overE+ (E−) are single points where the coordinatew is equal to 0 and 1 respectively.If A is large enough, thenM ′ = C×D \K is pseudoconvex [2].

The nonuniformizable skew cylinder we are looking for is the universal covering overM ′ (denote thiscovering byM ) equipped with the natural projection toD. Indeed it is a skew cylinder: it satisfies thecondition 1) of Definition 1; for anyN large enoughN ×D ∩K = ∅, then a lifting toM of the naturalmappingD→N ×D⊂M ′ is a well-defined section. The manifoldM is Stein by pseudoconvexity ofM ′

and the classical theorem due to Stein [9] saying that a covering over a Stein manifold is Stein.Let us prove thatM is nonuniformizable by contradiction. Suppose the contrary, i.e., there exists a

uniformization g : M → C × D. Let π : C × D → C be the standard projection,f = π ◦ g be thecorresponding coordinate component ofg. The fibers of the cylinderM overE± are conformally equivalentto complex line. Letw be the coordinate inC (we consider it as a coordinate onM ′ ⊂ C ×D). Considerthe chartlnw on the fibers ofM . It is a well-defined 1-to-1 chart on the fibers overE+, and this is not thecase for the fibers overE−, where this chart is multivalued and has branch points wherew = 0 . Therefore,for any z ∈ E+ the restriction to the fiber overz of the functionf (which is univalent) is Möbius in thechartlnw, and this is not the case forz ∈E−. LetSf be the Schwartzian derivative off along fibers in the(multivalued) coordinatelnw. It is a well-defined holomorphic function onM \ {w = 0}, since any twodistinct branches oflnw differ from each other by constant. It vanishes identically on all the fibers overE+ and on no fiber over no point inE− by the previous statement. The first one of the two last statementsimplies thatSf ≡ 0 (the setE+ contains the limit point1/2), which contradicts the second statement. Thiscontradiction proves thatM is nonuniformizable.

The nonuniformizable cylinderM is locally uniformizable. This is proved by using the fact that for eachpoint z ∈ D there exist its neighborhoodV andc ∈ C such that{w = c} ∩ (C × V ) ⊂K . One can alsoachieve thatM be Stein and locally nonuniformizable at the unique point 0 by choosing the correspondingfunctionu in (1) to be:

u(z) = ln |z|+∞∑

m=0,n=1

2−(m+n) ln

∣∣∣∣z − 2−2m n

2n+ 1

∣∣∣∣+∞∑

m=0

2−m ln∣∣z − 2−(2m+1)

∣∣

(which is harmonic and is equal to−∞ exactly on the setE = 0 ∪⋃∞

m=0 4−mE+), choosing appropriatefunctionψ(z) in (1) that is constant on each set4−mE+ so thatψ(4−m1E+) �= ψ(4−m2E+) form1 �=m2,and choosing in (1) a sufficiently largeA. Theorem 2 is proved.

2.4. Proof of Lemma 1

The sketch of the proof of Lemma 1 in the affine case presented below repeats that due to Yu.S.Ilyashenko.2 Let (M,p,D) be a Stein skew cylinder. We consider thatM is embedded as a submanifold inCn (one can putn= 5 by the Bishop–Narasimhan embedding theorem). LetB ⊂M be a subcylinder withcompact closure. Without loss of generality we consider that its basep(B) is the unit disc. We fix a ballBcontainingB and approximateM andp in B by a piece of smooth affine algebraic surfaceS ⊂ Cn anda polynomialR respectively. The possibility of approximations follows from results of the paper [1] andthe approximation and extension theorems for functions on Stein manifolds. As it is shown below, if theapproximations are accurate enough, then there exists a biholomorphismh :B→ S (not onto) that forms acommutative diagram withp andR. In the proof of this statement we use the following fact:

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(S) each Stein skew cylinder admits a holomorphic function that is locally univalent along each fiberand vanishes at the section.

The proof of statement (S) uses the fact that for any contractible Stein manifold each divisor is aone of meromorphic function [4]. To construct the biholomorphismh, consider a function onM fromstatement (S) and its holomorphic extension toCn (denote the last extension byf ). By construction,the level curves off in Cn are transversal to the fibers ofM . The biholomorphismh we are lookingfor is defined by the pair of equationsp(x) = R(h(x)), f(x) = f(h(x)), so thath tends to identity, asS ∩ B→M ∩ B,R|

B→ p|

B. Consider the inverse mappingi :D→ S,R ◦ i= Id, that is the pushforward

underh of the section of the cylinderB. The corresponding algebraic skew cylinder from Example 1 is theone we are looking for. This proves Lemma 1.

Acknowledgements.I am grateful to Yu.S. Ilyashenko, who attracted my attention to the problem. I am grateful toG.M. Henkin, who informed me about the paper [2] (one of results from [2] became the base for the proof). I wish tothank both them and M. Gromov, R. Gunning, C.D. Hill, N.G. Kruzhilin, I. Lieb, S. Nemirovski, A.A. Shcherbakov,D. Zaitsev for helpful discussions. I am grateful to the referee of the paper for helpful remarks. The paper was writtenwhile I was visiting the Institut des Hautes Études Scientifiques (Bures-sur-Yvette, France). I wish to thank the Institutefor hospitality and support. Research supported by CRDF grant RM1-2086, by INTAS grant 93-0570-ext, by RussianFoundation for Basic Research (RFBR) grant 98-01-00455, by the State Scientific Fellowship of the Russian Academyof Sciences for young scientists, by the European Post-Doctoral Institut joint fellowship of the Max-Planck Institut fürMathematik (Bonn) and IHÉS (Bures-sur-Yvette, France).

1 Everywhere by ‘closure’ we mean the one in the usual (not Zariski) topology.2 From unpublished paper by Yu.S. Ilyashenko (late 1960s).

References

[1] Banica C., Forster O., Complete intersections in Stein manifolds, Manuscripta Math. 37 (3) (1982) 343–356.[2] Berndtsson Bo, Ransford T.J., Analytic multifunctions, the∂- equation, and a proof of the corona theorem, Pacific

J. Math. 124 (1) (1986) 57–72.[3] Bers L., Simultaneous uniformization, Bull. Amer. Math. Society 66 (1960) 94–97.[4] Gunning R., Rossi H., Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, NJ, 1965.[5] Ilyashenko Yu.S., NondegenerateB-groups, Soviet Math. Dokl. 14 (1973) 207–210.[6] Ilyashenko Yu.S., Shcherbakov A.A., Skew cylinders and simultaneous uniformization, Proc. Steklov Math. Inst.

213 (1996) 112–123.[7] Nishino T., Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (II). Fonctions entières

qui se reduisent à celles d’une variable, J. Math. Kyoto Univ. 9 (2) (1969) 221–274.[8] Shcherbakov A.A., The exhaustion method for skew cylinders, Algebra i Analiz 12 (5) (2000) 178–206.[9] Stein K., Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. 7 (5) (1956) 354–361.

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