Several Complex Variables and Complex Geometry (Proceedings of Symposia in Pure Mathematics, Vol.52) (Pt. 2)

Proceedings of Symposia ifl PURE MATHEMATICS Volume 52, Part 2 Several Complex Variables and Complex Geometry Americ...

Author: E. Bedford | J.P. D'Angelo | Steven G. Krantz | R.E. Greene

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Proceedings of Symposia ifl

PURE MATHEMATICS Volume 52, Part 2

Several Complex Variables

and Complex Geometry

American

oci.ty

Proceedings of Symposia in

MATHEMATICS Volume 52, Part 2

Several Complex Variables and Complex Geometry Eric Bedford John P. D'Angelo Robert E. Greene Steven G. Krantz Editors

American Mathematical Society Providence. Rhode Island

PROCEEDINGS OF TUE SUMMER RESEARCH INSTITUTE ON SEVERAL COMPLEX VARIABLES AND COMPLEX GEOMETRY HELD AT THE UNIVERSITY OF CALIFORNIA, SANTA CRUZ SANTA CRUZ. CALIFORNIA JULY 10—30, 1989

with the support of the National Science Foundation Grant DMS—8S 4802

1980 Mathen1u:ks Sul'jt'ci Clasri/Icazicn (1985 Revision). Primary 32A, 32D, 32E, 32F, 32H (Part 1) 32B, 32C, 320. 32H. 32J, 32K, 32L, 32M, 53C (Part 2) 35N15, 32F20. 538. 32A, 32F, 32C (Part 3)

Library of Congress Cataioging-in-Publkstion Data Summer Research Institute on Several Complex Variables ar'J Complex Geometry (1989: University of California, Santa Cvu7.) Several complex variables and complex geometry/[cditcd byj Eric Bedford... [Ct al.). cm.—(Proccedings of sYmposia in pure mathematics, ISSN 0082-0717; v• 52) p. of the Summer Research Institute on Several Complex Variables and Complex Geometry, held at the University of California, Santa Cruz, Santa Cniz. California. July 10—30, 1989"—T.p. verso. I nclucic, bibliographical references. 1. Functions of several complex variables—Congresses. 2. Geometry, Dift'crential—Congresses. I. Bedford, Eric, 1947—. II. American MathematicaL Society. Ill. Title. IV. Series. 91-11227 QA331.7.S86 1989 CIP 515' 94—dc2O

ISBN 0-8218-1489-3 (part 1) ISBN 0-8218-1490-7 (part 2) ISBN 0-8218-1491-5 (part 3) ISBN 0-8218-1488-5

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969594939291

Several Complex Variables and Complex Geometry

Contents xiii

Preface PART 1

for the Dirichiet Problem for Harmonic Maps from the Annulns into the Space of Planar Dücs

Uniqueness

DAVID E. BARRETT

1

CR Maps Between Hypersurfaces in STEVE BELL

Bounds

for

13

the Degrees in Polynomial Equations

CARLOS A. BERENSTEIN AND ALAIN YGER Lagrange Interpolants for Entire Functions on

THOMAS

23

v"

BWOM

29

An Inequality for Analytic Functions 37

URBAN CEGRELL

Some Solutions of the Beltrami Equation with J. A. C1MA AND W.

R.

=

1

DERRIcK

On the Problem of the Lincarization of the Stability Group of a. RealAnalytic Hypersurface V. EzHov

41

45

Mappings of Strongly Pseudoconvex Cauchy-Riemann Manifol4s FRANC FORSTNERIC

59

Rational Curves and Extensions of Holomorphic Mappings S. M. IVASHKOVICH

Applications of JACOB

93

Capacities 105

CONTENTS

Convexity in Complex Analysis 119

STEVEN 0. KRANTZ

Solutions of 82u/ôzOZ = e2" and Analytic Multivalued Functions 139

DONNA KUMAGAI

Local Versus Global Convexity of Pseudoconvex Domains 145

ALAN NOELL

The Scaling Method and Holomorphic Mappings SERGEYPINCHUK

151

Plurisubharmonic Functions as Solutions of Variational Problems EVOENY A. POLETSKY

163

On Complex Analysis in Tube Cones A. G. SERGEEV

173

Separately Meromorphic Functions and Separately Holomorphic Mappings BERNARD

191

Some Aspects of Weakly Pseudoconvex Domains NESsIM SIB0NY

199

Unions Minimales de n-Plans Reels d'Enveloppe Egale a c' PASCAL

J.

233

Separately Subharmonic Functions: When are They Subbarmonic JAN WIEOERINCK AND RnN ZEINSTRA

245

Functions of Bounded Mean Oscillation in the Bergman Metric on Bounded Symmetric Domains KEHEZIu

251

PART 2

The Geometry of Complex Manifolds: An Overview ROBERT E.

I

Angular Derivatives in Strongly Pseudoconvex Domains MARCO ABATE

23

CONTENTS

Degeneracy Points of the Kobayashi Pseudodistances on Complex Manifolds YUKIN0Bu ADACHI AND MASAKAZU

On the Construction of the Moduli Space for Strongly Pseudoconvex Domains 53

TAKAO AKAH0RI

Qn the Holomorphic Sectional Curvature of Complete Domains in C" that are not Stein ANDREW BALAS

Normal Forms for Convex Domains J. BLAND AND T. DUCHAMP

65

Homogeneous Vector Bundles and Families of Calabi-Yau Threefolds. II CIPRIAN BORCEA

83

Holomorphic Morse Inequalities JEAN PIERRE DEMAILLY

93

Some Problems PIERRE DOLBEAULT

Jnvariants of CR Densities 0. EASTWOOD AND C. Roamr GRAHAM

Convex Symplectic Manifolds YAXOV EL1ASHBERG AND

GRoMov

Simple Partial Connections and the Einstein Condition H. R. FISCHER AND R. J. FISHER, JR.

163

Applications of Affine Geometry to Geometric Function Theory in Several

Complex Variables Part I. Convergent Rescalings and Intrinsic Quasi-Isometric Structure SIDNEY

A Lie Algebra Character and Kihler-Einstein Metrics of Positive Scalar

Curvature AIUTO FurAKI

209

CONTENTS

On the Ends of Complex Manifolds Homogeneous Under a Lie Group

217

BRUCE GILLIGAN

Determinant Line Bundles for 1-lermitian Foliations and a Generalized Quillen Metric 225

JAMES F. GLAZERROOK AND FRANZ W. KAMBER

Sharp Constants for the Koebe Theorem and for Estimates of Intrinsic Metrics on Convex Domains 233

IAN GRAHAM

Rigid Immersions of G-Structures and Analyticity of CR Mappings into Spheres 239

CHONG-KYU HAN

Optimal Analytic Disks 251

J. WILLIAM HELTON AND ORLANDO MERINO

Some Remarks on the Product Property for Invariant Pseudometrics 263

MAREK JARNICKI AND PETER PFLUG

Smoothing of Currents and Moisezon Manifolds 273

SHANYU ii

Biholomorphic Mappings Between Quasicircular Domains iii C" 283

KANG-TAE KIM

Characterizations of Certain Weakly Pseudoconvex Domains in C" from the Viewpoint of Biholomorphic Automorphism Groups AKIOKODAMA

291

Complete Ricci-Flat Kähler Metrics on C" Need Not Be Flat 297

CLAUDE LEBRUN

On Meromorphic Maps into Varieties of Log-General Type STEVEN SHIN-Yl Lu

305

Aspects of Kahler Geometry on Arithmetic Varieties NGAIMING MOK

335

Moduli Spaces of Holomorphic Mappings into Hyperbolic SiSaces and its Applications J.

'

397

CONTENTS

ix

The Analytic Radon Transform

403

SALOMON OFMAN

Technique in Applications of the Käh let Manifolds

I-lodge Theory on Complete 413

TAKEOOHSAWA

Bounds on the Derivatives of Holomorphic Endomorphisms 427

MARIUS OVERHOLT

Functions with Large Center 435

GIORGIO PATIUVO AND PIT-MANN WONG

A Characterization of Affine Varieties 449

MATHIAS PETERNELL

Compactilications of

A Survey 455

THOMAS PETERNELL AND MiCHAEL SCHNEIDER

An Infinite-Dimensional Point of View on the Weil-Petersson.Metric 467

T. RATIU AND A. ToDoRov

The Cartan Conjecture for Moving Targets 477

MIN Ru AND WILHELM STOLL

On Kollár's Conjecture 509

M0RIHIK0 SAITO

An Introduction to L2-Cohomology 519

LESLIE SAPER AND STEVEN ZUCKER

On the Compactification Problems for Stein 3-Folds

On One of Calabfs Problems

Ti,n

543

Multi-Dimensional Analytic Structures T. TONFY

The

Algebras 557

on Complex Spaces J. VAROUCHAS

563

CONTENTS

Fixed Points of Holomorphic Mappings in a Bounded Convex Domain in 579

JEAN-PIERRE ViovE

Holomorphic Functions Taking Their Values in a q-Spacc and the CauchyFormula 583

L. WAELBROECK

Schwarz's Lemma and Hermitian Manifolds with Constant Holomorphic Curvature 593

B. WONO

Polynomial Functions on Complete KAhier ManifoUs

H.Wu The Chern Numbers of Projective Algebraic Hypersurfaces KICH0ON YANG

A Review of Complex Differential Geometry 619

SHINO-TUNG YAU

PART 3

Minimality and the Extension of Functions from Generic Manifolds M. S. BAouENDI AND LINDA PREISS ROTHSCHILD

Holomorphic Mappings of Real Analytic Hypersurfaces M. S. BAOUENDI AND LINDA Paaiss ROTHSCHILD

Extension of CR Structures DAVID CATLIN

Regularities for N1S Operators on Weakly Pseudo-Convex Domains

of Finite type in C2 DER-CHEN E. CHANO AND STEVEN G. KRANTZ

Global Regularity of the i-Neumann Problcm in Dimension Two So-CHIN CHEN

On the 8b Equation for Three-Dimensional CR Manifolds MICHAEL CHRIST

-

63

CONTENTS

Necessary Conditions for the Local Solvability of the Tangential CR Equations PAULO CORDARO AND

83

TREVES

Operator Theory on Reinhardt Domains 93

RAUL CURTO

Finite Type and the Intersection of Real and Complex Subvarieties 103

JOHN P. D'ANOELO

Local Analyticity for the a-Neumann Problem and for

in Some

Model Domains without Maximal Estimates MAKHLOUF DEuuDJ AND DAVID S. TARTAKOFF

On 9 Estimates for 129

ERIK FORNAESS AND NESSIM SIB0NY

Approximate Projections and Analytic Pseudolocality of the SzegO Projection on Domains in C2 DARYL GELLER

On an Example of a Subelliptic Boundary Value Problem PENOFEI GUAN

Degenerate Surfaces in C2 GARY A. HARRIs

Counterexamples to Newlander-Nirenberg up to the Boundary C.

179 -

HILL

191

Finite Type Analytic Partial Differential Operators and Analytic Hypoellipticity A. ALEXANDROU HIMONAS

199

Singularities of Harmonic Functions in C' D. KIIAVINSON

207

A Study of the Bergman Projection in Certain Hartogs Domains CHRISTER 0. KISELMAN

219

The Frame Bundles of CR Structures and the Bergman Kernel (I) MASATAICE KURANISHI

233

CONTENTS

AlL

du 0b a Support Compact et Phénomëne de Hartogs-Bochner dans les Variétés CR 239

CHRISTINE LAURENT-THIEBAUT

On the Regularity of Coanalytic Toeplitz Operators 251

EWA

Obstructions to Embedding of Real (2n — 1)-Dimensional Corppact CR Manifolds in V

HING-SUN LUK AND STEPHEN S.-T. YAU

261

On Sharp Holder Estimates for the Solutions of the b-Equations 277

JEFFERY D. MCNEAL

Equivalence of Analytic and Plurisubharmonic Phragmén-Lindelof Conditions REINHOLD MEISE, B. A. TAYLOR, AND DIETMAR VOGT

287

The Cauchy-Riemann Equations in Convex Domains 309

JOHN C. POLKING

Examples of Smooth CR Structures with only Nonsmooth CR Functions 323

JEAN-PIERRE ROSAY

Noncompactness of the s-Neumann Problem and Toeplitz 329

NORBERTO SALINAS

L.ocal Solvability and Estimates for

on

CR Manifolds

MEl-Cm SHAW

335

Rigid H ypersurfaces in C2 NANCY K. STANTON

347

The Integrability Problem for CR Vector Bundles S. M. WEBSTER

355

Preface The American Mathematical Society held its thirty-seventh annual Summer Research institute at the University of California at Santa Cruz, Santa Cruz, California, from July 10 to July 30, 1989. The focus of this institute was several complex variables and complex geometry. The members of the AMS Committee on Summer Institutes that selected this program were Eric Friedlander, Steven L. Kleiman, Paul H. Rabinowitz, Thomas C. Spencer, Robert B. Warfield, Jr., and John Wermer (chair). The institute was supported in part by a grant from the National Science Foundation. The Organizing committee for the institute consisted of Eric Bedford, John D'Angelo, Robert E. Greene. and Steven G. Krantz (chair). All of the organizers edited these proceedings. The institute explored recent developments in the geometry and function theory of several complex variables. An attempt was made to create a dialogue among the different methodologies in the subject:, differential geometry, algebraic geometry, partial differential equations, harmonic analysis, and classical methods. Each week there were two series of several connected lectures (called the Principal Lectures) representing important directions in the field of emphasis for that week. These were intermixed with single hour lectures. Both the connected series of talks aad the hour lectures were presented in the mornings and early afternoons. The last three hours of every day consisted of two parallel sessions of briefer seminar lectures. Principal lectures Week One Nessim Sibony, Weakly Pseudocon vex Domains Sergey Pinchuk, The Scaling Method and Holomorphic Mappings Week Two Ngaiming Mok, Problems in Kdhler Geometry Related to Hermit Ian (Locally) Symmetric Spaces Jean Pierre Demailly, Holomorphic Morse inequalities Week Three David Catlin, Extensions of CR Structures

John Erik Fornaess, L" Estimates for XIII

ziv

PREFACE

Hour lectures Steven R. Bell, CR Maps and Local Regularity Jean Martinet, Wild Cauchy Theory and Applications Bent Stensanes, Aspects of Envelopes of Holomorphy

David Barrett, Uniqueness for a Nonlinear Dirichiet Problem Connected with One Dimensional Function Theory Jean-Pierre Rosay, About the Embeddability of CR Structures B. A. Taylor, Equivalence of Analytic and Plurisubharmonic Phragrnen-Lindelof Conditions Franc Forstneric, Mappings of CR Manifolds Vladimir Ezhov, The Problem of Linearization of the Stability Group S. T. Yau, Uniformization Sergeii Ivashkovitch, Rational Curves and Extension of Holomorphic Maps Jung Hsi Wu, Holomorphic Functions of Polynomial Growth on Kdhler Manifolds Michael Schneider, On Two Conjectures of Hartshorne Steven Zucker, A Survey of L2 Cohomology Leslie Saper, L2 Cohomology of Algebraic Varieties Y. 1. Siu, Global Nondeformability of Compact Ifermitian Symmetric Manifolds Takeo Ohsawa, Convexity, Concavity, and L2 Cohomology on Noncompact Manifolds Akito Futaki, A Lie Algebra Character and Kdhler-Einstein Metrics of Positive Scalar Curvature Edward Bierstone, Local Resolution of Singularities James Carison, Harmonic Mappings on Kdhler Manifolds Gang Tian, On One of Calabi's Problems Linda Rothschild, CR Functions on Manifolds of Higher Codimension Sidney Webster, The Integrability Problem for CR Vector Bundles M. Salah Baouendi, General Reflection Principles J. J. Kohn, Pseudoconvex CR Manifolds with Diagonoiizable Levi Form Francois Treves, On the Local Solvability of the Tangential CauchyRiemann Equations Yakov Eliashberg, Topological Characterization of Stein Manifolds of Dimension> 2 F. Michael Christ, Estimates for on Finite Type CR Mamfoldr Armen Sergeev, Complex Analysis in Tube Cones Elias M. Stein, Analysis on Finite Type Domains

The purpose of the papers in these volumes is to exhibit and describe the current state of the field. The papers are organized according to the week in which the corresponding lectures were presented, and are given in alphabetical order. The first week was devoted to function theory, the second to complex geometry, and the third to analysis and partial differential equations.

PREFACE

The University of California at Santa Cruz provided pleasant facilities and a beautiful environment in which to conduct our sessions. Wayne Drady of the American Mathematical Society made the entire event run both smoothly and harmoniously. The American Mathematical Society was a great help at every stage of the planning. We dedicate these volumes to the memory of Jean Martinet. E. Bedford J. P. D'Angelo R. E. Greene S. G. Krantz

sympoela in Pure Volutne 52

Part 2

The Geometry of Complex Manifolds:

An Overview ROBERT E. GREENE

One of the avowed purposes of the 1989 American Mathematical Society Summer Institute on several complex variables and complex geometry was the promotion of a stronger association between the analytic and the geometric. The analytic theory of functions of several complex variables and the geometric theory of complex manifolds of a right ought to be viewed as a unified whole. In practice, the intrinsic unity has tended to dissipate somewhat under the usual pressures toward technical specialization. Still, in their separate progresses toward technical goals, function theory and complex manifold theory have maintained deep connections. The purpose of this article is to provide a view of some aspects of complex manifold theory that emphasize connections with function theory. In particular the L2 — theory as an aspect of geometry (as well as analysis) is a main theme. In an article of reasonable length, such a goal can be accomplished only in restricted terms; a detailed treatment of the whole would require something more on the order of a book than a short paper. But it is hoped that the topics to follow will enable the analyst to see a path into the geometry of complex manifolds that at least passes for a while through familiar territory and never strays too far from that. On the other hand, for the complex geometer, the reminders of how closely the techniques of geometry aie related to complex analysis may prove useful and perhaps stimulating. This article is intended to serve as a partial introduction to the specialized papers that follow. In particular, the reader who wishes to see the most recent developments about the generalizations of Riemann surface uniformization ideas to higher dimensions should continue by reading S. T. Yau's survey in this volume. 1980 Mathematics Subjc'cz Classification (1985 Revision). Primary S3C55, 32C10; Secondary 32E10, 32325. Research supported, in part, by a grant fiom the National Science Foundation.

This paper is in final form and no version of it will be submitted for publication elsewhere. 1991 An;cricin Mathematical Society 0082 0717/91 51 00

$25

per page

R. E. GREENE

2

1. Complex structure and metric. The underlying idea of global Riemannian geometry is that curvature controls topology; from hypotheses on curvature, one hopes and expects to obtain conclusions about the topological

nature of a Riemannian manifold. The natural extension of this idea to complex manifolds is that curvature should also control the complex ture. For this extension to have any possibility of working, there must be some relationship between the metrics considered and the complex structure of the manifold. The required relationship is most easily expressed in terms of the almost complex structure. Specifically, if M is a complex manifold and p a point of M, we define a (real) linear endomorphism J of the real tangent space of M to itself by setting

(o\

a

(o\_

a

for some holomorphic coordinate system = defined in a + neighborhood of p. Note that J is in effect the real version of multiplication by i; in particular, J a = multiplication by —1. The Cauchy-Riemann equations show that this definition is independent of the choice of a holomorphic coordinate system. Similarly, it is an exercise in the Cauchy-Riemann

I

equations to show that 'M' i.e., I,, Vp E M, determines the complex

structure of M: a C°-function f: M -,

is holomorphic if and only if df °1M = JCO df. This way of expressing the complex manifold structure of' M in terms of a collection of real mappings is a standard and useful device (for further details, see, e.g., [KObNJ and [Cr2J).' C

The most obvious way to relate metric and complex structure on a complex manifold M is now to suppose that I is an isometric endomorphism relative

to M's metric, i.e.,

g(JX,, JX2)= g(X,, X2), PEM. Suchametric g iscalledHermitian(for

forall I). An additional condition that forces a still more intimate relationship

between complex structure and metric is to suppose that the almost complex

structure tensor I is parallel with respect to the Riemannian connection determined by the Hermitian metric g, i.e.,

D1(JY) = JD1Y, for all (local) vector fields X, Y. A Hermitian metric with this additional property that .1 is parallel is called a Kãhler metric. This is equivalent to A C°° familyofendomorphisms,

T,M-. TIM,

JoJ=

multiplication

dim1 M> 2, not every almost complex structure arises from a complex manifold structure on M. There are inlegrabilhty conditions, by — i is called an almost complex structure on M. In

which ate neceseary, and also sufficient INNJ. The integrabiity conditions are automatically satisfied for dim1 M 2, and every almost complex structure comes from a complex structure in that dimension.

GEOMETRY OF COMPLEX MANIFOLDS

0, where the Kahler form the usual definition in terms of forms, dw co(X, Y) = g(JX, Y) (see [KobN] or [Gr2J). It is interesting to note that the concept of what is now known as Ki.hler metric was in fact first introduced

by J. A. Schouten and D. v. Dantzig [SD) some three years before Kähler's paper on the subject [Kili]; I am indebted to H. Wu for bringing this to my attention. The present "Kãhler" terminology was solidified by A. Weil [We), with a rather hedged bet as to its historical validity (cf. the introduction to (We]). By now, the error is unfortunately too deeply embedded in the literature to be rectified. Sic transit gloria mundi. For "K.ählcr" manifolds, the expectation that metric geometry should determine complex structure can be seen to be reasonable in a very precise

sense: Because I is parallel, I at one point of a (connected) manifold determines I at every other point as well. Thus the metric together with the linear algebraic choice of I at a single point suffices to determine the whole complex structure. Since the behavior of parallel translation is itself controlled by curvature according to the principles of Riemannian geometry, the philosophical position that curvature controls complex function theory is thus validated at least for KAhier manifolds.

2. Curvature and Rlemann surfaces. Historically, the relationship between

curvature and complex structure was first explored in detail for Riemann surfaces. On a Riemann surface, a Hermitian metric (or a conformal metric, as it was called classically) is automatically Kiihler: this is obvious from the form viewpoint in that d (KAhier form), being a 3-fonn, is zero for dimension reasons alone. To see it from the parallelism of I viewpoint,

note that the linear-algebraic choice of I at one point is restricted to a choice of orientation only, if the metric is given: IX, I a (real) tangent vector, must have the same length as X and be perpendicular to I, so there are only two possibilities for ii in the 2-dimensional case, and once one

such is chosen, J is determined at that point since I, JX form a basis in the 2-dimensional case. From this depiction it is clear why I is parallel: I is just rotation by ,r/2 in the direction determined by the orientation this is parallel because the metric itself is parallel. This argument also shows of course that there are at most two complex structures (on a connected 2manifold) compatible with a given metric. That some compatible complex structure exists (for an orientable 2-manifold with metric) is the goal and consequence of the "isothennal parameter" problem of writing a given metric g as A(x, y)(dx2 + dy2) for some local coordinate system (x, y). Then

x + iy becomes a natural choice as a complex local coordinate. This was clarified quite early in the development of the subject (see, e.g., (Bar, Chapter 21).

The next question was naturally to try to relate the complex structure attached to metric plus orientation to curvature. Here the situation for compact Riemann surfaces is deceptively simple, compared to higher dimensions, es-

4

R.E.GREENE

pecially for positive curvature: A compact Riemann surface with nonnegative curvature (for some Kähler metric) that has positive curvature at one point

must h.' topologically CP1: by the Gauss-Bonnet Theorem, it has positive Euler characteristic and by classification of compact surfaces, the only onen table compact surface of positive Euler characteristic is S2(= CP5. Then by the Uniformization Theorem., the Riemann surface must be biholomorsphere." Of course, the curvature hypothesis phic to CF' , the here can be relaxed to J K dA > 0, as one sees from the proof just given. 0, or indeed if The Gauss-Bonnet argument also shows that if curvature j. K c/A = 0, then the Ricmann surface is a torus; and if K I) is +x. I

For purpose of generalization to higher dimensions, it is useful to replace the integrated curvature idea with pointwise estimates of curvature behavior. In the case of manifolds with a rotationally symmetric (around...one point)

metric, this can be carried out quite directly in great detail: a 2-manifold with metric of the form dr2 + f2(r) dO2 in geodesic polar coordinates, with f(r) > 0 for all r > 0, can be mapped conformally onto the disc in C of radius (which may be +oo):

by the map, with R = euclidean polar-coordinate radius:

(r, 0)—. (R, 0),

R=

In this situation, it is clear that f(t) being large for large t is associated

to f I /f(r) di being finite and hence to biholomorphism to a disc of finite radius, hence to On the other hand, relatively smaLl 1(t) corresponds to f 1/1(z) di being too so that the complex structure is that of C. Along the lines of this observation, Milnor [Mi] showed the following:

If the curvature K(r) at distance r is —l/(r2 logr) for r large, then the

If, for some e > 0, K(r) —(1 + e)/(r2 log r) for large r and also f(r) is unbounded above, then the manifold is biholomorphic to These results of Milnor suggest that as far as powers of r are concerned,

manifold is biholomorphic to C.

quadratic decay of is a natural boundary between hyperbolic and parabolic C) The following result of the author and H. Wu [CrW7] gives precise substance to the part of this expectation (in all dimensions): Let M be a Riemann surface with Hermitian (conformal) metric having

R. E. GREENE

6

Gauss

curvature K satisfying

K-A/(l+r2), distance from some fixed point p M and A is a positive constant. Then M is hyperbolic, i.e., has universal cover

where r

This result, as shown in [GrW7], holds in higher dimensions as well; in this

case, the bound above is to be imposed on holomorphic sectional curvatures, and the conclusion is that the manifold is hyperbolic (in the generalized sense of there being bounds above on derivatives of maps of into M, taking 0 to a given point of M). See (GrW7J for further details; this reference also contains some conditions under which a given manifold admits no nonconslant bounded harmonic functions which give a higher dimensional analogue of the biholomorphic to C conditions of [Hu], [BIFIJ and [MI]. The relatively simple situations for relating curvature and Riemannian surfaces just described become much more complicated in higher dimensions.

New features that arise include, in no particular the failure of the Euler characteristic to determine topological type, the failure in general of both existence and uniqueness for a complex structure relative to which a given Riemannjan metric is Käbler (or Hermitian), and the failure of the Uniformization Theorem. So much goes wrong that it is natural to wonder if anything goes right in trying to generalize the Riemann surface results. But in fact some things survive. First of all, the behavior of holomorphic maps of into a complex manifold can be studied via the "pull back" of the manifold metric to be a Hermitian metric on in this approach, it is really irrelevant that the "target" manifold is In particular, the Ahlfors Schwarz Lemma and related characterizations by curvature of when a manifold is hyperbolic continue to apply in higher dimensions, as already noted. Second, a certain set of general expectations arising in the case of Riemann surfaces continues to oiler useful guidance. Among these are: that there are relatively few complex manifolds of positive, or of nonnegative curvature; that negativity of curvature in the noncompact case is associated to boundedness of domains in C"; and that nonnegativity, or the existence of only a little negativity, is associated in the noncompact case to being like C", rather than like a bounded domain. These expectations aie so general as to provide little specific guidance, it would seem. But over a long time, developments have occurred that give specific substance to this vague picture, and in certain directions, at least, the situation has become very well understood, though, in other directions, major questions remain open.

3. Compact manifolds and holomorphic line bundles. Like the theory of Riemannian manifolds in general, the theory of complex manifolds divides naturally into the compact and noncompact cases. The theory of compact manifolds is, at first sight, less closely related to the analytic theory of several complex variables than is the noncompact manifold theory. In particular,

GEOMETRY OF COMPLEX MANIFOLDS

many questions about noncompact manifolds are well illustrated by the specific case of domains in C" , while the natural questions in the compact theory do not have such obvious function theoretic analogues. On the other hand, as it happens, the techniques that arise in the compact theory are closely related to, or even equivalent to, in some cases, important proof methods in function theory. Thus it is well worthwhile to consider the compact case, even if one is primarily interested in function theory. As noted, every compact Riemann surface is algebraic. But already in complex dimension two, there are compact complex manifolds that are not algebraic. Thus the question of which are and which are not algebraic becomes fundamental. The most obvious restriction that a manifold must satisfy in order to be algebraic is that it must admit a Kähler metric. This follows from two facts: first, a complex submanifoki of a KihIer manifold is a K.ahler manifold in

its induced metric, and second, CP" has a Kahler metric. (The standard Kähler metric on known as the Fubini-Study metric, is obtained by —' CP" a Riemannian submersion, i.e., dir making the fibration it : to be isometry on the orthogonal complement of the kernel of dir at each point.) Riemann surfaces are automatically KAhler manifolds: as already noted, every Hermitian metric is Kihler metric. But in complex dimension 2, there are compact complex manifolds which admit no Kähler metrics (see (CEJ for some basic examples). The necessary condition that a compact complex manifold M be a KAhier manifold turns out not to be sufficient to imply that M be algebraic. A further necessary condition arises from considering holomorphic line bundles. Recall that a holomorphic line bundle L —. M is a holomorphic complex vector bundle with fibre dimension 1. Think of this as a collection of "transition functions" n : —. C relative to some covering {U} of M, with holomorphic, = l/a,,, aIJaJkakI = 1. Then a Hermitian metric on the bundle corresponds to a collection of positive C°° functions h.: U1 —. R satisfying h. = Since a section s of the bundle is a collection S.: —+ c satisfying = this property of the h1's makes setting 11s112 = well defined in changing from U1 "coordinates" to coordinates. The Levi form is

(*)

—>2

is globally defined, as one sees by calculation; we call it the curvature form of (L, h). We call a bundle positive if there is a Hermitian metric on the bundle for which this form is positive definite. (For further details, see, e.g., [Gr2].)

It is easy to see that a positive line bundle on a manifold restricted to a submanifold is again a positive line bundle. This Is a consequence of the fact

R. E. GREENE

the Levi form along a (complex) submanifold is just the restriction of the manifold Levi form to the submanifold. has on it a natural line bundle, as indeed does every comNow plex manifold, namely the bundle of differential forms of type (n, 0), i.e., in holomorforms that have the expression f(z1, ... , z,jdz1 A. A phic local coordinates (z1, ... , Zn). This bundle is called the canonical line bundle. On CPA, the dual of this canonical line bundle is positive. The Hermitian metric required by the definition of positivity is obtained by usto have A. . . A itself: one defines ing the Fubin i-Study metric on A A or, equivalently, where length = to have length I if Vi,..., Va is an orthonormal basis of the linear span of The fact that this satisfies the positivity condition follows by

in the direct calculation, it is associated to the positive curvature of usual sense of Riemannian geometry. In fact, one can show that the metric induced on the dual of the canonical bundle by a KAhier metric on the manifold satisfies the positive line bundle condition if and only if the Kähler metric has positive Ricci curvature. Combining the observations of the two previous paragraphs, one sees that a compact complex manifold that is algebraic has a positive line bundle. This necessary condition is in fact sufficient, as proved by Kodaira (cf. [MorK] and references therein). More precisely, Kodaira constructed an embedding

of such a manifold M into CP", k large, as follows: of L, the ratios For any line bundle L M, and sections s1, ... , are well defined: If e 0 is an element of ir(P) then = 0, one can set sjsp = C; so, if C, and E C. ç This ratio is independent of the choice of e. More generally, along the same (p) are 0, then the numbers t1 lines, if not all s1 (p), ... , (.P) obtained as before by setting t = s/e determine a well-defined point of (p) as "homogeneous coordinates." Thus one can by taking 11(p), ... , CP' well defined everywhere, provided the sections obtain a map M SI. ... , do not all vanish simultaneously at some point of M. This idea can be used to construct embeddings of Al into CFk provided one knows how to construct holomorphic sections of some line bundle L, the Sections not having any common-to-all zero. From the viewpoint of complex analysis, there is a natural way to construct holomorphic sections. First, operates on sections and a section s is holomorphic if and only if =0. Thus one looks for holomorphic sections = , for some (nonholomorphic) section of the form s1 — u where Here one hopes to control u so that s1 -. u is nonzero at some particular point of M; for example, one could take s1 to be nonzero at the point and try to force the solution u of = to vanish at the point. The familiar L2 — method of analysis applies here. Historically, the

GEOMEtRY OF COMPLEX MAN!FOi.DS

reasoning we arc discussing predates by a considerable margin the use of the method in analysis. however. It is just that tile analytic form is more

lamiliar to analysts, not that it was developed lust, in fact, the essentials method for the noncompact case (e.g., domains in C2) were i.e.. [AnVI—AnV4J predate a'so developed first via the geometry 0 if, for some [Hörl. As usual, we can solve •f, f a 1-form with + > Cf(a, a) or C > 0, an a itrary I-form, of the L2 —

i)

C f(a, 4. Then one obains a solution with 1. The first fact is that the notion of nontangential limit is not anymore the right one to use. As first discovered by Korányi and Stein ([Ko, KS] and [S];

see also [Krl, 2]), the natural approach regions for the study of boundary behaviors are not cones, but regions which approach the boundary nontangentially along the normal direction, and tangentially (at least parabolically) along the complex tangential directions. Here we shall use a particular kind of approach regions defined by means of the Kobayashi distance (see [Ki, 2] and [A4] for definition and properties), called K-regions, which in strongly pseudoconvex domains are comparable to the standard admissible approach regions defined by Stein (see § 1). In particular, we shall say that a function L defined on a domain D c C' has K-limit L at x E OD if f(z) as z —p x within any K-region with vertex at x; in strongly pseudoconvex domains, this is just a restatement of the notion of admissible limit. By the description we gave of approach regions, it follows immediately that a bobmorphic function having K-limit at a point x E ÔD has nontangential limit there too; the converse is, in general, false (see [Rj for a counterexample in the unit euclidean ball B" of C"). It turns out that to generalize the Julia-Wolff-Carathéodory theorem to several complex variables one needs another, slightly weaker, notion of limit. The reason is that another theorem enters the picture: the Lindelöf principle. The one-variable version of it states that a bounded bobomorphic function having limit along a curve ending at a boundary point actually has nontangential limit at that point. In several variables, then, one can hope that the existence of the limit along a curve would entail the existence of the admissible (or K-) limit. Surprisingly, this is not true: as discovered by (irka ([(]; see also [CX, DZ]), it only entails the existence of the limit along curves approaching the boundary slower than K-regions, roughly speaking. Of course, this takes in all nontangential curves; so this kind of limit, called restricted Klimit (or hypoadmissible limit), is stronger than the nontangential limit, and strictly weaker than the K-limit (see [R] for the relevant counterexamples). To be more specific, let D c C" be a strongly convex C2 domain, and take x We shall denote by n. the Outer unit normal vector to ÔD at x, and by the orthogonal projection of C" onto x + (which is the normal complex line to ÔD at x), i.e.,

f

—x+(z—x, where (., •) is the standard hermitian product on C". Let y: [0, 1) D be a curve such that y(t) x as I —. 1; we shall say that y is n-restricted if y goes to x nontangentially as t , and that 1

ANG(JLkR DERIVATIVES IN STRONGLY PSEUDOCONVEX DOMAINS

y is n-special if urn

(0.2)

where

:—.i .

is

—

I(y(t) —x,

=

the euclidean norm. Note that (0.2) exactly says that y is

approaching x slower than parabolically. (Actually, this is not the definition of n-special curve we shall use in this paper, but it is essentially equivalent to it; see Lemma 1.3).

We shall say that a function f: D C has n-restricted K-limit L at for all n-restricted n-special curves y in L as I D ending at x. Then a several variables version of the Lindelöf principle x E OD if f(y(t))

1

is the following: if a bounded (and not only; see Theorem 1.4) holomorphic

function f: D — C has limit L along a n-restricted curve in D ending at x E o D, then f has n-restricted K-limit L at x. It is worth noticing that this is only one kind of Lindelöf principle: a large class of such statements has been introduced in [A3]; see also § 1.

The second new feature a several variables versãon of the Julia-Wolff-Carathéodory theorem depends on a much more simple-minded fact: in one variable one has only one derivative, whereas in several variables one has a whole bunch of them (a—jacobian—matrix of them), and they have no reason to display all of them the same behavior near the boundary—and indeed they do not: the behavior of the normal component of the differential is quite different from the one of the complex tangential components. To

be specific, let D c Cm and D' c be two bounded C2 domains, and f: D — D' a holomorphic map having K-limit y E 00' at a point x 00. of) = Then the differential df of f has a normal component dl, where

= (z,

and a complex tangential component

Furthermore both components may be applied to a normal vector like n, or to a vector which is complex tangential to 00 at x. It turns out o that the normal-normal component has nonzero K-limit at o x; the tangential-normal component d(f — has K-limit zero if multiplied by the weight d(z, where d(., OD) is the euclidean distance from the boundary; the normal-tangential component has K-limit zero even divided by the weight d(z, OD)L'2 ; and the tangentialtangential component d(f — o is only bounded in K-regions. Summing up, the aim of this paper is to prove the following Julia-WolffCarathéodory theorem for strongly pseudoconvex domains:

Let D c Cm and D' c C" be two strongly pseudoconvex C3 domains, and f: D —+ D' a holomorphic map such that d(f(w),OD') urn inf O so that PROPOSITION 1.1.

c

c2M)fl V,

for any M> 1. PROOF. The inclusions on the left follow easily from [A3, Lemma 1.3] and from the boundary estimates for the Kobayashi distance [A4, Theorems (2.3.5 1) and (2.3.52)1.

To prove the inclusions on the right, we first remark (see [AS)) that the estimates of the Kobayashi metric proved by Forstneric and Rosay [FR) and

Graham [G] yield a neighbourhood V cc U of x and a constant a1 > 0 such that E

Dli V,

kD(z, w)

w)—a1.

Thus, using once again the boundary estimates, we get a constant a2 > 0 depending only on z1 such that kD(z, w) — kD(zl, w) + kD(zI, z)

w) —

+

w)

z) — a2,

for every z, w E D n V, and the assertion follows from 1A3, Lemma 1.3).

Let Dcc C" beabounded domain, and fix z0€D and x€OD. We L as shall say that a function f: D —, C has K-limit L at x if f(z) z x within Kf (x, M), for any M > 1; it is easy to check that this definition does not depend on z0. If f, g: D C are two functions such that for every M> 1 there is a constant CM)O sothat

f If

f = (t(1), we shall say that f

QX(g)

K-bounded at x. Again, it is easy to check that these definitions do not depend on z0. Note that we stated these definitions using Kf instead of Hf for convenience only indeed, is

in strongly pseudoconvex domains, big. and small K-regions are comparable

MARCO ABATE

28

near

the boundary (thanks to Proposition 1.1 and the equality of big and

small K-regions in strongly convex domains). have to define the notion of restricted K-limit suitable to our Now needs. Let D cc C'1 be a bounded C2 domain, and x E 3D. A x-curve x as 1—' 1. In (A3] D is a (smooth) curve y: [0, 1)—' D such that y(t) we defined a projection device for D at x as a set of data of the following A —* D which extends C' to so that form: a holomorphic immersion is transversal to 3D; a neighbourhood U0 of x in C'1; = x and and a device associating to every x-curve y contained in U0 a x-curve contained in — or, which is the same, a 1-curve contained in U0 fl

Given a projection device at x E 3D, we say that a x-curve y is restricted if y, —s x nontangentially, and that y is special if

=0.

limkD(y(t),

We say that a function f: D —' C has restricted K-limit L at x (with respect to the given projection device) if f(y(i)) —. L as I —' 1 along any special restricted x-curve. Using these definitions it is easy to prove the following general Lindelöf principle (see [A3J):

Let D cc C'1 be a C2 domain equipped with a projection Let f: D —' C be a bounded holomorphic function, and assume there is a special x-curve y° such that THEOREM 1.2.

device at x E 3D.

limf(y°(t)) f—si

exists.

Then f has restricted

= L

K-limit L at x.

should be remarked that the same theorem holds (with a very similar proof) for functions which are normal in the sense of Cima and Krantz [CKI. In this paper we shall need three different projection devices. We already introduced the first one: let D c C'1 be a strongly convex C2 domain. For any x 3D, let denote the outer unit normal vector to 3D at x, and the orthogonal projection of C'1 onto x ÷ It

=x+(z—x, (note that = (w, for any z E C'1). Since D is = strongly conve; we can find a biholomorphism between A and which extends C1 to in such a way that (1) = 1 and ! (1) is transversal to 3D. Then the normal projection device at x is given viking = U0 = C'1 and setting y for any x-curve y. Using this projection = ° device, we shall talk of n-special and n-restricted curves, and of n-restricted K-limit.

ANGULAR DERIVATIVES IN STRONGLY PSEUDOCONVEX DOMAINS

29

For the second projection device, let D c be again a strongly convex —. D is a complex geodesic if C3 domain. A holomorphic map = C2 E kD(cp(CI), Lempert [LI, 2, 3] (see also where w is the Poincaré distance on [A4]) has shown that every complex geodesic in a strongly convex C3 domain and that for any z0 E D and x E 8D there is a unique extends C' to (1) = x. Furthermore, (0) z0 and complex geodesic such that there exists a holomorphic function C' up to the boundary, such —. ° in particular, is a holomorphic retraction that = = (i.e.,

Then the geodesic projection device at x (with pole z0) is given taking = y for any x-curve y. Using U0 = C and setting this projection device, we shall talk of special curves, restricted curves, and, restricted K-limit without any other specification (cf. for instance §2). The third projection device is the localization of the first one. Let D cc (C"

be a C2 domain, and x E OD. A convexijication of D near x is

given by a neighbourhood U0 of x in C" and a biholomorphism 111: U0 —' 'P(U0) cc C' such that P(U0 n D) is a strongly convex C2 domain (in particular, U0nD is a strongly pseudoconvex C2 domain, and D is strongly pseudoconvex near x). Now fix a convexiflcation (U0, 'I') of 11 near x, fl D) with the normal projection device at 'l'(x). Then the and equip local normal projection device of D at x (with convexifi cation '1') is given = ° U0 and setting = taking V)v(X)) for any x-curve y. We shall say that a function f: D —, (C has normal restricted K-limit L at x if f(y(t)) —. L as t —, 1 along any x-curve y which is special restricted with respect to some local normal projection device at x (it is probably true that a x-curve y which is special restricted with respect to a given local normal projection device is special restricted with respect to all of them, but we shall not need this fact). Our aim in this section is to prove a LindelOf principle for the normal restricted K-limit; but first we need to briefly investigate the normal projection device in strongly convex domains. In particular, we need the following: Let D cc C" be a strongly convex C2 domain, andfzx z0 E D and x E OD. 'Let y be a x-curve. Then: (i) If y is n-restricted, then it is n-special jff LEMMA 1.3.

2

(1.1)

tim

'if (1.2)

Jim

IIv(t) — X112 — x,

= 0.

MARCO ABATE

30

(ii) If y(t)

K2(x, M) eventually for some M> 1, then

K.(x,

(iii) If y is n-restricted and n-special, then y(t) all M large enough.

is n-restricted. M) eventually for y

(iv) If y is nontangential, then it is n-restricted and n-special.

(i) Up to an isometry of C" , we can assume n = x; in particular, x)x. Then the first equivalence follows arguing as in the proof (y, = of [A4, Proposition (2.7.1 1)] (actually, the computations are much easier), noticing that (1.1) implies that y(t) belongs eventually to a euclidean ball PROOF.

internally tangent to OD at x. For the second equivalence, one has moreover, being

— x112

=

—

— I(y(t) — x ,

;

nontangential, there is c1 > 0 such that

ÔD)

— XII

aD),

I(y(t) — x,

and so (1.2) is equivalent to (1.1).

(ii) We can assume D c B" and D and B" tangent to each other at x. Then [A3, Proposition 1.13] yields y(t) E (x, M1) eventually for a suitable M1 > M. Since in B" the geodesic projection device with pole at 0 coincides with the normal projection device, the assertion follows from [A4, Lemma (2.7.1 2.i)1.

(iii) We can assume B" c I) and D and B" tangent to each other at x. Then a computation using (1.1) yields y(t) E B" eventually; therefore ([A4, Lemma (2.7.12.ii)] in B") y(t) E KB(x, M1) for M1 large enough, and the assertion follows from [A3, Proposition 1.11]. (iv) y is dearly n-restricted. Furthermore, being nontangential, jIy(t) — xli

— xli =

—

x,

for a suitable c> 0, and the assertion follows from (1.2). 0 We are finally able to prove the Lindelöf principle we sought:

Let P cc C" be a strongly pseudoconvex C2 domain, equipped with a local normal projection device at x E EID. Let f: D —. C be a holomorphic function K-bounded at x. Assume that f(y(t)) L as t —' 1 for a given special restricted x-curve y. Then f has normal restricted KTHEOREM 1.4.

limit L at x. PRooF. Assume first that D

is

strongly convex C2. Then the assertion

follows arguing as in the proof of [A3, Theorem 2.5], using Lemma 1.3 instead of (A3, Proposition 2.3].

Now let D be a generic strongly pseudoconvex C2 domain. Then the previous argument (and Proposition 1.1) shows that f has restricted Klimit at x with respect to the given local normal projection device. But nontangential x-curves are special restricted with respect to any local normal projection device, and the assertion follows. 0


2. The strongly convex case. In [A3) we proved the following version of the Julia-Wolff-Carathéodory theorem. THEOREM 2.1. Let D c Cm domains, and fix z0 E D and

and D'

be

D'. Let f: D

two strongly convex C3 D' be a holomorphic

map such that (2.1)

liminflkD(zQ, w) —

for a suitable

f(w))] =

loge < +oo,

E OD. Then:

(i) f has K-limit y E OD' a: x. (ii) The following maps are bounded in every K-region of vertex x: (a)

o

(b) [1 (c) (d)

is any complex tangent vector to OD at x. at x, and the maps (b) and (c) have restricted K-limit 0 at x. We call this theorem the invariant Julia- LVolff-Carathëodory theorem (in short, invariant JWC) for strongly convex domains. Several remarks are in order. First of all, the boundary estimates of the Kobayashi distance in strongly pseudoconvex domains (see [Al, 4J) show that (2.1) is equivalent to where

(iii) The function (a) has restricted K-limit

hmlnf w-.x

d(f(w),OD')

d(w,OD) the tim inf in (2.1) has the advantage of being

invariant under biholomorphisms, as well as the rest of the statement. Second, in Theorem 2.1 is proved only for self-maps of a strongly convex C3 domain, but the same proof (with obvious changes) works as well for the general case. Third, the statement is essentially uniform with respect to the map 1. In other words, the bound for maps (a)—(d) in M) depends only on M and and the rate of convergence of maps (a)—(c) to their limits along a restricted special x-curve y depends only on y and the rate of convergence of (2.1). This follows from a careful (but not difficult) inspection of the proof of Theorem 2.1. Finally, the invariance of Theorem 2.1 has a disadvantage. Since it is based in an essential way on global objects like complex geodesics, it is not suitable to be localized, and thus it cannot be directly used to prove statements for strongly pseudoconvex domains. For this reason, in this section we shall prove a version of Theorem 0.2, easily localizable. Clearly, the idea is to apply Theorem 1.4; so we first need to replace the maps in the statement of Theorem 0.2 by holomorphic maps. This is accomplished by

MARCO ABATE

32

PROPOSITION

fix z0

Let D cc

2.2.

be a strongly convex C3 domain, and

D and x ÔD. Then

d(z, ÔD) = Paoop. First of all,

and

1

1

Ii

= Ox(d(z, aD)).

1

= tanh kD(zo, z) , and the first assertion follows from the upper boundary estimate [A4, Theorem (2.3.51)1. where

Next,

if z

M) and so

M) then i%f(l

Ii

Now

[A3, Lemma 3.111 Theorem (2.3.52)] yield

= A'f(l

—

and

the lower boundary estimate [A4,

cM2d(z, OD),

< 2M2(1

1

—

for a suitable c> 0, and the second assertion follows. 0 Therefore in a hypothetical statement of Theorem 0.2 for strongly convex everywhere. domains we can replace d(z, ÔD) by — On the other hand, Theorem 0.2 is, essentially, the invariant JWC with 1

the normal projection device instead of the geodesic projection device. To properly deal with this replacement, we need to know something more about the geodesic projection device. We recall some facts. Let z0

2Y be a complex geodesic such that there is the dual map = x. Associated to

and

such that VT

where p:

E 8A,

is never vanishing. g, is uniquely determined up to a (1) = ii. It turns Out that the scalar positive multiple; so we can assume product IR4

q'(C)) = is a positive constant independent of The geodesic projection and the associated function —. are then defined as the unique solutions of the equations

(z

0, px =

For details and proofs see [Li, 2, 3] and [A4].

PRoPosITIoN 2.3. Let 1) cc be a strongly convex C3 domain; fix D and x OD. Then for any z E 1) we have (2.2)

=

ANGULAR DERIVATIVES IN STRONGLY PSEUDOCON VEX DOMAINS

and (2.3)

= is given by

where A7 E

(2.4)

A7

In particular, A0

—

(z

A, for all

A.

C

PROOF. By definition, = {v E C"

ker

(v,

=

is never vanishing, and and (2.3) for a suitable A7 E C. Furthermore, and so, since

are given by (2.2)

VCEA,

and so A, A,.. It remai'ns to prove (2.4). Fix z E D and define y(t) = clearly y(t)

z

+

D for )1J small enough. We have 0;

(y(t)

hence differentiating with respect to t and evaluating at t = 0 we get —

+ (z

0.

.

Then using (2.2), (2.3), the definition of

A,

a COROLLARY 2.4. Let D E D and x OD. Then

cc

—

A7J

PROOF.

C" be a strongly convex C3 domain; fix -

=

(d(z, aD)"2).

By (2.4) we have —

but [A3, (3.4)J yields liz

—

A71 = 0(liz —

aD)"2), and the boundary

=

estimates yield

8D) =

0(11

hence the assertion follows from Proposition 2.2. (3 Now we can begin to study the relations between = (z, so =

and

Set

MARCO ABATE

34

2.5. Let D cc C be a strongly convex C3 domain; fix E D and x e OD. Then

PROPOSITION

= OX(d(z, aD)'12),

—

where Ii

II

denotes the operator norm.

PROOF. For v E C" we have

_i)

(2.5)

we have

Since

= 0(11

hence the assertion follows from Corollary 2.4, Proposition 2.2, and (2.5). 0

Let D cc C be a strongly convex C3 domain; fix z0ED and xEOD. Then PRoPosITIoN 2.6.

cf(d(z, aD)'12). PROOF.

For v E C we

have

'1X)c:(P(z)))n

= thus —

o

ic —

+

> 0. We already showed in the proof of Proposition 2.5 thai the first addendum is (d(z, aD)). The second addendum can be c0

estimated as follows:

c)t

—

IA1—

—ç)f,

+

and the assertion follows from Corollary 2.4.

0

The last two preliminary results involve the behavior of the holomorphic map in question and of its differential: PROPOSITION 2.7.

Let D cc C", D' cc C" be strongly convex C3

domains, and fix z0 E D, such that

E D'. Let 1: D

D' be a holomorphic map

liminf[kD(zO, w)— kD(z'o, f(w))] =

0 so that aR) cc U', where a is given by (3.2).

Now, fix M> 1; it suffices to consider z E (3.3)

kD(zo, z)>

M) such that r.


R) (where Bk(zO, r) is the ball Note that Hf(x, 2M) \ Bk(zo, r) C of center z0 and radius r with respect to the Kobayashi distance of D), and so f(z) F?(y, aR) cc U' (by Julia's lemma (A3, Corollary 1.5]). in strongly convex domains are strongly convex near the Since boundary (1A3, Corollary 1.17)), by Proposition 1.1 in '1'(D fl U) we can single out a strongly convex C3 domain Q such that

It follows that 'F' 010 'I's:

'P'(U' fl D') is a holomorphic map between strongly convex C3 domains, where we can apply Theorem 2.9 (and Cowllary 2.10). Then we can invoke Proposition 1.1 and Theorem 1.4 (noticing —b

that a x-curve in D which is normal to OD at x is eventually contained in any K-region of vertex x), and we are done. REFERENCES [Al] (A2J [A3j [A4J [AS) [B)

M. Abate, Boundary behavior of invariant distances and complex geodesics, Atti Accad. Naz. Lincei Rend. CL Sd. Fis. Mat. Natur. 80 (1986), 100—106. , Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), 225—238.

—, The Linde/of principle and the angular derivative in strongly convex domains, 3. Analyse Math. 54 (1990), 189—228. —, Iteration theory of holomorphic maps on taut manifolds, Mediterranean Press. Cosenza, 1989. —, Two remarks on the Kobayashi distance, Preprint, 1990. R. B. Burckel, An introduction to classical complex analysis, Academic Press, New York, 1979.

(C)

C.

Carazhéodory, Uber die Winke/derivierten von beschrânkten analytixhen Fwikiionen.

Sitzungsber. Preuss. Akad. Wiss., Berlin, 1929, pp. 39—54.

[CK)

J. A. Cima and S. G. Krantz, The Linde/Of principle and normal functions of several

(CJ

complex variables, Duke Math. .1. 50 (1983), 303—328. E. M. The Linde/of and Fa:ou theorems in C", Math. USSR-Sb. 21(1973), 619—641.

(JJ (KI)

lu. N. Drothnov and B. 1. Zav'jalov, On a analogue of a theorem of Lindelof Soy. Math. DokI. 25 (1982), 51—52. F. Erwe and E. Peschl, Uber beschrankie Sysieme von Funkiionen, Math. Ann. 126 (1953). 185—220. F. Forsineric and J.-P. Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987), 239—252. I. Graham, Boundary behaviour of the and Kobayashi metrics on strongly pseudoconvex domains in C" with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219—240. M. Hervé, Quelques propriélés des applications analyliques dun.e boule a m dimensions dans elle-mème, J. Math. Pures App!. 42 (1963), 117—147. 0. Julia, Extension nouvelle d'un lemme de Schwarz. Ada Math. 42 (1920). 349—355. S. Kobavashi, Hyperbolic and holomorphic mappings, Dekker, New York,

[1(2)

—,

[DZJ

[EP) (FRI [G)

[H)

1970.

[1(o)

(KS)

Intrinsic distances, measures, and geometric function theory. Bull. Amer. Math. Soc. 82 (1976), 357—416. A. Korányi, Harmonic functions on hermitian hyperbolic spaces, Trans. Amer. Math. Soc. 135 (1969), 507—5 16. A. Korányi and E. M. Stein, Fatou 's theorem for generalized half-planes, Ann. Scuola Norm. Sup. Pisa 22 (1968), 107—112.

MARCO ABATE

[Kr)) (Kr2] [LV)

[Lii

S. Krantz, Fatou theorems on domains in C", BuU. Amer. Math. Soc. 16(1987), 93-96. Invariant metrics and the boundary behavior of holomorphic functions on domains in C" , Preprint, 1989. E. Landau and G. Valiron, A deduction from Schwarz 's lemma, J. Load. Math. Soc. 4

(1929), 162—163. L. Lempert, L.a métrique de Kobayashi et la representation des domaines sur Ia boule. Bull. Soc. Math. France 109 (1981), 427—474.

[U)

—. Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257—261. —, Intrinsic distances and holomorphic retracts, analysis and applications

[N)

1981, Varna, Bulgarian Academy of Sciences, Sofia, 1984, pp. 341—364. R. Nevanlinna, Re,narques sur le lemme de Schwarz, C.R. Acad. Sci. Paris 188 (1929),

[RI

W. Rudin, Function theory in the unit ball of C" , Springer, Berlin, 1980.

{S)

E. M. Stein, The boundary behavior of holomorphic functions of several complex variables,

(WJ

Princeton Univ. Press, Princeton, 1972. I. Wolff, Sur une ge4ne4ralisaiion d'un zhCorèmc de Schwarz, C.R. Acad. Sci. Paris 183

1027—1029.

(1926), 500—502. SECONDA UNIVERSITA DI ROMA. ITALY

of Symposza in Pure

Volume 52(1991), Put 2

Degeneracy Points of the Kobayashi Pseudodistances

on Complex Manifolds YUK1NOBU ADACHI AND MASAKAZU SUZUKI

Introduction. Let M be a complex manifold of dimension n and dM(p, q) be the intrinsic pseudodistance of two points p and q of M introduced by Kobayashi [6). We say that dM degenerates at p if there exists a point q p such that dM(p, q) = 0. In the present paper, we shall deal with the set S of degeneracy points of dM. To simplify the matter, let us assume that M is compact. We have then the following questions: (A) Is S a (closed) analytic subset of M? (B) If S is a complex analytic curve in M, then is each irreducible com-

ponent of S of genus
(Jd

+.

÷

do2 is the metric on S induced by ds2. Put U

{x E ,('(V)Ip(x)

the distance from x to S by ds2. If one takes e > 0 sufficiently small, we have then U C W and each U,, 1'(p) fl U, p V, where p(x)

is

is homeomorphic to a ball of

—2 dimensions. In particular,

K = {x E 7J1P(x) = e}

compact. We have (X) fl K = 0 by (I). Therefore, by Lemma 2, there exists a sequence of hoLomorphic mappings : A(R,) —, M which satisfies is

=p E V,

(7)

(6) 1ImRV =

and

(8)

=

I,

=0.

be the connected component of f'(U) which contains z =0 and V. We consider the sequence of the composed mappings : Let

have then, by (7) and (9), (10) Suppose that to

and has

a subsequence

the constant mapping F(z)

p on

which

=

converges uniformly

Then, if one takes

Y. ADACIU AND P4. SUZUKI

which contains p, there exists a positive integer such that C > Then it follows from (9) that the image is contained for all

whichisisomorphictoasubdomainof VaXfCC" We

in U

have further } has a subsequence (0) = p E J',. Therefore, U, on any which converges uniformly to a bolomorphic mapping 1: compact subset of Thus, we get a holomorphic mapping g = o f of 1 with g(O) = (0). Hence, by the + Izi < 1 to f: Iw,12 + Schwarz lemma, we get 1, where is the euclidean norm of the vector g'(O).

} converges to the constant F

On the other hand, since obtain from (5) and (8) IJg'(O)H

p, we

211f'(O)II = 2.

This is absurd. Hence, {F,, } has no subsequence which converges to the constant F(z) p on Therefore, by Lemma 4, there exists a sequence

of positive integers and a sequence of positive numbers r, 0 < rK 1. As one would expect from Shiffman's result, the HSC of this metric is positive in every neighborhood of 0. One can see this by calculating the curvature of holomorphic planes orthogonal to the radial direction. The completeness condition places stringent conditions on Kähler metrics in n> , so that every such metric has certain essential similarities to Grauert's example. Thus with little additional effort one can show directly that for any radially symmetric complete K.ähler metric on n> 1, the HSC is positive arbitrarily close to 0, and one proves this by calculating the curvature of holomorphic planes orthogonal to the radial direction. 1

Ii. The HSC is simply the sectional curvature restricted to complex planes through the origin of the tangent space. If V = E a1Z1 is a unit complex

vector at a point P (summations are taken over repeated indices and run from to n), where Z = 0/021 for coordinates z1,..., in C, then the HSC at V is defined to be 1

(1)

HSC(V) = 1980 Mathematics Subject Classajication (1985 Revision). Primary

32C 10.

This paper is in final form and no version of it will be submitted for pubLication etsewbere. © 1991 Ameticaa 0082-0717/91 8100+5.25

59

Society

ANDREW BALAS

60

where

are the curvature components 2g11

—

R ijkJ

—

O:kOZj

g

—

mp

Geometrically, where (g") is the inverse transpose of the metric g the HSC at V is the supremurn of the Gaussian curvature of the family of analytically embedded disks tangent to the plane spanned by V and V. Thus the HSC of a submanifold does not exceed that of the manifold. If P E fl, then HSC(P) 0 means HSC(V) (li, VV E 0

YPeQ.

means HSC(V)0,

If g and h are Kähler metrics with nonpositive HSC at a certain point P. then their sum again has nonpositive HSC at that point (Wu]. In symbols, HSCg(P) 0, HSC,,(P) 0 HSCg+h(P) 0.

Ill. We will call the metric g complete at the point P

8fl if P is

More precisely, infinitely far from every point of DEFINiTION. Let P E OQ. Then the metric g is complete at P if, whenever a C' path such that (0, 11 —. C", cr(t) = (a1(t), ... , VaE[0, l),then and

dt = oo. That is, the path from a(0) E c to P has infinite length.

Let g = (g13) be a radially symmetric Kdhler metric on n > 1, which is complete at 0. Then g has positive HSC

PROPOSITION.

somewhere in every neighborhood of 0. PROOF.

The hypothesis implies there exists a potential IP(Z) 1(1(Z)), Z =(z1, ... , Z,J)EC". One calculates

where f: (0,oo)—.R,

1

f'=

(2)

is radially symmetric, we can restrict our attention to a point of the form (u, 0, ... , 0), u > 0. At such a point, Since

0 (3)

,

0

1' Thus g is positive definite if and only if

tf'>O, Vt>0, and

(zf')'>O, Vt>0.

HOLOMORPHIC SECI1ONAL CURVATURE

For g to be complete at the origin, it is necessary and sufficient that the length of any radial line segment from the origin be infinite. We calculate the length of the segment a(u) = (u, 0,... , 0), h 0, there is a u E (0, u0) and an embedded analytic disk tangent to (u, 0, ... , 0)

if',

with Gaussian curvature K > 0 at (u, 0,..., 0). Thus the HSC of g is positive somewhere in every neighborhood of 0. 0 NOTE. A second way to prove the proposition is to use the formulas for HSC in [KN) and then take the HSC in a direction orthogonal to the radial.

ANDREW BALAS

62

/7

L,

/I(i.7)

(r. /ii')

(S.

FIGURE 1

Let h be a real-valued C2 function on (0, oc). Suppose h and is unbounded in every neighborhood of 0. Then for are positive and

LEMMA.

everpt>0, ac, 0<s 0, then PROOF.

and the conclusion of the lemma holds in this case. Therefore, assume h(0) = 0 and fix t. We seek

an s, 0< s < 1,

such

that (h(s)/s)' 0 the B

da(x).

This result c'n be also interpreted in terms of the counting function = #{j;

A)

and of the spectral density measure (a sum of Dirac measures)

HOLOMORPH1C MO1(SE INEQUALITIES

In these notations, Corollary 3.2 can be restated: f+OO

e

—a

djukn(2)

f

B1(X)

= Jfl

We thus see that the sequence of measures ,Uk

da(x).

Converges weakly to a measure

whose Laplace transform is given by the right-hand side. Inverting the formula, we obtain: For almost all 2 E

COROLLARY 3.3.

(]— where VB(4(2) is

the function on ill x R defined by s—rn

= 4.

÷2)dc(x),

2]) =

—rnf2

.•.B5

—+

E

[A —

+

Proof of the holomorphic Morse inequalities. Let X be a compact com-

plex manifold, E and F holomorphic hermitian vector bundles of rank and r over X. If X is endowed with a hermitian metric co, Hodge theory 1

F) can be identified shows that the Dolbeault cohomology group with the space of harmonic (0, q)-forms with respect to the Laplace-Behrami = operator acting on 0 F. We thus have to estimate + the zero-eigenspace of in terms of In order to apply Corollary 3.2, we first have to compute the hermitian connection Vk On Ek oF deduced from the Chern q is what now plays the connections of E, F, TX. The bundle F o A°'

rx

role of F.

The relationship between and Vk is most easily obtained by means of the Bochner-Kodaira-Nakano identity. In order to simplify the exposition, we assume here that the metric co on X is Kähler. For any hermitian holomorphic line bundle G on X, the operators and &' of G are related by the B-K-N identity is"

Here we have c(Ek

oF) =

kc(E) ®

A].

+ c(F), thus

+ k(ic(E), A] + [ic(F), Al.

At a given point z0 E X, we can find a coordinate system (z1, ... that is an orthonormal basis of TX and

,

such

.1. P. DEMAILLY

... , are the curvature eigenvalues of c(E) at z0. A standard formula gives the expression of the curvature term [ic(E), Aju for any (p, q)d z1 A we have form u. With u = ® where

([ic(E), A)u, u)

—

In the case of a (0, q)-form u = Eu1 2dY1 øeA we = and

where

= simply have E4u

V' + (ic(F), Aj,

=

(4:2')

(V'u, u) =

(here I = 0).

This is not yet what was needed, since only the (1, 0) part

appears. To

get the (0, 1) component, we consider u as a (n, q)-form with values in

0 F® ®e2

A

=

in normal coordinates. Thus (4.2")

=

and

+ ky'1 +[ic(F ®

(V"u, u)

Al2

(here!

A),

(I, ...

,

If the metric w is non-Kähler, we get additional torsion terms, but these terms are independent of k. A combination of (4.2') and (4.2") yields (4.3)

=

-V+

where 0 is a hermitian form independent of k and (Vu, u) =

— cxj)luJ AL

Now apply Theorem 3.1 and observe that 0 does not give any significant = + contribution to the heat kernel as k +00. We write here so

that B = ic(E) = EL<J0. At this point, the main idea is to use the eigenspaces in order to construct a finite-dimensional subcomplex of the Dolbeault complex with the same cohomology groups. This was already the basic idea in Witten's analytic proof q of the standard Morse inequalities. We denote by resp. the A—eigenspace of acting on F), resp. the direct sum of eigenspaces corresponding to all eigenvalues A. As = and commute, we find that , thus and are finiteC dimensional subcomplexes of the Dolbeault complex

yk.• we find that Xtk has trivial = kAld Since i9kc9k + akak = cohomology for A 0. Since is the space of harmonic forms, we see that Zj' has the same cohomology as the Dolbeault complex for 2> 0. We will call this complex the Witten We need an elementary lemma of linear algebra. LEMMA 4.5. #4

Ek

Set #4 —

øF). Then for every 1>0,

+ ... +

PROoF. The left-hand side is the contribution of the 0 eigenvalues in the right-hand side. All we have to check is that the contribution of the other eigenvalues is ? 0. The contribution of the eigenvalues such that 2> 0 yk.• is e'2 As is exact, one easily sees that the last sum is equal to the dimension of hence 0. • Combining Theorem 4.4 with Lemma 4.5, we get

#4—hr' q

+ 1=0

a.

E Jv /

fl

I

.

fa1j)

e'fr'CJ

'

,i

A

&

J. P. DEMAILLY

This inequality is valid for any t > 0, so we can let t lend 10 +00. It is is always 0, thus the integrand tends to 0 at clear that —E — every point where s < n. When s = n, we have — = < 0 for every and > 0 for every j E Cf 0 if and only if

j e J. This implies x E X(€, E); in this case there is only one multi-index J satisfying the above conditions and the limit is

By the

monotone convergence theorem, our sum of integrals converges to

=

X(1,E)

(=0

X( 0, one can construct many sections of high tensor powers Ek (e.g. by HOrmander's L2 estimates). For k k0 large enough, the "base separate any two points of X and is empty, the sections in locus" Z ( gives an embedding of X in some generate all 1-jets at any point. Then projective space This gives the famous Kodaira embedding theorem: a compact complex manifold X is projective algebraic if and only if X has a hermitian line bundle E with positive curvature. CE Vk such

The Grauert-Riemenschneider conjecture [G-R] was an attempt to characterize the more general class of Moishezon varieties in terms of semipositive line bundles. Let us first recall a few definitions. The algebraic dimension a(X) is the transcendence degree of the field of meromorphic functions on X. A well-known theorem of Siegel asserts that 0 a(X) n. A manifold (or variety) X is said to be Moishezon if a(X) = n. GRAUERT-R1EMENSCHNE!DER CONJECTURE (1970). A compact complex va-

riety Y is Moishezon if and only if there is a proper nonsingular modification X —, Y

and a line bundle E over I such that the curvature of E is > 0 on

a dense open subset.

When Y is Moishezon, it is well known that there exists a projective algebraic modification X; therefore E can even be taken > 0 everywhere on such an X. Siu [Siu2] solved the conjecture by proving the converse statement in 1984; he even showed that I is Moishezon as soon as ic(E) 0 everywhere and ic(E) > 0 in at least one point. We will see that Morse inequalities give a still stronger criterion, requiring only the positivity of some curvature integral.

Since 9(Y)

9(1), we only have to show that X itself is Moishezon. This will be done by producing many sections of Ek. For x = 1, the strong

HOLOMORPHIC MORSE INEQUALITIES

105

Morse inequality (1.4) gives

hI(Ek)

- hO(Ek) < P2.

fX(I,E)

+

2ir

In particular, we get the lower bound hO(Ek)

(2ir)

,

f

X(O (notethat ing t = kt with e small, one can check that all estimates remain uniformly valid and that the contribution of the nonzero eigenfunctions in x) becomes negligible in C° norm. Then Theorem 4.4 shows that K1k(x, as

t=

—.

+00. For

x)

(x) .. .

we get therefore the C° convergence

E

k 2iz

,,

Iai(x)"cr 'I (X)1 ask4+Oo.

As a consequence, the Fubini-Study metric on E induced by (Dk converges to

the original metric. 0. Tian (Til has proved that this last convergence statement

holds in fact in norm C4. 6. The case of q-convex msnifolds. Thierry Bouche [Boul] has obtained an extension of the holomorphic Morse inequalities to the case of strongly q-convex manifolds. We explain here the main ideas used. A complex (noncompact) manifold X of dimension n is strongly q-convex in the sense of Andreotti and Grauert if there exists a C°° exhaustion function on I such that has at least n — q +1 positive eigenvalues outside a compact subset of X. In this case, the Andreotti-Grauert theorem shows that all cohomology groups H"(X, with values in a coherent analytic sheaf are finite-dimensional for m q. THEOREM 6.1. 1, rank

E=

Let E, F be holomorphic vector bundles over X with rank

F=r.

Assumethat Xis strongly q-convexandthat E hasa

metric for which ic(E) has at least n

—

p+

1

nonnegative eigenvalues outside

HOLOMORPHIC MORSE INEQUALITIES

a compact subset. Then for all m p + inequalities hold:

the

following strong Morse

+

X(m,E)

•

For every c E R, we

1

f

® F) PROOF.

q —

107

consider the sublevel sets

w(x)

> a5+1 =

where Sm = r. As

a,., and more specifically that >

=

= a5,

we see that

®

the

Q

va the obvious of partial flags V D = ... M5(V). On we a formula completely : M(V) projection analogous to (7.2), where the only indices (j, k) involved are those for which in particular a1 > ak. Thus Q is ample and QQ = Qa) = ray. (7.3) H°(M5(V), Q') = H°(M(V), Now let E, F be holomorphic vector bundles over a compact manifold X and let

r=rankE,

r'=rankF.

We want to get asymptotic estimates for the dimension of cohomology groups ® F) as k +00. For that, we introduce the flag bundle —i X, where s = ... , Sm) IS defined as above, and we consider

the universal line bundle Q over M5(E). As Q is ample along the fibers of p3 : M5(E) X, the higher directimages ®pF), q 1 , vanish ®pF) = for k k0. By (7.3) we get F. The Leray spectral sequence gives the isomorphism

Qka®P*F) and we are reduced to applying Morse inequalities to tensor powers of the line bundle Q. We still need a formula for the curvature of Q. Let z0 E

be a point in M3(E). Choose a holomorphic frame (e1, ... , e,) of E such that the flag z0 is given by the basis (e1 (x°), ... , e,(x°)) (supposed to be

i. P. DEMAILLY

orthonormal). Assume also the curvature tensor = that

chosen such that DeA(x°) — 0 and consider It can be shown dx3 A dxk e

Adxk +

=

a >a

k

Finally, let the affine coordinates along the fiber Using the isomorphism (7.4), N(s) be the dimension of the fibers

where (zAJI) are

the strong Morse inequality becomes

j

>(_l)Q_rndimHrn(X

niN(s)

r(fl+N(S))! JY(O. Suppose the above field 5 on S has a periodic orbit, say G C S, such that the homology class [G] E H1 (S) vanishes. Then the integral y(G) = fG y for the orientation of G defined by 5, does not depend on a particular Liouville form y. (Compare (We2].) 1 .6.A. AN INTEGRAL CRITERION FOR coilvExiry.

If S contains a

characteristic curve G homologous to zero then y'(G) > 0 for all (Liouvile) forms y' Ofl S satisfying dy' = cv. In particular, S cannot be simultaneously convex and concave, where "concave" means "convex" for the reversed orientation.

1.6.B. REMARKS. (a) The unit circle S1 C R2 is simultaneously locally convex and concave. (b) Using 1 .6.A one can easily construct a closed hypersurface S E for all n 2 which is neither convex nor concave as it contains two homo-

logically trivial periodic orbits, say G1 and G2, such that y(G1) (3n/2)+2. and = Therefore we can apply Theorem L13.A. Q.E.D. 2.5.3. PRooF OF l.12.C. First note that the manifolds and are adimplies their quasisymplectic missible. Then observe that of and diffeomorphism, and apply Lemma 2.4.1. Q.E.D. 2.5.4. PRooF OF 1.l2.A. This follows immediately from Theorem 1.12.C. Q.E.D. 3. More Discussion

phismof

3.1. Convexity, contact type and pseudoconvexity. We discuss, in this section, different conceptions related to the notion of an co-convex hypersurface

in a symplectic manifold X, co. 3.1.1. Contact structure dominated by a symp/ectic one. Let be an oriented bypersurface in a symplectic manifold (X, oi) .A contact structure on L is a completely nonintegrable tangent hyperplane field. In other words, can be (at least, locally) defined by a 1-form cx such that the form dcx is

CONVEX SYMPLECTIC MANIFOLDS

Note that the multiplication of a by a function causes nondegenerate on the multiplication of by the same function. Let us fix the coorientation and the sign of a which defines the given coorientation. Then the of depends only on and will be conformal symplectic class of the form denoted by We say that the symplectic structure w on the ambient manifold X D or, in other words, if on dominates the contact structure

is conformally (with a positive if symplectic structure w restricted to factor) equivalent to the symplectic structure which the contact structure defines on itself. Note that this implies, in particular, that the characteristic (see §1.6) is transversal to line bundle K on 3.L1.A. EXAMPLE. Let (V, g) be a Kählerian complex manifold and be the complex tangency to be a coorien ted real hypersurface in it. Let is a maximal complex subspace of at the point x E (i.e. Then = is a real codimension 1, canonically oriented (and therefore, cooriented) tangent distribution to which can be defined by a 1-form a on The quadratic form E. whose sign is determined by the coorientation of

L(X, Y) =

JY), X, Y e

which is defined up to a conformal The (strict) pseudoconvexity of (see

factor is called the Levi form of § 1.5(1)) is equivalent to the positive definiteness of the Levi form L. Clearly

is a contact structure in this case. Let w(X, Y) = g(JX, Y), X, Y E T( V), be the symplectic form on V defined by the Kählerian metric g. The and L are symplectic structure w dominates if and only if the metrics conformally equivalent.

Note that if dim doconvex L

=

3

then

is always dominated by co for any pseu-

3.1.2. Relations with co-convexity. If

is a (locally) co-convex hypersur-

face in a symplectic manifold (V. co) then it carries a contact structure which is dominated by co. Indeed, if 0 is a contracting field transverse to then the Liouville form y = —(9 1 co restricted to Z defines a contact structure dominated by w. If dims> 3 then the converse is also true: If c V admits a contact structure dominated by w then is coconvex. Namely there exists a coniracting vector field 0 transversal to such (A simple argument which proves this was that the form —0 1 define be found in [McD3].) told us by V. Ginzburg and does not necIf = 3 then the condition that cv dominates on is exact and, therefore, does not imply co-convexity essarily imply that of In fact even in the exact case co-convexity does not follow from the exis-

tence of a contact structure dominated by w (see the next section). 3.1.3. Pseudoconvexity and co-convexity. Pseudoconvexity does not neces-

158

YAKOV ELIASHBERG AND MIKHAEL GROMOV

sarily imply co-convexity as the following example shows.

In C", with the standard complex and symplectic structures, consider a unit circle S c C' c C". Let N be a round tubular neighborhood of N of a radius e < 1. Then if is small enough then the boundary ON s pseudoconvex. Let S c ON n C' be the circle of radius 1 — e. Then S is a closed leaf of the characteristic foliation on ON (see § 1.6) and for any Liouville form A of the symplectic form a, in C" we have f5 A < 0 for S being oriented as in § 1.6. Then the criterion l.6.A shows that ON is not convex in C". In fact one can modify this example in order to get even the local nonconvexity. First consider the case n > 2. Let us find a Legendrian (i.e. tangent to the contact structure) circle S c ON' S isotopic to S, then attach a totally real 2-disc C C"\ mt with = S iicn is perpendicular to ON. This is always possible (see [Gr21) for n > 2 n = 2 there is a homological obstruction for that). Finally we can use (E12) to find a pseudoconvex neighborhood N of N U in C". The new boundary ON is diffeomorphic to (2n — 1)-sphere and still contains S as a characteristic. Therefore global nonconvexity for ON implies its local nonconvexity. If n = 2 the embedded disc c C2\ mt N with required properties does not exist. Instead we take an immersed totally real disc C mt N with one intersection point. We can arrange, in addition, that the disc near the intersection point is invariant under the multiplication by Then applying [E12J we can find a pseudoconvex neighborhood N of N U which

still contains S as a characteristic. Despite the fact that the boundary ON is not simply connected, the homology class of S in H' (ON) is trivial and, therefore, the criterion 1.6 shows that ON is not co-convex even locally. Note that this example also proves the existence of a non- co-convex hypersurface in C2 which admits a contact structure dominated by ai. 3.1.4. Fillable contact structures. Let (V, be an oriented closed contact manifold. We say (V, is (symplecticaily) fl//able if there exists a compact symplectic manifold (W, co) with boundary 8 W = V such that:

the given orientation of V coincides with that induced from V; is transversal to the characteristic vector field r (see § 1.6) of cv on V; cv dominates for the coorientation of defined by r. Symplectic fillableness has its holomorphic counterpart: we say that (V,

is holomorpijically fihlable if V arises as a pseudoconvex boundary of a complex manifold W and is the distribution of complex tangencies to V = 8 W. Note that such a W always admits a Kiihler metric whosc imaginary part dominates Thus holomorphical fillableness implies the symplectic one. We do not know examples when these two notions are different. See [Eli) and [EU] for further discussion of fillable contact structures.


159

3.2. Convex and Weinstein manifolds. We show in this section that the class

of convex symplectic manifolds is larger than the class of Stein manifolds. We start with the following example of D. McDuff (see [McD3J). 3.2.A. PRoPosmoN. There exists a 4-dimensional compact symplectic manifold (X, w) with w-convex disconnected boundary OX.

The manifold X constructed by D. McDuff' is the product of the unit cotangent bundle of a closed surface M of genus > 1 with (0, 1]. By completing X as in § 1.8.4 we get a complete convex 4-dimensional symplectic manifold X with 113(X) = Z. In view of 1.1.2 this means that I is not Weinstein. Taking the product Y = X x . . . x X we get (see § 1.9) an w-convex symplectic manifold of dimension 4n which has homotopy type of a 3n-dimensional CW-complex and, therefore, it is not diffeomorphic to a Weinstein manifold. 3.2.B. QUESTION. Does there exist a complete convex symplectic manifold of dimension 2n > 4 with nontrivial (2n — 1)-dimensional homology?

3.3. Stein and Weinstein. As it is shown in Theorem 1 .4.A, a Stein complex structure on a smooth manifold X defines a unique up to isotopy Weinstein symplectic structure on X. As it follows from 1E12] all Weinstein structures on X can be constructed by this procedure: 3.3.A. THEOREM. Any Weinstein symplectic manifold (X, w) admits a compatible Stein complex structure. Another result from [E121 gives a complete description of the topology of Stein and, therefore, Weinstein manifolds.

3.3.B. THEOREM. Any almost complex structure J on a 2n-dimensional manifold X which admits an exhausting Morse function with all critical points of index n is homotopic to a Stein complex structure I such that p is J-plurisubharmonic. 3.4. The effect of noncompieteness. There exists a w-convex (but noncomplete! ) symplectic manifold which is diffeomorphic to contains a symplectic image 3.4.A. PROPOSITION.

of

but it is not symplectomorphic (even stably) to K2".

The required manifold X is a boundary connected sum of two halfspaces. Namely, let X÷ = {x1 > 1 }, X_ = {x1 0 there exists a smooth function a (0, 1] —JO, e( such that the domain

is w-convex (see (E12J or (We3J).

The fact that X is not symplectomorphic to follows from the obvious fact that the space of symplectic embeddings of the ball R

YAKOV ELIASILRERG AND MIKHAEL

into have:

is connected (in fact, homotopically equivalent to U(n)) while we

3.4.B. CAMEL LEMMA.

The space of synzplectic embeddings BR

—'

X is

disconnected if R > e. SKETCH OF TUE PROOF. We will show that embeddings f_ : BR f_(BR) C X_ are not symplectically isotopic. Suppose with C

X, t E [0, 11, which connects that a symplectic isotopy h, 11R and f_ does exist. Consider a family of almost complex structures, :

.1k,

such that J, coincides with the standard complex structure I E (0, 11 , on everywhere, near L)X for all t [0, and is equal (h,),J0 for z = 0, [0, 1]. Then the theory of holomorphic curves in symplectic on t 1

manifolds (see [Gril) tells us that the sphere S = oX fl {x1 0} can be we filled by holomorphic discs for all t [0, 1). For any of these discs But there exists I E [0, 1] such that the image h,(0) of have IA < from the filling. Then the center 0 e BR belongs to a holomorphic disc passes through the center of BR and its the J0-holomorphic curve boundary is contained in OBR. Therefore,

= and, therefore R

e. (See o < f 0) = ire2 But /i, is symplectic, so [Gril for similar arguments.) Q.E.D. The Camel lemma can be sufficiently strengthened (see [EGJ): The Camel cannot pass through a small sphere. The result has especially simple formulation in the 4-dimensional case: the symplectic Let S E R4 be an unknotted sphere. We denote by and by A(S) the integral f5 subspace X5 R4\S C Let M(BR, X5) Let e: M(BR, be the space of symplectic embeddings BR — be the evaluation map, i.e. e(f) = f(O). The homomorphism : irl(M(BR, X5)) an epimorphis,n for 2irR2 > A(S) and it is trivial otherwise. 3.4.C. THEOREM.

'

it1(X5) is

3.5. Convex contact manifolds. It seems that the analogous notion of convexity in contact geometry is also useful. Let (X, be a contact manifold. A vector field 0 on X is called contact of Namely, the Lie derivative if it preserves the contact distribution is equal If integrates to a one-parametric group : X —. X, I R, t are contactomorphisms, i.e. then all The following notion of a convex Contact manifold is analogous to the notion of a Weinstein symplectic manifold. is called convex if there 3.5.A. DEFINITION. A contact manifold (X, exists a contact vector field 13 on X and a Morse function f: X —+ K4 (see such that: 0 is complete, i.e. integrates to a one-parametric group of contactomor-

phisms of X;


a

is

gradient-like for ço, i.e. Of>

0

outside critical points of F.

with the structure 3.5.B. EXAMPLES. (a) The standard contact dx1 is convex. Indeed, the vector field dz — y1 defined by the form

fl(

0

a —

is contact, complete and gradient-like for the function n

(b) The above example can be compactified which shows that standard is convex. contact structure on have no Notice that closed convex manifolds (as the above contact

counterparts in the symplectic world. QuEsTioNs. Do there exist nonconvex contact manifolds? If they do, what is their relationship with convex symplectic manifolds? For instance let (V, be a fihlable (see 3.1.4) contact manifold. is it convex? Is the converse true?

We think that the understanding of contact convexity is essential for the symplectic geometry. REFERENCES [A]

[AG)

V. I. Arnold, First steps of symplectic topology, Russian Math. Surveys 41(1986), 3.-I 8. V. I. Arnold and A. B. Givental, Symplectic geometry, Encyclopedia of Math., Dynamical System, Vol. 4, Moscow, 1986, PP. 5—139.

V. Benci, Talk at the conference in symplectic geometry, Berkeley, California, 1988. Ya. Eliashberg, Filling by holomorphic discs, Proc. of the Conf. in Low-dimensional Topology, Durham, 1989. Internal. J. Topological characterization of Stein manifolds of dimension > 2, Math. 1 (1990), 19—46. [EU) —, On symplectic manifolds with some contact properties. J. Differential Geom. 33 [Be)

[Eli)

[EI2] _,

(1991), 233—238.

[EG) [EH]

[Gil)

Ya. Eliashberg and M. Gromov, in preparation. I. Ekeland and H. Hofer, Syinpiectic topology and Hamiltonian dynamics. 1, Math. Z. 200 (1989); part H, Math. Z. 203 (1990), 553—568. M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307—347.

[Gr2) [GS)

[HI) [La) [Ma]

, Partial differential relations, Springer-Verlag, Berlin and New York, 1986. R. Green and K. Shiohama, Diffeo,norphisms and volume preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 225 (1979). 403—414. H. Hofer and E. Zehnder, .4 new capacity for symplectic manifolds, Analysis et cetera, Academic Press, New York, 1990. F. Laudenbach, Private communication, 1986.

B. Mazur, Stable equivalence of differentiable manifolds, Bull Amer. Math. Soc. 67 (1961), 377—384.

Applications of convex integration to [McDI] D. and contact geometry, Ann. Inst. Fourier (Grenoble) 31(1987), 107—133. [McD2] —, Blow-ups and symplectic embeddings in dimension 4. Preprint, 1989.

YAKOV ELIASHBERG AND MIKHAEL GROMOV

162

Symplectic manifolds with contact boundaries, Preprint, 1990; to appear in invent.

(McD3J

Math. IMO)

3. Moser, On the volume elements on a manifolds, Trans. Amer. Math. Soc. 120 (1965). 286—294.

[SI)

J..C. Sikorav, Rigidité symplectique dans le cotangent de T" , Duke Math. 3. 59 (1989),

[Sufl

D. Sullivan, Cycles for the dynamical study offoliazed manifolds and complex manifolds, Invent. Math. 36 (1976), 225—255.

759—763.

A. Weinstein, Lectures on syntplectic manifolds, CBMS Reg. Conf. Ser. Math., Vol. 29, Amer. Math. Soc. Providence, R. 1., 1977. [We2J _, On the hypotheses of Rabinowilz s periodic orbit theorems, .1. Differential Equations 33 (1979), 353—358. (We3J , Contact surgeries and symplectic handlebodies, Preprint, Berkeley, California, jWeiJ

1989. STANFORD UNIvERsUY

INSTITUT DES HAUThS ETUDES SCIENTIFIQUES, FRANCE

Proceediup of Sympouta in Pure Mathematics Volume 52(1991). Pan 2

Simple Partial Connections and

the Einstein Condition H. R. FISCHER AND R. J. FISHER, JR.

In [4) the concept of an Einstein-Hermitian connection is defined for a smooth complex vector bundle E over a complex manifold M. In [12) the notion of a simple partial connection is defined and is examined in terms of the set of all isomorphism classes of simple holomorphic structures on E. The goals of this paper are as follows: to show that every simple partial connection is the (0, 1)-component a smooth Einstein connection; to address the problem of determining when simple partial connections exist; and to generalize the Chern class inequality for holomorphic Einstein-Hermitian bundles to a 0. IntroductIon. In the differential geometry of complex vector bundles, Einstein-Hermitian bundles play a central role in as much as they are the differential geometrw counterpart of stable bundles in algebraic geometry. For a thorough treatment of this topic, see the recent book of Kobayashi [10). Other work on this topic that discusses Einstein-Hermitian connections from the viewpoint of Yang-Mills theory include [3, 8. 9, 12, 16). In [12] Lubke

and Okonek introduced the notion of a simple partial connection, which generalizes the concept of a simple holomorphic vector bundle; the latter plays a fascinating role in some important questions in algebraic geometry. See [13] for an interesting discussion of this concept. Briefly then, let E —' M be a C°° complex vector bundle of rank r over a compact complex manifold. Next, let denote the affine Fréchet space of all partial (0, 1)-connections in E, that is, all C-linear maps D': A°(E) —o A°"(E) satisfying the Leibniz rule; namely,

for all 1€ A°, SE A°(E). A partial (0, 1)-connection D' E

is said to be simple if the induced partial connection in the endomorphism bundle 1991 Mathematics Subject Classification. Primary 53C07; Secondary 32L07, 53C55, 53C56. This paper ii in final form and no version of it will be submitted for publication dsewhere. 1991 An,erican Mathematical Society OO$20717/91

163

H. It. FISCHtR AND

164

R.

J FISHER. JR.

L(E) = E* ® E, also denoted by D', has minimal kernel, i.e.,

ker(D': A°(L(E))

A°''(L(E))) = {cIEfc

C},

where 'F denotes Lhe identity endomorphism (cf. [4, §4J). Continuing, the usual smooth right action of the complex gauge group of E on its space of connections is compatible with the typing of connections determined by the base manifold and so it induces the obvious smooth right action on := In of i.e., the set g gE simple partial (0, 1)-connections is shown to be C°°-open (i.e. open in the iopology) in and invariant under the The main purpose of this paper is threefold: to show that there is a basic interaction between Einstein connections and simple partial connections (see Proposition 1); to begin to address the problem of determining some general criteria for guaranteeiiig that is nonempty; and to generalize the Chern class inequality for holomorphic Einstein-Hermitian bundles due to Lübke [111 to a nonholomorphic setting. See Theorem I in §3 for the complete statement. Finally, in an effort to thread together the various topics, examples are presented that examine this Chern class inequality; namely,

f((r as

— 1 )c1 (E)2 —

2rc2(E)) A

a possible necessary condition for the existence of simple partial connec-

tions.

1. Simple partial connections and the Einstein condition. In §4 of [4] the context of an Einstein connection was enlarged and studied in relation to the isotropies of partial connections. We bnefly recall tile definition given in [4). Let E —' (M, g) be a smooth complex vector bundle over a hennitian manifold. Given any connection D in E, let Rb' denote the (1, 1)-component of the curvature R(D) of D. As befci'e. the g-trace of denoted by I

is an endomorphism of E. We define D to be an Einstein connectionon E—.(A1, g) if =C'JF forsome constant CEC. If,in Trg

addition, D is compatible with some hermitian metric in E, then D is called Einstein-Hermitian. The purpose of this section is to prove the following:

(M, g) be a C>' complex vector bundle over PRoPosITIoN I. Let E a connected coin pact hermilian manifold. For each simple partial (0, 1)connection D' there exists an Einstein connection 13 in E extending 13'. To begin, fix a hermitian metric h in E. Next, given E s., D0 be the unique connection in E, which is compatible with h and has as its (0, 1)-component. See [4) for details on this connection. Also as long as no confusion arises, no notational distinction is made between a connection (resp. partial connection) in E and the corresponding induced connection (resp. partial connection) in L(E) E ® E. In order to produce PRooF.

let

SIMPLE PARTIAL CONNECTIONS AND THE EINSTEIN CONDITION

a connection D such that D' = a E A0' '(L(E) such that if D =

and Trg

=

c

'E'

suffices to find

+ a, then Trg Rb'' = CIE. Continuing,

observe that the curvature of D is given by RD = R0 + D0a + [a, a] where R0 is the curvature of D0. Thus Rb' = and hence, in order + to prove the proposition it is sufficient to understand To this end, observe that at the section level the Laplacian of is an elliptic operator. Explicitly, the symbol of at a point x is exterior multiplication by the (1,0)-cotangent vector v1'0 corresponding to each nonzero real cotangent vector v E Thus v): L(EA.) is injeclive. See [15, Chapter IV, §2] for basic details. The metric h in E in and hence one on L(E), namely, k = Ii ® Ii. In

yields a metric

turn, k and g induce hermitian inner products on the various Let (Dy: A1' °(L(E)) —. A°(L(E)) dejiote the corresponding formal adjoint of A°(L(E)) —. A''°(L(E)). It follows from the above observation on o that at the section level, Laplacian is a second order selfadjoint elliptic operator. Consequently, one has an orthogonal decomposit ion of A°(L(E)); namely, A°(L(E)) =

(1.1)

The next step is to observe that there is a relationship between the g-trace map Trg and This is a general fact stated below in the following lemma. Let F —÷

(M, g) be a

complex vector bundle over a connecied compact hennitian manifold. Let h be a herinizian metric in F and D any connection in F compatible with h If (D): A' '°(F) A°(F) denotes the formczl adjoint of D': A°(F) —+ A1 '°(F) with respect to the hermitian inner product on A• ' (F) constructed h and g, then Trg D' = — (D') LEMMA.

.

PRooF.

The proof of the lemma is a calculation based upon some prop-

erties of the g-trace map. It does not a priori require that (Al, g) be Kàhler. For ease of reading we provide an outline of the details involved in the calculation. Firstly, recall that g determines a Hodge star operator which enables one to define a hermitian inner product on the scalar (p, q)forms 4p,q Using h one extends. the innem' product to let h: by t

F

as follows:

be the C-antilinear bundle isomorphism given on seciions h extends the dual to a map With this, one defines the inner product on by (a, fi) = for complete details of this construction, see [15, p. 175] or Hirzebruch [7, p. 12!]. Note that our corresponds to Wells' Let = and let D be the dual to 1). Note Ihat D = hoDoir". Having said this, observe that the formal adjoint of D' is given by (D')' = To prove this last equality, prOceed by analogy F

=

H. R. FiSCHER AND R. J. FISHER. JR.

166

with the argument presented in [7, p. 121]. Since D is h-compatible, this last equation reduces to (D') = —(* 0 1) o D' o (* 0 1) or simply — * oD'* by the obvious abuse of notation. To prove the proposition, it is enough to demonstrate the desired equality on decomposable elements. First, for any w ® s E A' '°(F),

(D')(w®s) = (J'w)

(1.2)

the adjoint of observations about the g-trace map: where 6' =

—

is

*(*coAD's)

Second, there are the following useful

Tr8(coAO)= Trg(o. A 0) = *(*w A 0),

(1.3) (1.4)

where w

A"°, 0 A°" and

= g(•,

Note that under the usual

identification of T"° with TM, g is viewed as a hermitian metric in

A"° by = T"° and hence defines a C-anti-isomorphism g(., X). Equation (1.3) is readily verified in terms of a local unitary frame a,e, then locally = (e,, ... , ç) for T"o as follows: writing Tr,(UA 0) = EWA 0(e0, ë0) =

=

To prove (1.4) let 1 denote the fundamental form associated to g and then

note that it can be shown that for (1.5)

WE A"°

=

denotes the interior product with respect to Xe,,. Using (1.3) is a skew-derivation, it follows that and (1.5) and also that where

= 0(X0) = Tr,(w A 0).

*(*co A 0) =

Next let (sk) be a finite set of generators for the Then D's = Ek 0k ®Sk where 0k E A°". Applying (1.4), *(*w A D's) =

A°(F).

E(**w A 0k)) ®

= ETt,(WA Ok)sk = Tr,(WAD's). Recalling that

on A"0, J'is given locally in terms of (e,, ... , ç) by J'co =

=

(D')(w 0 s) =

it follows from (1.2) that

0 s - w A D's) = - Tr, D'(w Os). 0

Returning to the proof of the proposition, apply the lemma to the equation = Rh" + D'a; hence, Ti, = Tr, Rh" — Denote Tr, by K and Tr, Rb" by K0. Next, since is simple, it follows that (1.1)


167

'S Im(D0) . Continuing, there exists a by A0 (L(E)) = c A' '°(L(E)) such that K = CJE for some is given

C

EC

.L

C.

'E) =

K0).

Choosing c= (1E' Ko)/(IE, 'E)' we are done. 0 The above proposition appears as Theorem 3.8 in (8] under more special assumptions; namely, that be a simple holomorphic structure in E, i.e., is simple and = 0, and also that (M, g) be Kihier. In principle

our proof is analogous to what is presented on [8, p. 30]. The extra care needed to accommodate the more general hypotheses is necessary and yields the same conclusion as in [8).

In illustration of the above proposition, we point out two examples of simple holomorphic vector bundles which are of independent interest. For emphasis, we state these examples as corollaries.

Let M be a compact Riemann surface of genus g 2 1(M) denoting its Jacobian variety. For each s g, the rank s Picard

COROLLARY 1.1. with

bundle

—, J(M) admits an Einstein connection.

PROOF. For details on see [2] or [5]. In [5] it is shown that the holois simple. See Corollary 2 on p. 78 of [5] for morphic structure on

details. COROLLARY 1.2.

Tango's example of a simple (n

—

1)-bundle over P'

admits an Einstein connection. PROOF.

Tango's example is somewhat involved. For details, see [13, pp.

81—86].

2.Constrectlonsofsimplepartlalconnectlons. Let E-.M bea

com-

plex vector bundle of rank r over a connected compact complex manifold. In this section we make a start at addressing when the space of simple partial connections is nonempty. Specifically, it is shown that S 0 under the

following assumptions:

2.1. E = L

e L (r copies) where L -. M is a

complex

line

bundle.

2.2. There exist N = 2() linearly independent (0, 1)-forms on U, say, that no nontrivial combination is i-exact, fi1' , fin. such i.e., f, = 0 and is constant. f1fi1 =

Our technique

for showing

0 is a constructive one and begins as

II. R. FISCHER AND R. I. FISHER, JR.

168

follows: let 0

D= D1

0

However, since = but D' A°"(L(E))}, under 2.1 and 2.2 it suffices to produce A where To this end, write A = 4°' '(L(E)) so that 0' + A E

Of course

where E {D' + AlA

E

40. '(L(L)). Similarly, if is an endomorphism of E, then A°(L(L)). Next, letling D'+A also denote E = (011) where the partial connection in L(E) induced by D'+A, an elementary calculation a.1

ker(D' + A) if Vi, j,

shows that

+

(2.3)

= 0.

—

Continuing, choose the a.1 E A0" for i j to satisfy 2.2. Then the diagonal equations + — OIkakI) = 0

is constant and 01k = 0ki = 0 for k 1. Next, for i j equation (2.3) reduces to This — = 0. Since 0, 0k,. = establishes the existence of A so that D' + A E As a companion to the above existence argument, we present a possible criterion for guaranteeing assumption 2.2. First, we note that if there exist (0, 1)-forms satisfying 2.2, then in practice there will be an infinitedimensional vector space of such (0, 1)-forms. The following observation indicates the nature of the problem: let ft A°". the equation = ff3 where p, .1 Under what ft does the above equation have only the trivial solution, i.e., constants and f = 0? Observe that if the support of fi, supp /1, is properly conthined in M, then f/I has nontrivial solutions; namely, 6 constant and any nonconstant such that supp f fl supp /1 0. Thus a necessary JE on ft is supp ft = M. Naively one hopes that any ft E 40. I such that supp ft = Al and ft meets the requirements of 2.2. To justify this expectation, the following list of examples is provided: hold itt

and let Z =

I. Let M

coordinates. Define g E

g(7) -

(Z0, Z,, as follows:

JZ)2)

...

,

Z,1)

denote

on U0 = (Z0

on(Z0=0).


169

Clearly, the function g has the following properties: (i) g is positive on U0; note U0 has holomorphic coordinates

=

Z1/Z0; (ii)

g and all its partials tend to zero as Z tends to the boundary of

U0; (iii) g is not holomorphic.

Next define

onU0,

on(Z0=O). 0, and supp $

By construction, fi A0'1 Then suppose that = f/i. Clearly on = 0, 2 —1,

VXEB,.

In particular,

AFFINE GEOMETRY TO GEOMETRIC FUNCTION THEORY

203

This implies

1. Then e(G/H) = 2 and there exist a parabolic subgroup P of G containing If and a nontrivial charP acter C such that H = Ker In particular, this implies that one has the homogeneous fibration G/H rational manifold.

G/P = Q, where Q is a homogeneous

The next result is, in some sense, the analogue of Ahiezer's result in the setting of complex Lie groups. (See [10, Theorem 4].) THEOREM 4. Let G be a connected complex Lie group and H a closed complex subgroup with a finite number of connected components such that

e(G/H)> 1. Then e(G/H) =

2

and Gill fibers either as a

over

a torus bundle over a homogeneous rational manifold or as a torus bundle over a C-bundle over a homogeneous rational manifold.

Without elaborating we note in passing that 0. M. Hosrovjan has made a detailed study of the spaces G/lI which have two ends in the situation where G is a real semisimple Lie group and H is an arbitrary closed subgroup. He

determines the possibilities for the pairs of Lie algebras (g, h) which can occur when the center of G is finite or when Gill = M x R with M being a sphere or having positive Euler characteristic. We refer to his work (12] and [13]. 4. GiN satisfies some analytic condition. We begin with the situation where the space G/H with more than one end has nonconstant bolomorphic functions. The proof for G a complex Lie group appeared in [8] and for G real in [14].

Suppose G/H is a homogeneous complex manifold which satisfies C and e(G/H)> 1. Then e(G/H) 2 and the holomorph ic reduction of GIN has compact connected fibers and its base is biTHEOREM 5.

holonwrphic to an affine homogeneous cone with its vertex removed.

We will not discuss this result in any detail. Incidentally one should note that if a homogeneous complex manifold has any nonconstant bounded holomorphic functions, then it has one end. This follows from the result in [9].

Next we present the characterization in the Kähler case which is given in THEOREM 6.

Suppose e(G/H)> 1. Then G/H is Kdhler if and only if

one of the following holds:

(1) G/H

Q

x T, where the bundle restricted to {q) x T is a Cousin

group with two ends for every q E Q;

(ii) a finite covering of G/H is a product (F Q) x T. Here Q denotes a homogeneous rational manifold and T denotes a compact complex torus.

ON THE ENDS OF COMPLEX MANIFOLDS

221

OUTLINE OF PROOF. Given a C-bundle over a Kähler manifold, applying a classical result of A. Blanchard [5] yields the fact that the total

space is also Kähler. The proof follows the three steps outlined above. In order to give an

example of the kinds of methods that are used .we look at the case where &(G/H) Let G/H —+ G/J denote the normalizer fibration, if G is complex, and the g-anticanonical fibration, if G is real. If G — J, one may assume that G is complex and H is discrete. The case G solvable is a consequence of the results of K. Oeljeldaus and W. Richthofer in [16]. The case G semisimple is eliminated using the work of F. Berteloot and K. Oeljeklaus [3]. (G/H Kähler H is algebraic, thus finite. So G/H is Stein and hence e(G/H) = 1. This is the desired contradiction.) The mixed cas; i.e., when G is either a direct product S x R or a nontrivial seniidirect product of a maximal semisimple subgroup S with the radical .R, gives rise to several subeases which are also shown to be impossible. For example, if G is a direct product, one needs the argument of K. Oeljeklaus and W. Richthofer concerning

the Ahiezer conjecture [2], but suitably modified in the Kähler selling by means of the results of [3]. If G $ J, the fibration lemma implies G/J Q, a homogeneous rational manifold. It follows that C and thus f/H = A is a Cousin group with two ends. Hence A has the form A C/d/r.2k...l 1(4 x R, where K4 is the maximal compact subgroup of A. Let be a maximal compact semisimple subgroup of G. Then Q where L is a subgroup of K4 x K5 acts on G/H with compact Ks. Now the compact group K real bypersurface orbits. in his thesis, W. Richthofer [18) analyzed such hypersurface in a Kähler lube (here the whole manifold G/H) by considering the representation p: L —' Aut(A), which defines the A-bundle over Q, and showed that the hypersurface orbit either splits as a product S' x M, where M is some complex manifold, or else there is a '4preferred S' -direction" so that one can "push down" the Kähler form. Applied to G/H this yields a C-bundle over a compact homogeneous KAhier manifold, i.e., over Q x T. This means that G/H is of the form (1) in the statement of the theorem. For the details in the case C one should see the paper (11J. The basic idea is to show that under the Kähler assumption in the setting of two ends one gets a torus bundle over an affine homogeneous cone with its vertex removed. The technical part is to show that a finite covering of this bundle splits as torus cross the cone as stated in part (ii) of the theorem.

(1) This result is false if one does not have a Lie group acting transitively. For, it follows from an idea of W. Kaup (151 that X := C2\{(z1, 0), (z2, 0)), with is homogeneous under Aut(X) and has three ends.

222

BRUCE GILLIGAN

(2) If G/H is locally separable by analytic hypersurfaces (or meromorphically separable—these two are equivalent in the setting of two ends!), then G/H is also Kãhler and the classification applies; see [11]. (3) The classification shows that if X G/H with G real, e(X) = 2 and X Kähler, then the associated P1-bundle X is an equivariant compactification of X. Therefore, X is also homogeneous under the holomorphic action of a complex Lie group, namely the stabilizer in Aut°(X) of the 0of this P, -bundle. We suspect that real homogeneous imand plies complex homogeneous (in the above sense) whenever X has more than one end. For homogeneous spaces with one end this is false, e.g. any homogeneous bounded domain.

In the Fibration Lemma it is essential that the fiber 5. Finite of the bundle be connected. One can ask what happens if that is not so, for example in the case of coverings. Only under the very restrictive hypothesis that the isotropy subgroup has a finite number of connected components are we able to say something, and in closing we would like to point out the following'. Basically the reason why one is able to handle this particular case is because of the classification which exists in this setting. PRoPosITIoN 1.

Suppose G is a connected complex Lie group and H is

a closed complex subgroup having a finite number of connected components. Suppose H is a subgroup of G consisting of some of the connected components

of H. Then G/H and G/H have the same number of ends. i.e. the finite covering G//l G/H preserves the number of ends. PRoOF. We only look at the case where e(G/H) = 2, because the other cases are obvious once this has been proved. Since "going up" by finite covers in the present setting preserves two ends (e.g., sec [1, Proposition 2, p. 296)), it follows that e(G/H°) = 2, where H° denotes the connected component of the identity of H. We have to show that this implies e(G/H) = 2.

Let N NG(H°) be the normalizer of H° in G and let N° denote the connected component of the identity of N. Since N°/H° is connected, it follows from the Fibration Lemma that there are two cases which have to be considered. First suppose G/N° is compact and e(N°/H°) = 2. Then G/N is also

compact and hence is a homogeneous rational manifold. Thus N is connected. Clearly N/H° is an abelian complex Lie group with two ends (by Theorem 1) and therefore N/H is too. Because it1 (G/N) = 0, it follows that e(G/H) = 2 by the last assertion in the Fibration Lemma! The other case occurs when N°/H° is compact and e(G/N°) = 2. Since

GIN is Kähler, so is G/N°. Thus by Theorem 6 it follows that GIN° is a C-bundle over a homogeneous rational manifold Q. If GIN° is a 1We thank M. A. Mishchenko for bringing to our attention the fact that the argument presented in [10, Proposition his not very convincing.

ON ThE ENDS OF COMPLEX MANiFOLDS

223

product C x Q. then the map GIN° —. GIN is clearly given by a finite subgroup of C, where N denotes the subgroup of N consisting of those connected components which meet F!. If the bundle is not trivial, then a maximal semisimple subgroup S of G acts transitwely on GIN° and one has GIN° — S/L. From [1] there exists a parabolic subgroup P of S one has GIN S/L. containing L such that P/L = C. Letting L := see [I, p. 296]. Thus P ' L, since L normalizes But, in fact, P and hence is L. But then L/L is a finite subgroup of the group ?/L = cyclic. This means that P/L C' and one has the diagram: G/If° G/F! I

GIN°

GIN

GIN

s/i 2, since Now ir1(S/P) = 0 implies e(G/N) = 2. Therefore e(G/H) N/H is compact and connected and this completes the proof. 0 REMARK. We now give three examples which point out how very delicate the behaviour of ends is with respect to finite coverings. This is also indicative of how powerful the normalizer libration can be in the homogeneous setting,

particularly along with some other assumption; in this situation that H has only finitely many connected components. (1) The space S' x [0, 11 which has two ends is a two-to-one covering of the one-ended Moebius band. This is a homogeneous fibration with real groups (e.g., see [12, p. 921) but there is no complex structure. (2) Let T be a complex torus and pick t0 T with t0 0 but = 0.

Define a to be the biholomorphic map of order two of C' x T given by cr(z, z) := (liz, Then I := C x T is a finite covering of X C' x T/ p-., where x a(x). It is clear that e(I) = 2, while e(X) = 1! Here one has a complex structure, but no homogeneous fibration,2 (3) The covering map SL(2, SL(2, C)/Fk is of degree I/k if k divides 1. See [10, §5J and the references given there. Here there is both a complex structure and a homogeneous but the isotropy subgroups r', and rk both have an infinite number of connected components. REFERENCES J. D. N. Ahiezer, Dense orbits with Iwo ends, lzv. Akad. Nauk SSSR Ser. Mat. 41(1977), 308—324; English transi. in Math USSR Izv. 11(1977), 293—307. 2. —, Invariant analytic hypersurfaces in complex nilpotenz Lie groups, Ann. Global Anal. Geom. 2 (1984), 129—140.

2We thank D. N. Ahiezer for pointing out this example.

BRUCE GILLIGAN

224

3. F. Berteloot and K. Oeljeklaus, Invariant plurisubharmonic functions and hypersurfaces on semi-simple complex Lie groups, Math. Ann. 281 (1988), 513—530. 4. L. Bianchi, Sui gruppi di sostiluzioni lineari con coefficienri appartenenhi a corpi quadratici imaginari, Math. Ann. 40 (1892), 332—412 (also 38 (1891), 313—333). S. A. Blanchard, Sur les analytiques complexes, Ann. Sci. Ecole Norm. Sup, 73 (1956), 157—202. 6. A. Borel, Les bouts des espaces homognes de groupes de Lie, Ann. of Math. 58 (1953), 443—457. 7. H. Freudenthal, 692—713.

Iieber die Enden :opologischer Räume und Gruppen, Math. Z. 33 (1931),

8. B. Gilligan, Ends of complex homogeneous manifolds having nonconstant holomorphic functions, Arch. Math. 37 (1981), 544—555.

9. _,

On

bounded holo,norphic reductions of homogeneous spaces, C. R. Math. Rep.

Acad. Sci. Canada VI (1984), 175—178. 10. B. Gilligan and A. Huckleberry, Complex homogeneous manifolds with Iwo ends, Michgan J. Math. 28 (1981), 183—196.

II. B. Gilligan, K. Oeljeklaus, and W. Ricbthofer, Homogeneous complex manifolds with more than one end, Canad. J. Math. 41(1989), 163—177. 12. 0. M. Hosrovjan, On homogeneous spaces with two ends, Ann. Global Anal. Geom. 2 (1984), 91—111. Homogeneous spaces with two ends and positive Euler characteristic, Ann. Global

13. —,

Anal. Geom. 6 (1988), 259—264.

14. A. T. Huckleberry and B. Oeljeklaus, Classification theorems for almost homogeneous spaces, Elie Canan, vol. 9. de Nancy, Nancy, 1984.

15. W. Kaup, Reelle Transformationsgruppen und invariante Meiriken auf komplexen Rdumen. Invent. Math. 3 (1967), 43—70.

16. K. Oeljeklaus and W. Richthofer, On the structure of complex solvmanifolds, I. Differential Geom. 27 (1988), 399—421. , Recent results on homogeneous complex manifolds, Complex Analysis. III, Lecture Notes in Math., vol. 1277, Springer-Verlag, New York, pp. 78—119. 18. W. Richthofer, Homogene CR-Mannigfaltigkeiien, Dissertation, Bochum, 1985.

17.

UNIVERSITY Of REOINA, CANADA

Proceedings of Symposia in Pure Mathematics (1991), Part 2 Volume

Detrminant Line Bundles for Hermitian Foliations and a Generalized Quiflen Metric JAMES F. GLAZEBROOK AND FRANZ W. KAMBER Abstract. A transversally symmetric pseudodiff'erential operator is defined on the basic sections of a Hermitian foliated bundle over a compact Herinitian manifold with compatible foliation. With respect to a paramcter space of basic connections, a determinant line bundle is defined and we outline the construction of its metric compatible with its unitary connection. We show that this line bundle is essentially holomorphic with respect to this connection.

1. Let M be a compact Hermitian manifold and (M, a hermitian foliation on M of complex codimension k with bundle-like metric (see, e.g., [9]). We recall the exact sequence (1)

Let E be a Hermitian foliated bundle on M with unitary structure that is taken to be a (complex) Clifford module over the transversal Clifford algebra of .? with multiplication denoted by ci: C°°(Q ® E) —' C°°(E). Taking ft to denote the projection

*:Coo(T*MØE)

-.

C°°(Q®E),

we define the transversal Dirac operator by (2)

where yE denotes the metric connection on E. In [3] it was shown that is a symmetric transversally elliptic differential operator where PC E is the mean curvature form of (M, gM) (cf. [11)). We note

4r

that the normal bundle Q to the foliation in (1) is naturally endowed with a SpinC(k) structure. Indeed, we may just as well consider E = S ® V where 1980 M'athemat (ci Subject Clggsification (1985 Revision). Primary 58030; Secondary 53C12. Key wordr and Hermitian foliation, Dirac operator, determinant line bundle, Quilen metric.

Work supported, in part, by a grant from the National Science Foundation. Aaedosn MIUIemaIiCII Society 0082.07t7f91 5100 + 5.23 per psgs

225

J. F. OLAZEBROOK AND F. W. KAMBER

226

is the associated Spinc(k) bundle and V is a foliated Hermitian vector bundle (serving as a "coefficient" bundle if one wishes). We consider the subspace of basic or holonomy invariant sections of E S

rb(E) = {s

(3) Since

= 0, X E

E

the real codimension is even, we have the usual decomposition E = E respecting the half-spin representations. In [31, the basic Dirac

operator (4)

"5b

r6(E)

rb(E)

was defined to be 4i.Ir(E). The operator 1J6 and its powers are essentially selfadjoint on and each possess a discrete spectrum. For k < dim M, the operator is not elliptic but does in fact have the following ellipticlike regularity [3, 8): let JibU = v with v rb(E), u E (Sobolev space completion), then u E As for the index, we have (5)

=

=

—

denotes supertrace. It is a straightforward matter to deduce that these operators are unbounded Fredhohn operators which may be converted to bounded ones by appropriately altering norms in the Sobolev spaces. where

2. In (81 we considered the space of basic connections 4 on E under the free action of the "gauge" group preserving the foliated structure

andtakethistobeaparameterspace.

onE. We shall let

where 4i we have the covariance [8] (4)0.4 = denotes the lift of 4, to E and A E 4. Letting Z = M x Y we take the

For 4,

projections

zLy

(6)

M We note that Z and Y are effectively infinite-dimensional complex Hubert manifolds, but this poses no essential problem in our case. The bundle

£ = JrE is now a Hermitian foliated bundle on Z. For each choice of A E 4, we extend the above basic Dirac family to a family of operators C°°(E')

over Z. We are interested mmnly in the orbit classes under i.e. elements of Y. For convenience we shall simply by ii and suppress A in what follows (when this is understood). denote Over Y we have the infinite-dimensional continuous Hubert bundles {(lJb)A}:

=

4

LINE BUNDLES FOR HERMITIAN FOUATIONS

equipped with their L2-inetrics. The working principle is that in practiee, one considers finite-dimensional smooth subbundles and so these continuous infinite-dimensional objects are just handled in the same way, formally speaking. The connections defined on these and their tensor products, etc., will be taken to be unitary. These facts are verified in [7] for the absolute case and we shall proceed along similar lines. We remark that although the setting is almost geometrically equivalent, the analytic nature of our basic Dirac operators and associated C-functions is somewhat different family, we defined in [8] Employing the elliptic-like regularity of the the following element in the K-theory of Y: The element

PRoPosITIoN 1 [8, Proposition 5.1]. [ker..0J —

is a well-defined element

[Y x C']

of K(Y) , for some r.

The difference in (8) accounts for the possible "jumping" phenomenon.

We refer to (8) as the basic analytic families index of the 60 family. In our case Y is connected and by evaluation at [A] E Y, we obtain COROLLARY 2 (8, Corollary 5.2].

The index of the operators #ZJA

pendent of A E

Index(Ji)A Our "foliated" Dirac operators are of interest in so far that a heat expansion does exist and "basic" index theorems are expected (cf. (3]). Indeed, the resulting spectral analysis is that of the leaf space which in general is singular (for a general discussion of these topics, see the survey article [4]).

3. For a

spec(ff25), a >0, welet less than a (here,

denotethesumofeigenspaces is finite-dimensional and locally of of constant rank). We have the exact sequence 0—'

For fi > a, we set

—.

,y(a) P

=

= {(A1 E Y:a of Y, and we define a line bundle by

=

where for y

y(a) —, kerAb'

0.

yields Restricting.V to The open sets C Y,defincd determine an open covering on by

0

Y, we extract a canonical isomorphism

e The possible "jumping" of the kernels does not permit us immediately to define an honest line bundle via (12), although (11) does. To overcome this

J. F. GLAZEBROOK AND F. W. KAMBER

228

hitch we introduce some (smooth) patching isomorphisms. To proceed we define

= and on the overlap

®

fl

we have ,

we have the induced isothe above isomorphism given by b viewed as a nonzero section det morphism detM ' n we have the induced (smooth) isomorphisms On of

Using

given by

This achieves the appropriate patching for the smooth line bundle 2 —i Y. When lndexçfJ) = 0, = and in this case a canonical —. is produced. The multiplicasection det of correspond tive property of determinants shows that and by virtue of(15). We have then PRoPosITIoN 3.

There exists a globai

section det.1J of 2.

The first Chern class c1(2) = c,(Ind(.b)) is the topological obstruction to the existence of a gauge invariant determinant and, via transgression, defines a class in Z) [8, Theorem 6.1]. The corresponding local obstruction R), reflecting the question of the deflnability E H' of the logarithm of the determinant, relative to the regularity of the basic C-function in (17) below, was discussed in [8, §8].

4.Wenowproceedtoconstructametricon2. Fixing o>O,the subbundles inherit their metrics from and in turn inherits a metric If ft >a, then by (15), the ratio of the two metrics on Let e1, ... , consisting of is given by II be a basis of eigenfunctions with a dual basis = ... , Then we have

=

e,) =

j=I Now on the overlaps

, we have

=

fi

A.

A). To account

for this difference we define = dct(.V'..b)IA>0 = IlA>c,' and bring the C-function technique into play, but from the transversal point of view. In 131, the counting function Nb(t) = #{2

t:A E

LINE BUNDLES FOR HERMITIAN FOLIATTONS

22c'

was introduced. It was shown that there exists k', 0 < 0, and that Re w1 = r is the equation of M. It follows that Re to1 for all to The orthogonal projection of C" onto the subspace spanned by to0 is given by (w1, ... , to,,) (w1, 0, ... , 0). Hence all points in the image of Q under this projection satisfy Re to1 r. It follows from the assumption of the lemma that r to0

4. We shall now indicate how the results of [G2) may be sharpened directly. We begin with an improvement of the Lemma of Thorp and Whitley, which follows from a result of Hams EH1I. Harris begins with the following lemma, which is an easy consequence of the Schwarz lemma.

LEMMA 2 (Borel-Caratheodory lemma). If f: D C is a holomorphic function such that f(0) = 0 and Ref(z) for z E D then )f(z)I 1

21z1/(1

Using this result and the Hahn-Banach theorem, Harris deduces

LEMMA 3 [HI, Proposition 1, p. 36). Let j: D

—.

map. Then

1(0) + whenever z E D and

Izi

—

1—

f(0) E f(D)

C" be a holomorphic

IAN GRAHAM

Harris actually formulates this result for maps from D into locally convex topological vector spaces. For completeness I shall indicate the proof. PROOF OF LEMMA 3. Without loss of generality we may assume f(O) = 0. C such that Suppose now on the contrary that there exists E D and A I — but 1(D). The Hahn-Banach theorem gives a 21A01 1z01 continuous linear functional such that Re4(f(z)) 1 for

zED. But by Lemma 2, :

<j•

= 1A01

This is a contradiction.

Lemma 3, which was used by Harris as a replacement for the Lemma of Thorp and Whitley in the proof of the strong maximum principle, can actually be used to improve the constant in the Lemma of Thorp and Whitley:

Let f: D

holomorphic map. Then the closed convex hull of f(D) contains the disc f(0) + zf'(O) < LEMMA 4.

PRooF.

be a

To see that f(0)+af'(O)€f(D), where al
0 and a neigh-

borhood V of

in AN,suchthat

F(e'°, f) > sup F(, f0(e'°)) +

(V.1

—

E V).

In classical sup norm optimization in R' , the concept of strong uniqueness plays a fundamental role. It is known that strong uniqueness is a sufficient condition for second-order convergence rate in the Gauss-Newton method for sup norm optimization [Wa). The same result can be proved in the context of JJO0 optimization for quasicircular iteration algorithms. However, one can easily prove THEOREM IV. I.

Given any F, there are no strongly unique solutions to

OPT.

Local strong uniqueness does not tell the whole story in the (computational) H°° optimization. In fact, in this setting the failure of strong uniqueness has a very different character when N = than when N> 1. This results in different practical behavior of the algorithms. More appropriate is 1

the concept of directional strong uniqueness. DEFINITION. Given F and and p > 0, the function is an order p-strong directionally unique solution to OPT if for any lj in AN there exists

c(h) > 0 such that sup F(e'°, f0(e'°) + th(e'°)) > sup F(e'°, f0(e'°)) ÷ c(h)z",

for all tin (0, 1). Thus local strong uniqueness is equivalent to have p = I and c(h) independent of h. The following result makes clear one way in which the cases N = and N> I are different. 1

Given any F of class C', and f in AN a local

LEMMA IV.2 [HMer2J.

strict solution to OPT.

(I) if N = OPT.

1,

then f

is

an order

1

strong directionally unique solution to

OPTIMAL ANALYTIC DISKS

(2)

If N> 1, then f

is not a strong directionally unique solution to OPT.

We believe that directional strong uniqueness has a definite impact in quasicircular iteration algorithms, and possibly on many other applications. Extensive computer experiments show that for algorithms in this class, ORDER 2 CONVERGENCE RATE IS ACHIEVED WHEN N = I , AND N> I YIELDS ORDER OF CONVERGENCE

In addition, theoretical estimates [HMer2] strongly suggest that DSU is important. For example, consider the relation (2) sup F(e

/0

sup

,

f(e iO

))

f *(eiO))t(f(e:O) — I

{r(e'°, f(e'°)) +2 Re

+

Quasicircular iteration algorithms tend to quickly satisfy condition I, but are when close to a solution, produce update functions I for which nearly in the set N(a). In relation (2) above, this means that sup

r(e'°, f(e'°))

+ 0(111 —

0

Thus under these conditions, an error hf — fall

=

in the solution would be reflected as an error of order in the optimal value, and decreasing the optimal value corresponds to a substantially smaller decrease in the true error in the optimal function. Moreover, slow convergence is intimately connected with condition (II) of Theorem 111.1 being met slowly. Indeed, when N 2 one sees (flAG I) go to zero rapidly and (flAG II) goe to zero slowly. Thus we are currently experimenting with algorithms which enforce condition (II) in some special way. We suggest that second-order convergent algorithms must have this property. As far as our computational efforts are concerned we consider the main

open question to be that of adapting quasicircular algorithms to improve their behavior when DSU fails. Also determining the effect of DSU on other algorithms is intriguing. BIBLIOGRAPHY

(AWl) H. Alexander and J. Wermer, Polynomial hulls wills convex fibers, Math. Ann. 271 (1985), 99— 109.

[BGR] J. Ball, I. Gohberg, and L. Rodman, interpolation of rational matrix functions, Birkhauser [DymJ

Verlag, Boston, 1990. H. Dym, J contract/ye matrix functions, reproducing kernel Hilbert spaces and interpolation, Regional Conf. Series in Math., No. 71, Amer. Math. Soc., Providence, R.I.,

(F]

F. Forstncric, Polynomial hulls of sets fibered over the circle, Indiana

1989. 869—889.

3.

Math. 37 (1988),

J. W. HELTON AND ORLANDO MERINO

262

[FF1

C. Foias, A. Frazho, The commutan: lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 0T44, 1990.

B. A. Francis, A course in H°° control theory, Lecture Notes in Control and Inform. Sci., vol. 88, Springer-Verlag, Berlin and New York, 1986. [GMW] P. Gill, W. Murray, and M. Wright, Practical optimization, Academic Press, New York, f Fr)

1981.

[Hi)

.1. W. Helton, Optimal frequency domain design vs. an area of severai complex variables, Proc. Math. Theory Networks and Systems, 1989.

[112)

Worst case analysis in the frequency domain: an H°° approach to control, IEEE Trans. Auto. Control AC.30 (1985), 1154—1170. Operator theory, analytic functions, matrices, and electrical engineering, Regional Conf. Ser. in Math., No. 68, Amer. Math. Soc., Providence, RI., 1987. , Non.Euclidean functional analysis and electronics, Bull. Amer. Math. Soc. 7(1982),

[113)

[H4J

—, 1

[H5) [116)

1117]

—, Broadband gain equalization directly from data, iEEE Trans. Circuits and Systems CAS-28 (1981), 1125—1137. —, Optimization over spaces of analytic functions and the Corona problem, J. Operator Theory 15 (1986), 359—375. —, IEEE Conf. on Decision and Control, Los Vegas, 1983.

[HMeriJ .1. W. Helton and 0. Memo, Conditions/or optimality over H°°, preprint. IHMer2I —, A novel approach to accelerating Newton's method for sup-norm optimization arising in preprint. [Her) R. Herman, (to appear). [MHI

D. Marshall and J. W. Helton, Frequency domain design and analytic selections, Indiana Univ. Math. J. 39 (1990), 157—184. [Mer) 0. Merino, Optimization over spaces of analytic functions, UCSD Thesis, 1988. [L] L. Lempert, Complex geometry in convex domains, Proc. Internat. Congr. Math., (Berkeley), 1986, Amer. Math. Soc., Providence, R.I., 1987. [Sli) Z. Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), 25—260. [S12) Polynomial hulls in C2 and quasicircies, Preprint. [S13a] —, An analytic set-valued selection and its application to the Corona theorem, to polynomial hulls, and joint spectra, Trans. Amer. Math. Soc. 294 (1986), 367—377. [SUb] Analytic set-valued functions and spectra, Math. Ann. 256 (1981), 363—386. [S14) Polynomial hulls with convex fibers and complex geodesics, preprint. [Wa] G. A. Watson, Approximation theory and numerical methods, J. Wiley, New York, 1980. [Wi) J. Wermer, Banach algebras and several complex variables, Grad. Texts No. 35 (2nd ed.), Springer-Verlag, New York, 1976. [W2J Polynomially convex hulls and analyricity, Ark. Mat. 20 (1982), 129—135. IY) D. C. Youla, 4 new theory of broadband impedance matching, IEEE CAS, 1964. D. C. Youla and M. Saito, Interpolation with positive rca/functions, I. Inst. 204 [YS)

—,

—, —,

(1967), 77—108. [Yng]

N. Young, An introduction to Hi/bert space, Cambridge Univ. Press, New York4 1988.

UNIVERSITY or CALIFORNIA,

Dizoo

P,uceeduap of Symposia an Pant Mathematics

Volume Si (1991), Pitt 2

Some

Remarks on the Product Property for Invariant Pseudometrics


Introduction. Let c6 (resp. kG,

1.

Ko) denote the Carathéodory

pseudodistance (resp. Kobayashi pseudodistance,

pseu-

It is

dometric, Kobayashi-Royden pseudometric) for a domain G c known that they have the following product properties (cf. [6, 9, 11]):

z),

(*)

(**)

ÔGXG((Z1

dG(z, z')}, Z,ZEG2CCfl2, whered=cord=k; z'))=max{dG (z, X2)) =

,

X1), JG(z2, X2)},

whereô=yorJ=,c.

EG1

Conditions (*), (s*) reflect a general principle in complex analysis: an cxtremal function for G1 x G2 is the maximum of corresponding extremal functions for G1 and G2 (cf. 113, 141). The aim of this paper is to discuss some aspects of (*) (resp. (**)) for some intermediate invariant families of functions (resp. invariant families of pseudometrics), that is families of functions dG : G x G —p (resp. such

that (1)

VF E 9(G1,

(resp. (1')VF

G2): d0(F(z'), F(z"))

d6(z', z"),

z', z" E

tl(G1,G2) : ÔG(F(z),F'(z)X)

z

G, G1 C

X); (2) dE

(resp.

p =: the Poincaré (hyperbolic) distance on E := {2 E C:

< 1}

(2') 6EYE)•

— 1980 Mathematics Subject CIaujf,cation (1985 Revision). Primary 32R1 5. This paper is in final form and no version of it will be submitted for publication elsewhere. © 1991 Ametican Mathematical Society OO$2-0717/91

263

51.00+5.25 peipqc


264

It is useful (cf. f6J) to consider families of functions dG: G x G —. [0, which satisfy (1) and which are normalized by the condition (2*)

1)

:= tanh p := the Möbius distance on E.

dE =

It is clear that d = (dG)G satisfies (1), (2) if d =: (tanhdG)G satisfies (1), (2*). Moreover, d has the product property if d' does. In particular, the family of Möbius pseudodistances := (tanhcG)G has the product = property. It is known that for any invariant family of functions (dG)G (resp. invariant family of pseudometrics (ÔG)G) one has

(resp.

CG

dG

CG

dG

5G

kG,

if d6 is a pseudodistance,

KG), where

kG(z', z") := inf{p(0, t) : t E [0, 1),

39 EeI(E, G) :9(0) = Z', Note that k has the product property [11). Recall that CG = G is biholomorphic to a convex domain [10) and therefore the product properties for domains equivalent to convex domains are trivial. One can prove that if ô = (ÔG) is an invariant family of upper semicontinuous pseudometrics then the family f ô := (f ÔG)G, where

f ô((z, z") := inf{1JG(a(1). a'(t))dt: a: [0, 1]—. G, a(0) = a( 1) = z", a is piecewise C' }

,

z', z" E G,

is an invariant family of pseudodistances. Moreover, if 5 has the product property (**) then d = Jo satisfies (*). In particular, the family of inner Carathëodory pseudodisiances c' = :=

(f YG)G has the product property.

In this paper we will present examples of invariant families of functions and of pseudotnetrics with or without the product property. The paper was started when the second author visited the Jagiellonian University (Krakow) (August—September 1989) and it was finished during the stay of the first author at the University of Osnabruck (Vechta) (October— November 1989). We would like to express our gratitude to both universities for their hospitality and to the Deutscher Akademischer Austauschdienst for financial support. The second author thanks the American Mathematical Society for presenting him the possibility to attend the 1989 Summer Research institute at Santa Cruz.

265

INVARIANT PSEUDOMETRICS

2. Main results. Let us start with a general method of producing invariant families of functions and pseudometrics.

For a domain G c C' and a 6 we set := {u: G [0, 1): ulog-psh and u(z)

—

at

near a}

(cf. [7]).

Suppose that for any domain G and for any a

G we have a family

such that

(a) forany F

G2)

(b)

=

EXAMPLES. (1)

(PEN), with UEC2 neara}.

(3)

Define now

Observe that

d(a, z) := sup{u(z) =

is

u

an invariant family of functions in the sense of

(1), (2*).

The case 2 =

X.

Put g0 :=

The function log the pluricomplex Green function with a pole at a (cf. [3, 5, 7]).

)

is

called

The following is known (cf. [5, 7]).

PaoPosmoN 2.1.

a€G;

(l) (3)

If log

)

E

\ {a})

(for

example:

G is bounded), then

[ddClogg0(a,

(4) If G is bounded and hyperconvex, then

is continuous on G x

G

ith

urn g6(a, z) =

1,

whenever a E G, z0 8 G; (5) If G is a domain of holo,norphy, then

is upper-semicontinuous on

Gx G.

We will see in §4 that, in general, holomorphy. We will prove the following theorem. THEOREM 2.2.

is not continuous for domains of

Whenever G1, G2 are domains of holomorphy then b),

(z, to))

=

(a,z€G1,h,wEG2).

MAREK IARNICKI AND PETER PFLUG

266

This means has the product property in the class of domains of has the product propeny holomorphy. It seems to be unknown whether in general. REMARK. Recently, Bedford-Demaily [3] proved that there are very regular domains of holomorphy such that the function is not symmetric. From our examples (cf. §4) we get even a Reinhardt domain for which g0 is not symmetric. =

The case fore we call

dr'.

=

Observe that := the pM Mobi us function for G. Put

For any p 1 we have: n C(G) and, for any a, ext remal function for (a, 20); (2) if PRoPOsITION 2.3. (1) )

There-

E G, there exists an

is bounded.

G

The proof will be given in §3.

Note that, in general, for p 2 (resp. p 3) is not symmetric and continuous (resp. does not have, product property) (cf. Following the ideas of Azukawa (cf. [1, 2]) we are going to introduce an invariant family of pseudometrics associated to a family as follows:

1-—u(a+AX)

0G (a, X) := sup lim

:uE

1

when

Note that according to Proposition 2.1, (1) is an

LEMMA 2.4.

: G x C —.

invariant family ofpseudometrics (cf (1'), (2')).

PROOF. Apply Lemma 2.1 of (2]. The case = X. Put := (see (8]). It is known (cf. (1, 2, 8]) that:

AG

is called Azukawa pseudometric

(1) AG(a, X) = (2) If C is a domain of holomorphy, then AG: C x C —. PRoPosITIoN 2.5.

is upper-

semiconhinuous.

Our examples will show that, in general, AG is not continuous for domains of holomorphy. In view of Proposition 2.5, (1) and Theorem 2.2 we obtain: THEOREM 2.6.

For G1 c C' and G2 c

following formula hoJds:

domains of holomorphy, the

INVARIANT PSEUDOMETRICS

267

AGXG((zt , z2), (X1 , 12))

= max(AG

(z1

,

X1), AG(z2, 12))

E

E

The case = Put Observe that = the pth Reiffen pseudometric. We will prove in §4:

=

C"). We call

PROPOSITION 2.7.

(1) (p) 76(a,X)=llm 2-40 .

p 111

I/p

1

=sup and, for any (a, 1) E G x C", there exists an extremal function; (2) If G

is

bounded.

In general, for p 2 (resp. p 3)

is not continuous (resp. has no

product property).

The case 2' — £fS.

Put

:=

was introduced by N. Sibony

[12].

The following is known [8, 12]: PROPOSITION 2.8.

the supremum is taken

near a.

(1) SG(a, 1) = overall

u: G

(a)X1Y1)"2} where

[0, 1), log-psh, u(a) = 0 and u E C2

)]2

is C2 near a, then AG(a, )=SG(a, ). (3)11 GcC',then AG=SG. (2) If [g0(a,

It seems to be whether S0 is upper-semicontinuous on G x C" even for domains of holomorphy G C C". We will show that, in general, SG is not continuous. Product property for and SG remains open.

3. Proofs. Ideas of the proof of Theorem 2.2 are taken from [14]. PRooF THEOREM 2.2. According to Proposition 2.1, (2) it suffices to consider G1 c C"' , G2 c C"2 to be bounded hyperconvex domains. Take cf. the addendum

MAREK JARNICK1 AND PETER PFLUG

268

a

G1

and b

and set

G2

b), (z, w)),

1(z, w)

r(z, w)

z), log

w)).

Jtisclear,that Ir. According to Proposition 2.1, (4) if we define / and r to be zero on the x G2. Moreover boundary of G1 x G2 then I and r are continuous on = 0 on (G1 \ {a}) x x G2 \ {(a, b)} and = 0 on (cf. Proposition 2.1, (4) and Lemma 3.1, p. 52 in (14)). (G2 \ {b}) =: are bounded, for any e > 0, there exists a ball B8 c G1 x G2 Since with center (a, b) such that (1 + c)I r in Now we apply the domination principle of Bedford-Taylor (4) and obtain min[r —

(1

+ e)/J = min[r

—

(1

+ e)I)

0,

where

Therefore, on

we

have r (1 + e)I. For e —.0 and because of the

continuity of I and r the remaining inequality 1 r holds on G1 x G2. PRooF OF PROPOSITION 2.3. By Montel arguments one easily gets (1), (2)

and the upper-semicontinuity in (3). So it remains to verify the continuity if G is bounded. E), bean extremal funcFix z0, W0E 6 and let fE tion for w°). For any z E G set

a

max(1,

c(z) Note that c(z) —, Since

( we

1

_2, and denote by r(a) := are relatively prime; let b(z) ... C"). In this case the invariant functions orda((b—D(a)) (a = (a1, ... , and pseudometrics introduced in §2 can be effectively calculated, namely where

I/r (B)

r=r(a), a,z€G; r = r(a),

z) =

G,

It] ÷ 1 otherwise;

E N and

where

a, z E

r = r(a), a

(C)AG(a, X) =

G, XE C",

X) :=

where

iPkr (D)

G

X) =

I (YE(ø(G), 4),(a, X))f'1' if r(p, 0, otherwise,

r=r(a), aEG,XEC"; X))J I/p

S(a, X) =

if at most one of the a,'s is zero and S(a, X) = 0 otherwise, r =r(a), a E G, X E C". REMARKS. (a)

is not continuous in the first argument and not sym-

0) whenever b =

metric, e.g. gG(0' b)

(b1,

... , b,,) E G with

Therefore, by Proposition 2.1, (2) one can easily construct a bounded Rejnhardt domain of bolomorphy D with arbitrarily regular boundary such that is not symmetric (cf. [3]). (b) For p 2, is not continuous and not symmetric. The same method as in (a) gives a regular bounded Reinhardt domain D of bolomorphy with being not symmetric. (c) SG : G x C" is upper-semicontinuous; 4G' SG are not continuous as functions of the first variable.

)2isC2neara.

)ESG(O, (e) gQ(a, ) (f) AG(a, ) = SG(a, particular,

)

)

) ). 1) whenever a1

AG(a, ) (p

Now we will give the proof of the above formulas: For (A),let u(z) := for z E

therefore SG(a, )u.

for p 2 and

G then u

a,,

E

0; in

_________________ MAREK JARNICKI AND PETER PFLUG

270

then there exists a Iog-psh function v on the Unit Now let u disc such that u(z) = v a Q(z) if 'I'(z) 0, and i/ Hence u(z)

obvious. It is known that any f 9(G, E) where Ii E), in particular with ord0fp is of the form f= For (B), the inequality "" is

f

r

h

p; hence

Therefore

(C) resp. (D) follows directly from (A) resp. (B) and Proposition 2.5, (1) resp. Proposition 2.7, (1). For (E), the first part of the formula follows from (A), (C) and Proposition 2.8. (2). Without loss of generality we can assume that a1 a5 0 and

(sn—2). Inviewof(C)weobtain: SGa, X) where

JIG(a, X) =

r(a) = r = a3÷1 +

+

..

.

Since SG(a,

)

is

a seminorrn we finally

get

We would like to add some information about the yr's (without proofs) where P = P(R) {A C: (1/R) < 0, there exists 0 < e" <e' such that

forO<e<e" as a quadratic form at every point of closure(U). Thus for any b > 0, there exists 0 < <e" such that

forO<e<e1 as a quadratic form at every point z E — closure(O). This proves (2). Next we prove (1). Consider any point Z E M — closure(U). Remember that on Al we have the formula for currents cot

—

*

= *

= We

—

—

want to show that for every 1 0, such that for any

a as a quadratic form at evely point of M.

PRooF. FromSiep 1 intheproofofLcmma 3.1, Since co isaKãhlercurrentand P is smooth on M, there exists > 0 and a1 > 0 such that for any 0 < e < 82, ? a1 as a quadratic form at every point of closurc(U). Then by the proof of Lemma 3.1, the Corollary foilows.

Q.E.D.

3.3. Estimate for

Let M, S, W,

be as in §3.1. For any on M a }Iermitian line bundle

with curvature 0L such that

In order so apply Siu's theorem in §2.5, we need to have the following lemma. LEMMA.

There exist 83>0 and C >0 such that

C fi,ranyo<e<e3.

*

By Lemma 3.1, for fixed {Uj}j<j 0 such that

Then we

of closure(O)) C.

CM(l +

> 0 such that for any 0< e < w1 —b as a quadratic form at every point of closure(O). It follows from Lemmas 2.5 and 3.3 that for any fixed 0 < e < min(e0, Again applying Lemma 3.1, there exists

dim H°(M, for k sufficiently large. By Lemma 2.4, it follows from (ii). Next we prove (i). If M is projective algebraic, we can take w to be a Kihier form and take S = 0. Conversely, if there exists such a KJhler current w and e smallenough, w1 is positive definite with twi H2(M, Z). Then from §2.3 and from Kodaira's embedding theorem, M is projective algebraic. Q.E.D.

Proof of Theorem 1. We fix coordinates (z1, ... , z,,) for C", and ak, dzk Ad2,, where ak, are periodic distributions on C" with respect to the lattice A. Let *f) + P as in §2.2, where x1 is the usual approximation identity with respect to A. Notice that here we did not use any partition of the unit. By a similar method, 4.2.

write co =

By (i) in Theorem 2, it follows that is a positive definite real closed (1, 1)-form with H2(M, Z), so M is projective algebraic, i.e., it abelian.

Q.E.D. REFERENCES

JDJ

(GRJ

J.-P. Dcinailly, Champs magnetiqueei I liter de Morse pour la d"-col,o.nologie Ann. Inst. Fourier (Grenoble) 35 (1985), 189—229.

H. Gauert and 0. Riemanadmeidrr,

(LG)

aaalprist*e aitf komplexen Rdwne, Invent Math. 11(1970), 263-292. P. Lelong and L Gruman, Entire fw,ction of several complex variables, Sprioger-Vcda,

EM)

New York, 1986. Y. Miyaoka, Extension theorem

giegruppen

Kãhler metrics, Proc. Japan Acid.

409. [ShJ

B. Shiffman and A. I. Sommese, Vanishing theorem on complex Boston-Basel-Stuttprt, 1985.

(1974), 407-

SHANYU

2S2

(Sit) (512)

1W)

ii

Y.-T. Sin, A vanishing theorem for semipositive line bundles over non-Kdhler manifolds, J. Differential Geom. 19 (1984), 431—452. —, Some recent resuh3 in complex manifold theory related to vanishing theorems for the semiposilive case, Lecture Notes in Math., vol. liii, Springer-Verlag, Berlin and New York, 1985. R. 0. Wells, Modezon spaces and the Kodaira embedding theorem, Value Distribution Theory: Part A, Marvel Dekker, New York, 1974, Pp. 29-42.

UNIVERSITY OE HOUSTON

in Pure Mitheinatics Proceedinp of Volume 52(1991). Pan 2

Biholomorphic Mappings Between

Quasicircular Domains in C" KANG-TAE KIM Abstract. We prove that any two holomorphically equivalent bounded convex quasicircular domains are equivalent under a biholomorphic mapping which extends holomorphically past the boundary. Hence, two such quasicircular convex domains are not biholomorphically equivalent unless their singular loci are equivalent by a biholomorphic mapping of their neighborhoods. As an application, we construct families of inequivalent domains and demonstrate how to use inequivalence to compute the automorphism

groups of certain domains in C'.

1. Introduction. Call a bounded domain in C' quasicircular if there exists a n-tuple (a1, ... , of positive rational numbers such that is invariant under the map laO

laO

foreach OER. Thenweprove

Let and Q2 be bounded convex quasicitcular domains in C". If they are holomorphically equivalent, there is a biholomorphic mapping between them which extends holomorphica/ly past the boundary. THEOREM 1.

Let a1 and Li2 be as in the theorem above. Assume further that the boundaries and are piecewise C1 smooth. Then their singular loci are homeomorphic to each other by a biholomorphic mapping defined on their tubular neighborhoods in C". COROLLARY.

These statements are useful when one tries to compute the automorphism groups of bounded convex domnins by the so-called "scaling method" ((81; see also [4, 121). We will give a demonstration on this in later sections. In our proof of the theorem above, we use a simple observation on maxima) compact Subgroups of the identity components of the automorphism groups 1980 Mathematics Subject Classification (1985 Revision). Primary 32A40 Secondary 32F1 5. This paper is in final form and no version of it will be submitted for publication clscwhei'e. © 1991

00*24117/9% 51.00+5.25 pee

283

KANG-TAE KIM

of the convex quasicircular domains, which also gives a simple proof of the following (weaker, but in some sense more general) analog of the theorem of Burns, Shnider, and Wells [2]. To state the result we first give a

is said to admita DEFINITION. Let Q be a domain in C', n 2. sphere envelope, if there exists a constant r> 0 such that for any q E there exists a point q0 C' such that Cl c Bq(T) and q OCl n where Bq0(r) denotes the open ball in C' with radius r centered at q0. Note that the boundary OCl in the definition above is not necessarily smooth as one can see from examples such as the domain in C2 defined by denote the collection of all circular the inequality + IwI < 1. Let Ck(R), as usual, the collection of all domains with a sphere envelope, and smooth functions from R to R equipped with the usual uniform topology on compact subsets. Then we state THEOREM 2. Let Cl = {z Then for any k 2 andfor (1)

C any

0. there exists 0
0 gives a metric agreeing isometric to the above Taub-NUT metric up to dilation and rescaling. Because our method depended upon so little of the fine detail of the formula for the metric, it can also be be applied to analyze more general metrics arising from (3). For instance, let a finite collection := {X, =

(x1

,

y1, z1),

... , XN = (xN, YN' ZN)},

of points in R3 be given, and let V :=

ic2

-,

+ j=1

J

COMPLETE RICCI-FLAT KAHLER METRICS ON C' NEED NOT RE FLAT

denotes the Euclidean distance from the point Choose the direction of the x-axis to be generic in the sense that u = y + iz is one-to-one on .9'. Then, letting where

=

—

the same argument shows that the complex manifold built from our data (with the N points deleted) is biholomorphically equivalent to the one constructed in the form [(C — .9') x

(12)

II (C —

Y)

C2 molov [B2] (see also [Deal) to show that S contains only a finite number of algebraic C-curves. Let M4 be the set of nontrivial strictly ratIonal maps2 from an algebraic variety A to S not mapping to algebraic C-curves. We show that MA can be endowed with an algebraic structure of dimension at most one. Moreover, S admits a finite number of dominating morphism from products of algebraic curves

F,:C1xB,—'S, so

i=l,...,q,

that given an algebraic variety 4, any element in the compkment of a

finite subset of MA factors through F In b, for some i and b, E then there are positive constants m and fact, if H is a polarization of a such that for any algebraic curve C in S with closure C in H. + a,

where Xt is the Euler characteristic of the normalization of C. Furthermore, if S is minimal (in the sense that its log-canonical bundle K1 (1)) is numerically effective) with > 2c2, then a meromorphic map from an algebraic variety A to S is in fact a strictly rational map as long as its image is not an algebraic C-curve and M4 must be finite. This extends the results of (LYI to the case of noncompact targets and also gives the finiteness result for nontrivial strictly rational and meromorphic maps. Furthermore, these multi are displayed and worked out in the larger framework of "pseudoample cotangents" in order to cover targets of arbitrary dimensions in this paper. 1980 Mathematics Subject Classification (1985 Revision). Primary 321120, 32H25, 32H IS, 32H30.

Research supported in part by an NSERC Postgraduate Fellowship. This paper is in final form and no version of it will be submitted for publication elsewhere. 'Implicit here is that D is a normal crossing divisor in a smooth projective S. 2A rational map is said to be strict in the sense that its graph projects properly to the first factor. This is automatically satisfied in the case where the target variety ii complete. Amaicaa 00*2.0717/91

305

Society + $.2S per

S. S.-Y. LU

1. Introduction. Investigations on the behavior of holomorphic maps from

a noncompact manifold, the quintessential examples being C or D, to an algebraic variety has been going on since the turn of the century. Much evidence was found suggesting that holomorphic images of C in a variety of (log-)general type be algebraically degenerate, especially after the works of M.

Green [Gi, G2, G3], T. Ochiai [Och], J. Noguchi [Ni, N3], and the seminal paper of Green and Griffiths EGG] in the past decade. Motivated by Diophantine geometry, S. Lang conjectured that the Zanski closurc of the union of holomorphic images of the complex line in a variety of (log-)general type must be a proper algebraic subset whose complement is mordellic [U. In fact, a bolder conjecture dispenses with taking the Zariski closure, and states that this algebraic subset is the same as the union of positive-dimensional subvarieties not of (Log-)general type or more strongly as the union of nondegenerate (strictly) rational images of group varieties3. We verify this conjecture of Lang for a log-general surface whose log-Chern numbers satisfies > 2c2 and the analogous mordellic properties over complex function field for the case of trivial families (the split case) with the weaker hypothesis of > c2 on the Chern numbers. Although the results focus on dimension two, the main theorems are worked out in a more general setting and we list them here (after some definitions): Let D be a divisor with normal crossings4 in a nonsingular projective

variety V with an ample line bundle H. We call such a pair (V, D) a smooth completion of V := V \ D and a smooth model for the proper birational class of V. The log-cotangent bundle := D) of V is then a well-defined vector bundle over V. Let

of P :=

V. We note that Pj,, :=

be the canonical projection is the projectivized

tangent bundle of V so that holomorphic maps from curves to V have natural lifts to P. A subvariety of P is called horizontal if it dominates

V via m. Given a subvariety E of P. we let a : normalization of

—. P denote the composed with the inclusion of E in P. Set

{c(z)I s(z) = 0 for all s

H°(a(

>(m)nH'))} U

the set of singular points of !. An algebraic variety properly birational to V is said to have almost e-ample cotangent (or pseudoample of P of codicotangent when e = 0) if for every horizontal subvariety #! for some in > 0. This condition mension no more than e, where

is

will be shown to be independent of the choice of a smooth completion and polarization and is hence a proper birational invariant. 3The consideration of group varieties here can be reduced, by the Chevalley decomposition theorem, to simply that of abehan varieties and the one-dimensional algebraic tonis, C, (cf. (LI, p.321). 4We do not require the divisor to have nonsingular components in our definition of normal crossing.

VARIETiES OF LOG-GENERAL TYPE

307

If V is an algebraic surface with nonnegative logarithmic then V has pseudoample > Kodaira dimension such that then V has > cotangent. If further, V is minima with THEOREM 1.

almost 1-ample cotangent.

be a subvariety of P. Let f be a nontrivial holomorTHEOREM 2. Let (via phic map of an algebraic curve into V which lifts to E but not to

its natural lift to P). Then f must in fact be algebraic and degfH —mx the topological Euler characteristic of the normalization of the curve. As a result, the number of such maps is always finite for a given curve and in particular there is no such map from the affine line C. where x is

Now let V be an arbitrary algebraic variety having pseudoample cotangent. We assume without loss of generality that V has dimension greater than one. THEOREM 3.

Given an algebraic variety A, any dominating meromorphic

map from A to V is necessarily a rational map and there can be at most a finite number of such maps. In particular, V cannot be dominated meromorphically by a group variety.

We remark that if V can be shown to be of log-general type, Theorem 3 would be a corollary of the logarithmic version of a theorem of Kobayashi and Ochiai [KOJ (cf., also, [CGJ). One good evidence for V to be of loggeneral type is that it follows from the validity of the conjecture, known in dimension two, that a variety not of log-general type admits a dominating, generically finite, and strictly rational map from a fibration of group varieties, that V cannot be so dominMed. and the fact, as we will show in THEOREM 4.

The number of codimension-one subvarieties of V not having

pseudoample cotangents is finite. In particular. V has only a finite number of codimension-one subvarieties not of log-general type if dim V < 4.

Let Z be the union of codimension-one subvarities of V not of loggeneral type nor having pseudoample cotangents5. Z is a proper algebraic subset by the above theorem. In the following theorem, we let MA denote the space of strictly rational maps from an algebraic variety A to V not mapping to Z and whose images have codimension one. THEOREM 5.

There are a finite number of one-para'neter families of codi-

mension-one subvarieties of V. so that given an algebraic variety A, any element outside a finite subset of MA factors through a member of one of these families. Hence, MA has an algebraic structure of dimension at most one. If flirt her, V is two-dimensional, then it admits a finite number of dominating morphism from products of algebraic curves

F;: C1 x B, —' V, 5With the remark above, this last condition is expected to be redundant in general.

S. S-Y. LU

so that given an algebraic variety A, any element in the complement of a finite subset of M4 factors through for some 1 and b. E B. b, THEOREM 6. If V has almost 1-ample cotangent, then a meromorphic map to V whose derivative has rank r dim V — I somewhere and which does not map to Z must, in fact, be a rational map. In particular, there can map from a group variety. Also MA isfinitefor any be no such algebraic variety A.

More generally, THEOREM 6'. If V has almost e-arnple cotangent with e = dim V — k, then a proper algebraic subset Z contains all rank r k meromorphic images of group varieties. Z can be taken to be the union of dimension d k subvarieties not of log-general type nor having pseudoample cotangent. Also,

given a variety A, rank r k ,neromorphzc maps of A to V not into Z are necessarily rational maps and their number must be finite. Hence, if V is compact and has almost e-ample cotangent with e then V is pseudohyperbolic in the sense of Lang [U. Similar theorems hold for higher order (logarithmic) jet bundles allowing

dim V — 1,

for a larger class of examples — possibly including the general case (cf. EGG, Lu]). One can also make a finer analysis of the "base locus" to obtain sharper results and consider exterior powers of the tangent bundle to cover interme-

diate dimensions. Further details of these, including interesting examples and the borderline case of = 2c2, will be relegated elsewhere. The pToof of these theorems proceeds by constructing, on a smooth model of V. semi-length functions with strongly negative curvature analogous to the construction of Carison and Griffiths [CG]. We note that the finiteness results we obtain for rational maps do not depend on such a construction, which is only used to force meromorphic maps to be rational maps. The author wishes to thank Professors Heisuke Hironaka and Mark Spivakovsky for setting him straight on the resolution of singularity theorems. Thanks also to Professors Barry Mazur, Serge Lang, Joe Harris, and Takahiro Shiota for help and useful discussions. Last but not least the author expresses his deep gratitude to Professor S.-T. Yau for his kind attention and keen interest, as well as his very patient guidance without which this paper would never have gotten off the ground.

2. Preliminaries. The ground field is C throughout this paper. A property is called generic if it holds outside a proper algebraic subset while it is called general if it holds outside a proper subset consisting of a countable number of algebraic subvarieties. We will use the same notations to denote locally free sheaves and their associated vector bundles. Also, the tensor product notation will be suppressed when tensoring with line bundles (if no ambiguity results). a.

Conventions on maps and two basic theorems. Let 1: A

B

be a

VARIETIES OF LOG-GENERAL TYPE

309

rational map between algebraic varieties A and B. Its graph is, by definition, a subvariety I' of A x B which projects generically one-to-one to the first

factor A. The map f is called strict, or strictly rational, if r projects properly to the first factor. It is called proper if F also projects properly to

the second factor. For f strict, the fundamental set of f is defined to be > 0 where p1 is the the collection of points z E A such that projection of A x B to A. The (algebraic) image of f, denoted Im(f), is the Zariski closure of the image of the generic point of the source variety or equivalently the Zariski closure of the image of F under the projection to the second factor. A strictly rational or meromorphic map between two varieties is dominating if its derivative is surjective somewhere. In this paper, a variety dominates another always strictly rationally if not otherwise indicated. Also, varieties are algebraic by default. Finally, we note that strictness is built into the definition of meromorphic maps, so that a rational meromorphic map is strictly rational.

Let f Xj_l

be a morphism between algebraic varieties X and Y. Then := (y), the fiber of f over y, is a closed X for any y Y. If f is sul)ective and if is connected for all y E Y, then we say that X is a fibration over Y (defined by f) with fibers X1,. Y

The following weaker versions of Zariski's main theorem [Mum, pp. 48— 49) and Hartog extension theorem will suffice for our purpose and are used mainly to establish the proper birational invariance of certain definitions. THEOREM (Zariski). The fundamental set of a rational map between smooth complete varieties has codimension two. THEOREM

(Hartog extension). A holoinorphic function defined outside a

codimension two subset of a smooth variety extends to a holornorphic function on the entire variety.

b. Smooth mode! and logarithmic Kodaira dimension. Let D be a normal crossing divisor in a nonsingular, complete algebraic variety P. We call (iv, D) a smooth completion of V = V\D and we call D the divisor at infinity. Also, we call (V. D) a smooth model for any variety properly birational to V. The logarithmic (or log-) canonical bundle of V (over is defined where Kp is the canonical bundle of V. We denote this bunto be when no confusion arises. The logarithmic Kodaira dimension of dle by V is defined to be —oo if 0 for all in > 0 := dim ji°(.)). Otherwise, it is defined to be the integer ?c appearing in the following theorem. THEOREM.

an integer ic (0

If

0 for so,ne positive integer m, then there exists

ic dim V) satisfying K

0—mi

mK, i.e.

am for some positive constants a, b, / and for all integers m > 0.

S.S..Y.LU

310

We denote the logarithmic Kodaira dimension of V by fc(V). Since the fundamental set of a rational map between nonsingular complete varieties has codimension two by Zariski's main theorem, it follows from the Hartog extension theorem that the dimensions of the space of sections over V of various positive tensor powers of are proper birational invariants of V.

In particular, the logarithmic Kodaira dimension is such an invariant. A variety is said to be of log-general type if it has a smooth model (V, D) with V) = dim V.

Given an algebraic variety V', Hironaka's global resolution of singularity theory tHu tells us that V' admits a proper birational morphism from a V = V\D where V is nonsingular and complete and where D is a normal crossing divisor with only nonsingular components. This means that D= (s,) where the s1's are sections of the line bundles associated to the components of D and that for each point z V these line bundles can be trivialized in a neighborhood of z so that those s's which vanish at z form part of a holomorphic coordinate system around z. We call V a desingularization of V. In fact, we may choose V = V' when V' is smooth and we call the pair (F, D) a very smooth completion of V' with this choice. A smooth model of an algebraic curve C is clearly uniquely determined as rational maps between smooth projective curves are just morphisms. It thus makes sense to speak of the desingularization of C, which is also called the normalization of C. The sheaf of logarithmic 1-forms and the log-cotangent bundle. Let (F, D) be a smooth completion of V. The log-cotangent bundle of V (with this completion) is a vector bundle over V defined by the following locally free sheaf, denoted (see Lii, §1 1)). (1) D) is a subsheaf of the sheaf of rational 1-forms of V. (2) the sheaf of regular rational 1-forms of V. = (3) Given a point p E V , we have by definition a regular local coordinate system (z1, ... , around p such that D is defined by setting the first 1 z.'s to zero there. Then w if and only if there exist a, E i 1, ... , n such that c.

1=1

With this definition, dim V since

D) is clearly a locally free sheaf of rank n = =

+> 1=1

is

isomorphic to

Note

E a,dz,.

1=14-1

1=1+1

We denote the corresponding vector bundle by

that the log-canonical bundle K(D) coincides with the top exterior

VARIETIES OF LOG.OENERAL TYPE

power of

dz

since

dz, +1

A•••Adz "

1

dz1A...Adz

Riemann-Roch and stability of vector bundles. Let E be a rank d vector bundle over a smooth projective variety Y. Let P = P(E) be the hyperplane d.

bundle with projection p. The Chern classes of E can be determined from over F: the following polynomial identity for the line bundle L := (*)

•pc1(E) + ... + (_l)dp*cd(E) = 0.

—

We recall that the dual of L is just the canonical subbundle of E', the dual of E. Hence, there is a canonical isomorphism L'\{O-section}

E\{0-section}.

The following result, see [Ha, pp. 68—71] for a proof, will be useful., THEOREM. Setup as above, there are natural isomorphisms

H'(P,pE) ZH'(Y, E), H'(P,

Z H'(Y, E®Sm(E)),

foreverym1 With this result, we will only need the following special case of the Riemann-Roch theorem, cf. [Hir] THEOREM (Riemann-Roch). dim Y,then

Let F be a line bundle over Y with n =

X(F) := h°(F) — h'(F) + h2(F) — ... + (F) + lower powers of c1(F). = n! For example, setup as above with d = n = 2 so that dimP = 3, we have by (*)

c(L) = Hence

for

A line bundle over a smooth projective variety Y is called numerially effective if it has nonnegative degree with respect to every curve in Y. Let F be such a line bundle. A vector bundle E over Y is called semistable with respect to F if every proper subsheaf of E satisfies

c1(F).c1(2') < c1(F).c1(E) rank(E) We recall one further useful result in algebraic geometry.

S.S.-Y.LU

312

THEOREM (Serre duality). Given a vector bundle E over a smooth projective variety Y of dimension n, we have (by choosing an ident(Jication of H"(K1) with C) 6 canonical iso,norphisms

H'(E) = h'(E)

(HPI_s(E* ®

h?I_i(E*

Kr),

for 0 i n. e. Pseudoampleness of bundles. Let L be a line bundle over a smooth projective variety A of dimension ii. We have natural flberwise linear maps P(HO(Lm)m). If H°(L) Lm —. Ho(Lmy which projectivize to :A separates points and tangent directions of A so that

L is called very ample. If Lm is very ample for some m > 0, then L is called ample. We call L big, or pseudoample, if some power of L admits a nontrivial homomorphism from a very ample line bundle H0. This as one sses is equivalent to the maximal growth condition ho(Lm) "s.. from the exact sequence

the fact (from the Riemann-Roch formula and the Kodaira vanishing = X(Hm) has maximal growth for any ample line theorem) that bundle H over a projective variety, where (H0) denotes a smooth hyperplane section. Note that having a nontrivial homomorphism s from H0 to Lm (as rational to be birational onto its image as 3' o entails = maps) for the rational projection 3' : P(HO(Lm)*) ——-i l?(Ii°(H0Y) defined by s. Conversely, we have the following fundamental lemma of litaka [Ii, §10]. and

L is pseudoample if and only if onto its image for some in (i.e., LEMMA I.

is a generically finite map = n).

This lemma provides an alternative definition of pseudoampleness, whic makes sense for line bundles over any complete algebraic variety Y; namely, L is a pseudoample line bundle over Y if its pull-back to a desingularization

of Y satisfies the condition given in the above lemma. This is easily seen to be independent of the desingularization chosen, again via Zariski's main theorem and Hartog extension. As Y is birational to a projective variety, we see that this alternative (more general) definition is again equivalent to the maximal growth condition on ho(Lm), where L is the pull-back of L to a desingularization. is usually done via the fundamental class of Y , but this paper will take a more general position.

One can use more generally a normalization instead of a desingulanzation establish, similarly, the independence of the definition from the choice of [Ue,

of Y and see


313

Analogously, a vector bundle E over a smooth projective variety is called for some m and pseudoample if there is a nontrivial section of an ample line bundle H. Let P := F(E) be the hyperplane bundle of E and let ir be the bundle projection. Let L be the line bundle associated with (LTh*H1y1) = and as LirH' is As

ample for I large enough (see Proposition 7.10 in Hartshorne [H, Chapter II §71), we see via the same arguments as above that E being pseudoample is equivalent to the maximal growth condition of ho(Sm(E)) and to the pseudoembedding (generic immersion) condition of E* in Hb(Sm(E)*) for some integer m. We note that L is a line bundle over P whose dual

is by definition the canonical subbundle of ir'E. A subvariety of P is called horizontal if it intersects every fiber of Note that a section a of corresponds naturally to a section a' of We denote the zero locus of a' in P by Z° and the intersection of the zero We say that E is almost k-ample locus of all such a's for a fixed m by (or pseudoample when k = 0) if L is pseudoample when restricted to every horizontal subvariety of P having codimension no more than k, k 0. We remark that this definition can be considerably broadened, without affecting any results in this paper, by considering only those horizontal components of which intersect the generic fiber of P only at multiple the 'iterated base linear subspaces. f. Almost k-ample cotangent and its birational invariance. DEFINITION. If an algebraic variety V' has a smooth model (V, D) so that is pseudoample (respectively almost k-ample), we say that V' has pseudoample (respectively almost k-ample) cotangent (0< k 0 for any x E V and 0 s E L!. More L such that which, over generally, a pseudometric on L' is a C°°-section of L any small enough open subset U of V, is of the form h > 0, and a metric g' a rational number For any open subset U of V on L1. Let g be a pseudometric on

I

ddClog(gjs(2) defines and a hobomorphic, nowherezero section s of and a real (I, 1)-current over U * Since 2irddC = 0 log 1f12 for any nowhere vanishing holomorphic function over an open subset, we in

fact obtain a well-defined (1, 1)-current over V by varying (U, s). This current is denoted by Ric(g). Given a complex manifold M, let L = (I). Note L' is the blow up of a zero section of TM, the tangent bundle of M. Hence, a (pseudo)metric on L' defines naturally a function on TM which is called a (semi)Iength function on M, cf. LL). Inequalities between semilength func-

tions on TM will always mean the corresponding inequalities between the pseudometrics on L h.

Group varieties and other definitions. A group variety consists of a pos-

itive dimensional variety Y together with a morphism Y x Y —. Y so that Y is a group with the operation given by Au and so that the inverse map y is also a morphism. By a fibration of group varieties, we mean a fibration whose general fibers are group varieties.

VAR IETIES OF LOG-GENERAL TYPE

For a group variety G, there is the classical Chevalley decomposition

l—4G0---'G--.A--sO (exact),

where G0 is a maximal connected affine algebraic subgroup and A is an abelian variety. If G0 happens to be an algebraic torus C*k, then G is called a semiabelian (or, sometimes, a quasiabelian) variety. It can be shown does not contain C as an algebraic subgroup and that this is the case if that G is commutative in this case, see [112, §7]. We have defined an algebraic C-curve to be a curve dominated by a holomorphic map from the complex line C. These are simply algebraic curves whose normalization have nonnegative Euler characteristics. Smooth curves with nonnegative Euler characteristics are easily enumerated. They are sim-

ply p1 and the one-dimensional group varieties consisting of C, C, and the one-dimensional complex tori. i. Flatness and the rudiments of Kodaira-Spencer theory. In this paper, a morphism p : X —' B of nonsingular varieties such that

dp:

-.

suijective for all x E X is called a smooth morphism. If p is smooth and proper and if the fibers of p are connected, then we say that p defines a smooth family. It follows, in this case, that the fibers of p are complete is

and nonsingular varieties. A well-known theorem of Ehresmann says that a smooth family is differentiably locally trivial. In particular, all the fibers are diffeomorphic.

Given a smooth family (X, p, B) with a base point b E B, we have a short exact sequence of sheaves

*(TB)

0.

The first connecting homomorphism of the long exact cohomology sequence of this short exact sequence H°(Xb, p*(TbB))

T1),

yields the Kodaira-Spencer map

Pb:

Tx),

since TbB is naturally isomorphic to H°(Xb, p(TbB)). We record the following theorem of Kodaira and Spencer [KS]. THEOREM.

Let

(I, p, B) be a smooth family as above. If h' (Xb,

constant and if the Kodaira-Spencer map Pb vanishes for all b E B • then the smooth family is locally analytically triviaL is

Here, a smooth family is called locally analytically trivial if every point of B has an open neighborhood U over which the family is analytically trivial;

S.S.-Y.LU

316

i.e., there is a commutative diagram

AxB .p2j

B,

B

where A is a nonsingular complete variety, p2 A x B —' B is the projection to the second factor, and where a defines an analytic (i.e., holomorphic) isomorphism of Ax B with := p'(U). We remark that the first hypothesis appearing in the above theorem can be dropped for all practical purposes due to the Semicontinuity Theorem in algebraic geometry, which we now come to.

Let 1: X —ø Y be a proper morphism of algebraic varieties. A coherent sheaf 8' on X is said to be flat over Y if is a flat for is flat over Y, then f is said to be flat. If 8' is locally allx E X. If free, then it is certainly flat over X. If further f is flat, then 8' is flat over Y by the transitivity property of flat modules. Finally, we observe that if f defines a smooth family, then f is flat. We now record two useful theorems.

f: I

THEOREM (semicontinuity). Let -. Y be a proper morphism of algebraic varieties, and let 8' be a coherent sheaf on I which is flat over Y.

Thenfor each

iO,

h'(y, 8') is an upper semicontinuous function with respect to the Zariski topology on

Y. THEOREM (generic smoothness).

Let

f: I

—.

Y be a surjective morphism

of varieties, and assume that I is nonsingular. Then there is a nonempty open subset U c Y such that f: f'(U) -. U is smooth. For a proof of these theorems, see for example (H, Chapter III j.

10,

§

12).

Blowing up, the process of desingularization. For the definition of the

blow up process, or equivalently monoidal transforms in algebraic geometry, we refer the readers to [II, §7.10]. Given a variety X with a subvariety Z,

The blow up or monoidal transform of I with center Z is a variety along with a map r: X which induces an isomorphism

QzXV'(Z) Z X\Z, restriction. This is not a definition, but will suffice for our purpose. If nonsingular, then D is a nonsingular codimension one subvariety of which is then also nonsingular. We call D the by

Z and I

exceptional divisor for the blow up. It is a theorem of Hironaka that any variety admits a desingularization via a sequence of blow ups with smooth centers. Also

VARIETLES OF LOG-GENERAL TYPE

317

THEOREM (Hironaka). Let D be a reduced divisor on a nonsingular varifor I j / such ety. There is a sequence of blow ups r1 with center

(1) V0—V, D0=D, (2)

(3) Z, is nonsingular and

c D3, (4) 0, is a normal crossing divisor with nonsingular components. (5) if r is the composition of the r1 's then the points where r fails to be one to one is a normal crossing divisor with nonsingular components, called the exceptional divisor of r. We will use the fact that if D is the exceptional divisor of a composition of blowing ups r: Y —+ X with smooth centers and with X smooth, then is relatively ample in the sense that (r_UJf)m(_D) is ample for some ample line bundle on X (assumed to be nonsingular projective) and for some integer m > 0; see Proposition 7.18 of [Ii].

3. The surface case (Theorem 1). Let V be a complete algebraic surface with a rank 2 vector bundle E and let K = dci E. We first give a lemma of Bogomolov (B]: LEMMA 1.

Assume

> c2(E) and Kmo is effective for some m0 > 0.

Then E is pseudoample. Pgoop. +

h2 (Sm(E))

c1(E)

; c2(E) m3 + 0(m2)

by Riemann-Roch. Using Serre duality, we have for m 0 mod (m0) that

h2(Sm(E)) =

= hO(Srn(E)K_mKV)

> c2(E), then either ho(Sm(E)) m3 or In the latter case, choose a very ample divisor r such that L = effective. We then have an exact sequence (of sheaves): Hence if

0

Sm(E)L_l —'

m3. is

0.

As hO((Sm(E)KV)I(ff)) = 0(m2) we see that ho(Sm(E)L_l)

sheaf

m3. But the injects to the sheaf Sm(E) via a section of L. Hence 0

This lemma yields the first part of Theorem I. The second part of the theorem follows from the semistability of with respect to and amis result goes back to Bogomotov FBI, Miyaoka (MIOJ, and Yau (YasJ.

S.S.-Y.LU

318

a lemma of Miyaoka [Mi] stated for the more general vector bundle case [S-TI:9

Assume E is pseudoample with > 2c2(E), and K is numerically effective. If E is semistable wit/i respect to K, i/len E is almost LEMMA 2.

1-ample.

be a horizontal codimension-one subvariety of P = P(E). PRooF. Let Let L = 9p(E)(l). As a divisor up to linear equivalence, = n(L) + ir'(F) for some line bundle F over V and n > 0, where we use the notation (G) is the canonical to denote a divisor representing the line bundle G, and V. We compute some numerical (intersection) data for projection P = LIT:

(L)2

(n(L) + ir(F))

- c2(E)) + (E) (F),

=

n(K)2 + (K) (F),

.

0. Now

where we have used the identity (L)2 — (L) ir'c1(E) + {0)

H°(P, L"irF)

H°(V, S"(E)F)

Cir*F =

is effective. So has a nontrivial homomorphism into But S"(E) is semistable with respect to K by a theorem of Maruyama [MaJ, since E is. Hence as

(

FK
0, — c2(E)) — (K).

(F)

— 2c2(E))

>0. A key idea introduced by Schneider and Tancredi is to use a result of Marnyatna [May that the symmetric powers of a semistable vector bundle over a projective variety are also semistabk. This fact can be seen directly for the case of log-cotangent bundle by the argument of Cheng and Yau of approximating by Kahler-Einstein metrics (see Tsuji fI'sj)) which yields indeed a further refinement of Miyaoka's analysis.

VARIETIES OF LoG-GENERAL TYPE

L = and k = r'irK. (f42 (?K) > 0. The Then we have = (L)2 > 0 and (L)• CR) first gives ho(Lm) — hI(Lm) + h2(Lm) in2 by Riemann-Roch and the latter Let

be a desingularization of

—.

CR) < 0 for m >> 0. By Serre duality, = which must vanish for in >> 0 as k is numerically effective.

shows (—m(L) +

m2. 0

Therefore

4. Pseudornetric with stTongly negative curvature. Let D be a normal crossing divisor in a nonsingular projective variety V having only nonsingular components, D1, ... , Di,. (V, D) is therefore a very smooth completion of V = V \ D. Let L, be the line bundle with section s, such that = (s,). Equip each L, with a smooth metric g. and let w, = Ric(g1). and r = r1r1• ••rq. Assume Let r. = ispseudoample so that there is a nontrivial section s of for an ample line bundle H and some in. We will use the same s to denote the corresponding section of where L = Sp(fl1 and r is the projection from P = gM to V Let be a metric on with positive definite Chern form ma. Then gL defines a pseudometric on with Chern form ,rcv + ?n'Z(5) where Z(3) denotes the cS-current defined by s. We also use gL to denote the corresponding semilength function on V. -

PRoPositioN 1. g = g1/r2 defines a semilength function on V with holoinorphic sectional curvature bounded from above by 1, after rescaling the

s1's and

s.

PROOF. As s, rescales without changing a,, we may assume the r1 'S dominate a large constant c> 1 so that w +2 w,/r, dominates

a positive-definite (1,1)-form cii. We then obtain, suppressing ,r and the 5-current that Ric(g) = Ric(gL)_ 2ddClogr

')w+2E Note that 2,rdr, A d'r1 = (1,1)-form, where'°

r, A r.

=

0g.

p. = _._L + '0Noie that

A c', is

r2

a positive semidefinite

ds.

s.

is independent of any local trivializatlois and so defines a global (0,1)-

form. Similarly, sr1ds, defines a rational I-form on P with a simple pole along (s).

S. S..Y. LU

320

Given any point o e V, let D., ... ,

be the components at infinity containing o. We trivialize the L. 's in neighborhood of o so that by where assumption there is a chart centered at o with coordinates z1, ... , = S. . Rescaling if necessary, we may assume irr2w' > = S. , ..., A d2,. Then, these taken together give

irr2Ric(g)> v':T

D,

dzE A d21 +

A

i=n+I

1=1

form a basis for nV over a neighborhood As i's, ... , ii,,, ... , there. As V is of o, it follows that r2Ric(g) dominates a metric on compact, we see that Ric(g) > g'/r2 globally for some Hermitian metric over V. Now g' endows naturally a metric on L which, by g' of compactness of P, dominates the pseudometric after a rescaling of s. These rescalings together give Ric(g) > = g as is required. 0 As a consequence, we obtain the following distance decreasing property for

maps. Let B be a Riemann surface with metric p. Let f be a nontrivial holomorphic map from B to V with natural lift f' to F. Asswne B is covered by a local isometry from the Poincaré disk D and f'(B) Zs. The usual Ahifors' lemma (cf. [LUY]) gives the following inequalities between metrics and pseudometrics on the tangent bundle T8 of B:

o> gjf•j2 >

(*)

a pseudometric on T8, is understood to be = where r is the section of Hom(T8, = (respectively, r' the section of canonically associated with L) the differential of f. Ditto for where

as

LEMMA3.

Jff:D

—'

V

holomorphic map f: D —. V.

doesno:lift:o Zs,thenf extends:oa

Paoop. Replacing H by H' and m by ml, we may assume that H is very ample. Let be a basis of sections for H. Consider the following diagram of maps: D

r

f"

I

f

m

s

ill

—i

I

I

V

V

,pN

is a nontrivial holomorphic vector field over D. This diagram commutes wherever compositions of maps are defined; i.e., anywhere not getting mapped to the origin of CN+I, as the only nonholomorphic map in where


0 by assumption, the the diagram, p, is undefined there. Since sf'" diagram commutes as meromorphic maps. So, it is sufficient to show that to a meromorphic map from D to CN+I. For this of so that mw is the pull-back under purpose, we set gM = = and With 4). = the Fubini-Study metric of we observe from (*) that 4) =

extends

>

{nIzI2(lc;1z12)2Fm

=

=

and integrating over 0, we get

Thus, setting z = r '

8

r2)2m > y—

f

=

azt. By comas a Laurent series 4)0(z) = paring the orders of growth of both sides in (f), observing that r —+ 0, we conclude that a, = 0 for i —m. It follows that is meromorphic over D and similarly for the other 4), 'a. Therefore 4) is meromorphic over D as desired. a Two remarks are in order. First, we have implicitly used the fact that Z3 Z" and replaced s by S' J1o(Lmhir# H_i) to begin the proof of this lemma. Second, had we started out with just a smooth projective completion of V, we could replace it by a very smooth completion by resolving the components at infinity and work with H replaced by (pH1)(—X) and s where we expanded

replaced by (ps')/, where p is the resolution morphism and X

(sX)

its exceptional divisor. This is possible since (pH')(—X) is ample for 0, (II, As Zj is clearly unchanged, the conclusion of the lemma holds without assuming the smooth completion to be very smooth. Henceforth, we drop the assumption that the normal crossing divisor D has only nonsingular components. The exact same argument works to give the extension property for meromorphic maps of higher dimensional varieties is I

not lifting to the base locus.

---' V does not lift to If a meromorphic map f: Dk x Z3 in tixe sense that 0, then f extends to a meromorphic map LEMMA 3'.

J:&'xD--4V. x I)' if necessary, By performing an automorphism of Q := we may assume that for v)))tm E C, where p maps Q to *, a point, and the trivial tangent bundle TQ is identified with the pull back, via p, of the bundle Ck x C —.*. Hence, replacing D by Q and by p(0 xv) in the proof of Lemma 2, we again get a commutative diagram of meromorphic maps and 4) extends to a meromorphic map from DA x D to V as before. 0 PRooF.

S. S.-Y. LU

322

COROLLARY 1. Let (1, D) be a smooth completion of a variety A. Any meromorphic map I from A to V not lifting to extends to a meromorphic map f from T to V. Such maps are therefore rational maps.

PROOF. f extends over the smooth part of D by the above lemma. As the singular part of D has codimension two or more in 1, the Hartog extension theorem applies to give the first statement. The last statement follows by invoking GAGA or Chow's theorem, which claims that meromorphic maps between complete algebraic varieties are in fact rational maps. a Before entering the main task of proving Theorem 2, we record the following BASIC FACT [Ii, § 11.1 ci. Let A, ö be smooth completions of smooth varieties A, B respectively. Let f: U B be a morphism from an open

f:

subset U of i to

then f induces a log-pullback and, dually, a log-pushforward df: T41 —' TB as vector bundle morphisms.

PROOF OF THEOREM 2. We will break the proof of Theorem 2 into two parts, the first assuming E = P while the second deals with the more general case

PART I. Let be a nontrivial map from a smooth algebraic curve C to V which does not lift to Z3. We have already concluded that it extends to a map f, necessarily a morphism, between their smooth completions. The log-pushforward of f,

canonically defines a lift J of f to P = of As the section (f_I of

together with a section is nontrivial by hypothesis, we have by computing the degrees over that — degf'H = = ÷ 0. Now

=

where

I) is the divisor at infinity. Therefore =

since

=

Here, xC and

—zC,

by Riemann-Roch (or more simply by Gauss-Bonnet). are

the Topological Euler characteristic of C and

respectively. Hence

degfH -mXC. From standard Hilbert scheme theory (see [Gro, exp. 221] or [Mmf, §51), the space of nontrivial maps from C to V not lifting to Z3 is then

an open subset of a quasiprojective variety Q parametrizing morphisms

f: C -.

V

such that dcgfH

One can arnve at the of memmorphic map. directly via some simplc formalisms in value dsstribution theory, which amount to integration by parts. Namely, one has a logarithmic bound on the 'characteristic' function using Ahifor's lemma, and this forces algabraicity (cf. ISha, LI)). A more classical, but equivalent, approach can be found in (SW).

VARiETIES OF LOG-GENERAL TYPE

323

Assume is not finite. Then the Zariski closure of in Q has a component of positive dimension and we may choose a curve A0 from this component intersecting Mc. Let A be the normalization of A0. Let A be AxC the smooth completion of A. We thus get a morphism F0 from Fl

to V, which extends to a rational map F from Ti := A x

to V. Note

is a smooth completeion of Fl over which the log-tangent bundle of

that

flis .=p1T4 eP2TC,

to its ith factor. Let SF be the where p. denote the projection of A x fundamental set of and let U = fl\SF. Then F restricts to a morphism F: U —, V. Hence, F has a well-defined Iog-pushforward dF : u Let d2F be the composition of dF with the obvious inclusion of vector over U. We observe that s((d2F)m) 0. This is bundles: PTc —, corresponds to an element of for z in an open because f1 := Fl zxC 0, for such a z. Since SF subset of A, so that s((d2F)f = is codimension two in TI, A x c is outside SF for a generic point c C. Thus, for a generic point c E C, F is well defined over a neighborhood

of Axc in TI,

and

hence so is JF and d2F. Now, for

E

is a nontrivial section (over Ax p) of which is a subbundle of It follows that for c chosen generic enough, we obtain a nontrivial morphism = from a compact variety A

x c to affine space

defined by

Sm(1v) We remark that it is sufficient to choose c E C so that '1 0 and so that is nontrivial. This contradicts the fact that holomorphic functions from a compact variety must be constant by the maximum principle. Hence, is finite.

II. We now treat the case when f lifts to a horizontal component

of Z, with the proviso that

pseudoample. Exact analogues of the preceding results derive as follows: By resolving the divisor at infinity in V, is

we first obtain a very smooth completion t' of V along with a morphism V. Let denote the log-cotangent bundle of V over and P = with canonical projection ft. y defines naturally a map y' : P — —. P. As is an isomorphism on V. it identifies V with a Zariski open subset of a horizontal subvariety of P. We thus get a birational map ii which is P(C211)

the projectivized log-tangent bundle over

! --.!

an isomorphism "over" V. We resolve t', via p' : —. to a birational morphism ii' : —' E. We may assume E' to be nonsingular by resolving further. Let = L and H0 = * it where ft is an ample line bundle on i'. Let be a metric on fl with positive definite Chern

S. S.-Y. LU

324

form. Note that L0 is pseudoample so that, for an integer n chosen large

enough, LgH' has a nontrivial section c. We thus get a pseudometric

on Li', where

gHo

=

Let L1 be the line

bundle with section defining the ith component of the divisor at infinity where is a metric on L1. Set P fl1?,. in Let P, = The same argument as with Chern form + Let g' in Propositon 1 shows that, for any hermitian metric Ii on =

dominates the log-metric defined by h after suitable rescalings of the si,ich that its associated metric on and o. Now take a metric g on As = gfrlyl2 is only a pseudometnc dominates the pseudometric we may choose our h above to dominate it. Summarizing, on h

R

gLo

the pull-backs are suppressed for clarity. This says precisely that g' has bolomorphic sectional curvature bounded from above by —1 and so the same analysis as before applies. 0 where

5. Algebraic foliations and subvarleties without pseudoample cotangents. In

this section we tackle the remaining theorems. For their proof, we may replace V by a smooth model V = V\D without loss of generality. We may also assume V to be projective with polarization H due to the fact that any algebraic model (V, D) admits a modification to a projective algebraic one. This is a consequence of Hironaka's resolution of singularity theorem, since algebraic varieties are bimeromorphic to projective varities (cf. Moishëzon P. [Mo]). Fix m such that PROOF OF ThEOREM 3. Dominating meromorphic maps from A to V are algebraic by Corollary 1. Blowing up further, if necessary, we assume that A is smooth with normal crossing divisor at infinity. Let J be a sequence

of distinct dominating maps from A to V. It is easy to see that the J 's remains distinct and strictly rational when restricted to a general byperplane section (of sufficiently high degree), and hence when restricted to a general curve in A. 12 But the general curve C will be such that is nontnviál

and does not lift to

for i =

1, 2

The existence of such a C

contradicts Theorem 2. 0 An algebraic foliation on an algebraic variety is given locally by nontrivial rational I-forms satisfying the compatibility condition =0 in the common domain of definition of and 1u,. An integral hypersurface of is a codimension-one algebraic subvariety such that the 's pull-back '21n fact, one can show that different maps remains distinct when hyperplane section.

to a gcncral


-

325

to trivial local 1 -forms under its canonical inclusion. The following propositions are key to the rest of the paper. If an algebraic foliation has an infinite num-

PROPOSITION 2 (Jouanolou).

ber of integral hypersurfaces, then these hypersurfaces belong to a pencil of integral hypersurfaces.

13

For a proof of this proposition, see [J]. PRoPosITIoN 3. Given a dominating morphism f: V —. .8, with connected fibres, the generic fibre of f has pseudoample cotangent. In fact, has a nontrivial section over for some 'b m.

PROOF. J: V —. B is smooth over a generic point b on B by the generic smoothness theorem, 14 [H, p. 272]. In particular, := f'(b) is smooth with smooth completion V,, = 1(b). So we get an exact sequence o

Note

1'?L, = €9'

—

—1—

°

I

b

b

—

—

—.0. b

where d = dime B. Since

0,

0 for the generic point b in B. Now

we have

has a filtration with factors

dime SJ(Cd) times

for i, j

0 and i + j = m. Hence

0< i+j=m

i.jo

must have a nontriyial section for a positive integer i m

So

as desired.

0

If a variety A admits a dominating, generically finite map to V. then A has pseudoample cotangent. LEMMA 4.

PRooF. We may assume A is smooth with smooth completion 1 without loss of generality. Let f be such a map. By virtue of the Hartog extension theorem, one has a well-defined pull-back map f': which, as I is dominating, is injective. So the lemma follows directly from the growth characterization of pseudoampleness. 0 '3The pencil here is used in the general sense of a one-dimensional linear system of divisors. I.e., Sard's theorem in algebraic geometry.

S. S.-Y. LU

326

Note that group varieties do not have pseudoample cotangents since their 15 This gives automorphism groups are infinite. V cannot be dominated via a generically finite and strictly rational map from a fibration of group varieties. 0 COROLLARY 2.

We now proceed with Theorem 4. of codimension-one subvarieties of V not having LEMMA 5. The pseudoample cotangents must be finite.

in V. We resolve PROOF. Let X be such a subvariety with closure along with its divisor at infinity, by a composition the singularity of V. This gives a desingularization X0 of blow-ups over V, say p V0 the proper transform of of X with a smooth completion X0 = ---p P of maximal rank. Let : induces a map Note PA :=

I,

k be the algebraic image of 4, which we call the lift of X. We first This can bc seen, for example, by appealing directly claim that c to litaka's lemma. Within our framework, this follows from the fact that is an ample line bundle over V0 for I >> 0, where 110 = is the exceptional divisor M is the line bundle with section S%f, and then there would be a section s of of p. For if 4 did not map to is nontrivial, and we would obtain

such that s0 =

I , contrary to our assumption that a nontrivial section of X does not have pseudoample cotangent. subvarieties Now assume there are an infinite number of The same procedure of V not having pseudoample cotangents, X, as above gives us, for each i, a desingularization X, of x: with smooth ---i P, of V. : X1 and a rational lift 4 completion :

H We have shown that the image P1 of p1 lies in Zm for each i. As the

number of components of Z is finite and as each nonhorizontal component defines a unique proper subvariety of V, it follows that a horizontal of component ! of must contain an infinite number of the images of these lifts. If one of these images lies in the singular locus of E, then it must be a component of the singular locus of ! as it is codimension-onc in !. Hence only a finite number of the X1's lift to the singular locus of! and we remove them from the list. The rest must lift further to some desingularization

of! and we denote these lifts by

Let a : !

:

-+!.

P be the desingularization map E —+ E composed with

A more general class of varieties without pseudoample cotangents are those with smooth completions having numerically trivial log-canonical classes. This is due to the fact that their cotangent bundles (and their symmetric powers) are then semistable so that any holomorphic section must be parallel.

327

VARIETIES OF LOG-GENERAL 1 YPE —. P. Consider the Stein factorization of it o projection from P to V)

the incusion

(it is the

f

has pseudoampic := is a finite morphism, we note that cotangents by Lemma 4. We will show that E defines an algebraic foliation on V.. with an infinite number of integral hypersurfaces, none of which has pseudoample cotangent. Observe fir5t that there is a natural morphism a: E —. in is necessarily a horizontal component of the divisor he this component and let ft be the bundle projection of fr'(s) is nonsingular, complete, and

whose image

P. Let Now

I =dim*'(s)— 1, := for generic s E 17 by the Generic Smoothness Theorem. As under a for such a is the image of the connected, and hence irrcducible is a codimension-one subvariety of the projective space *— '(s) point s, and thus has a well-defined degree > 0. This degree is clearly a semicontinuous function as one can see from elementary means, using for example

the Chevalley theorem on constructible sets, cf. [H). But we give a much more round about proof as follows. By the Generic Smoothness Theorem, ft is smooth over a Zariski open subset ?/ of j7• Hence, any !ine bundle over E is flat over ?/, and thus its degree when restricted to must be semicontinuous with respect to S E ?/ by the Semicontinuity Theorem. But is just the degree of the line bundle where L = as required. So d5 is some integer A over a Zanski open subset of We

show that this constant A must be one. Let T be the closure of X in : V. By construction, the map —. 17, which is a 'smooth' desingularization of 1 composed with the inclusion of in 17, has a natural lift making the following diagram of rational maps commutative p. :

4

4

I

I,,

•1

1*

P II

1* 17,

where all maps are morphisms except those mapping from the first to the second column. Let be the image of := a Let 1. := =

S. S.-Y. LU

328

is a hyperplane of the projective space ft(P). Observe that itl is a hyperplane (i.e., for any smooth point x of X. Hence := = for a degree one hypersurface) of 1.1(2) for generic 2 X. But generic x E 1, since otherwise would contain the component ft being irreducible and horizontal. Hence d5 = for generic contradicting as the X:'s are codimension-one in V. and i;. This forces 2 their number is infinite. Therefore, one may assign for generic a in the nonsingular part of Vt., a codimension-one linear subspace of f(a) and hence of Tr as f is generically a local isomorphism. As this assignment is algebraic, the process defines an algebraic foliation on V and clearly each X is an integral hypersurface of this foliation. By Proposition 2, this foliation defines a pencil of hypersurfaces given by a rational map p from V to a morphism p'. Taking the Stein factorization of p' gives a fibration of K = over some curve with an infinite number of fibers not havingpseudoample cotangents. But K has pseudoample cotangent by Lemma 4, contradicting Proposition 3. 0 This proves the first part of Theorem 4. The second part follows from 1

1

Corollary 2 and the fact that if an algebraic surface is not of log-general type,

then it admits a dominating, generically finite map from a fibration with elliptic curves, or C* as fibers. This fact, in the compact case, is a result of the following facts for a minimal algebraic surface S:

(I) If S has Kodaira dimension

= —00,

then S is either rational or

ruled.

(2) If 0

< 2, then S is either an elliptic fibration (i.e. a fibration

with abelian varieties as generic fibers) or a K3-surface. (3) A K3-surface has oc' elliptic curves by the work of (GGJ completed by Mumford et. al (see [MM]).

For the noncompact case, see (Lu]. A more direct approach, as indicated before, is to appeal to the semistabiity of the cotangent bundle of a minimal surface. We note, in passing, that the above fact remains a conjecture in higher dimensions. But it is known for large classes of varieties; notably, the following is true:

(1) If a subvariety of a

variety is not of log-general type, then it is a fiber bundle with a semi-abelian variety as fiber (cf. [Iii) and [112)).

(2) If a nonsingular projective variety has ample anticanonical bundles, then it is uniruled. That is, it admits a dominating, generically finite map from a fibration of P"s (see Miyaoka and Mon [M2fl. It is a


329

conjecture in general, that a projective variety with negative Kodaira dimension is uniruled.

THEOREM 5. Without loss of generality, we may assume A is smooth with a smooth projective completion A. Let ft : A ---' V be a sequence of distinct nontrivial strictly rational maps. The same proof as in Now assume Theorem 3 shows that only finitely many ft 's do not lift to that each ft has codimension-one image which is neither of log-general type nor has pseudoample cotangents. Theorems 3 and 4 and the well-known theorem of Kobayashi-Ochiai, the log-general version, immediately eliminate the possibility that an infinite number of the ft 's has the same algebraic image. So the ft 's must have an infinite number of distinct algebraic images which lifts to The same argument as in Lemma 5 now shows that, aside from a finite number, these must lie in one-parameter families of codimension-one subvarieties necessarily having pseudoample cotangents. Let 5" be the onedimensional scheme parametrizing these families. We thus get a morphism from M4 to 5" with finite fibers. Hence, MA is at most one-dimensional. The first part of the theorem follows since can define at most a finite number of such families. PRooF

We now focus our attention on the case dim V 2. Consider one of the above families containing an infinite number of algebraic images of the ft. 's. This family, by construction, is defined by a pencil of curves in a finite branched covering X of V. Let p: P' define this pencil pulled back of X. Let a V —' resolve the singularity to a smooth completion of p. By taking the Stein factorization of p o a, we obtain a fibration of Y = cr'(X) over a curve B. Let r: Y —. B define this fibration. Note that an infinite number of the ft 's lift to maps into distinct fibers of r. Now Y has pseudoample cotangent by Lemma 4 so that h°(Y, 0

for some k and an ample line bundle II over a smooth completion of y. Proposition 3 then shows that h°(Y,,, 0 where k,, S k for the =

It follows that C < —kxc for the general curve C in A. Hence, by the theory of Hubert schemes as before, we obtain a family of proper dominating morphism Ii, from to F, , fort in a Zariski generic fiber

open subset U' of P1. Let r

be

an extension of r as a morphism from a

smooth completion of Y. We claim that this forces the family F, , as defined

by r, to be locally trivial analytically on a Zanski open subset U of U', over which r is smooth. We invoke the theory of Kodaira-Spencer for this purpose:

Observe that given a vector bundle E over F,, t E U, there are natural homomorphisms h,' : H'(F,, E) —e h,'E), where h, is the extension of h, as a proper morphism between the respective smooth completions. Applying the Kodaira-Spencer map gives the following commutative diagram

S. S.-Y. LU

of homomorphisms: 1—

p

TUIb

L If(dn,) I

H

H

As the source family is trivial, we have p' = 0 and hence h o p = 0. Now is injective by Serre duality as is clear from the commutative diagram

H'(Y,, E)

H°(Y,, E .1

.1.

H'(?,

H°(?, h'E 0

using the fact that H' (Ky) —. is an isomorphism via pull-back. Hence p = 0 and so defines a locally trIvial fibration (see EKS]) as claimed. We now show that 1', form a locally trivial family: If is a curve of genus two or more, then h, is independent of t and we see that Y, is a locally trivial family. If Y1 = P', then Y has at least three points as its divisor D at infinity, since = (D) is positive. It is clear that over a small neighborhood of z one may trivialize this P' bundle over U, by taking part of the divisor at infinity of Y over U, corresponding to three such points. Then h, is independent of t with this trivialization by Theorem 3. This is because it dominates F' with three punctures which is a to belocally trivial in variety having pseudoample cotangent. This forces this case. The case when V, is an elliptic curve is similar. Thus the family can be trivialized by pulling back to a finite covering of U. This establishes our claim and ends the proof of the theorem. 0 The proof of Theorem 6 is completely analogous to our previous analysis. PROOF OF THEOREM 6. Let f be a meromorphic map from a variety A

of rank r n —1, where n dimV. We have taken care of the case when r = dim V. So assume r = n — and f is not algebraic, i.e., f to V. By Corollary does not extend to a meromorphic mapping from I, f must lift to a component of ! of We claim that f maps to 1

a codimension-one subvariety Z' of V. For this purpose, we may assume that ! is horizontal as it is otherwise clear. The analogue of Corollary 1 (with the same proof but using the metric constructed on L' applied to As maps lifting to ! shows that f must lift to a component E of

f hasrank r=n—l onaopensubset U of A,itmustmap U,andhence A, to the algebraic subset

W=(zj

VARIETIES OF LOGGENERAL TYPE

has a —. V is the canonical projection. Hence where p : = 2n — 3. As Z has dimension 2n — 3, component I' with dim as needed for W, and Z' = n — irreducibility. This forces dim W = by our claim. Now if Z' is of log-general type, or has pseudoample cotangent, then f is algebraic by Theorem 3 and the theorem of Carison and Griffiths [CG], contradicting our hypothesis. Hence Z' C Z, which yields tie first part of Theorem 6. We now show that MA is finite. For if not, then the same proof as in Theorem 3 shows that a sequence of distinct maps, f1, f2,..., belonging Each to MA, must lift to ZZ(E) for some horizontal component ! of one of these maps must define a component of ZZ(!) by dimension count. But the f1 's generate an infinite number of distinct subvanetics of V by Theorem 3 and the theorem of Kobayashi and Ochiai, since any of their images is either of log-general type or has pseudoample cotangent. This has only a finite number of components. ci contradicts the fact that 1

Theorem 6' is proved in exactly the same way and by induction on e. algebraic reduction of a com6. Concluding remarks. By going pact complex space (see, for example, [UeI), there is no loss of generality in replacing algebraic varieties in this paper by an analytic open subset of compact complex spaces where strictly rational maps are replaced by meromorphic maps which extend to meromorphic maps between the respective completions.

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[B2]

F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR lzv. 13 (1979), 499—555. , Families of curves on a surface of general type, Soviet Math. DokI. 18 (1977),

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Carison and P. Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557—584. sur une surface de type M. Deschamps, Courbes de genre ge5om#trique (d'aprls F. A. BogomolUv), Séminaire Bourbaki 519 (1977/ 78), 1—14. M. Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. 3. Math. 97 (1975), 43—75. , Holomorphic maps to complex tori, Amer. 3. Math. 1UO (1978), 109—113. , Some examples and counterexamples in value distribution theory for several variables, Compositio Math. 30 (1975), 3 17—322. M. Green and P. Griffiths, Two applications of oigebraic geomelty to entire holomorphic mappings, The Chern Symposium 1979, Springer-Verlag. New York, Heidelberg-Berlin, 1980, pp. 4 1—74. R. Hartshorne, Algebraic geometry, Graduate Texts in Math. vol. 52, Springer Verlag,

New York, 1977. H. Himnaka, Resolution of singularities of an algebraic variety over a function field of characteristic zero, Ann. of Math. 79 (1964), 109—326. F. Hirzebruch, Topological methods in algebraic geometry, Grundkhrcn Math. Wiss.,

vol. 131, Springer, Heidelberg, 1966.

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S. litaka, Algebraic geometry, Graduate Texts in Math., vol. 76, Springer Verlag, New York, 1982. , Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA 23

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(1976), 525—544. , On logarithmic Kodaira dimension of algebraic varieties, Complex Analysis and

[II)

Algebraic Geometry, Iwanami, Tokyo, 1977, pp. 175—189. Jouanolou, Ilypersurfaces solutions d'une equation de Pfoqff analytique, Math. Ann. 232 (1978), 239—248.

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S. Kobayashi and T. Ochiai, Meromorphic mappings onto compact complex spaces of genera/type, Invent. Math. 31(1975), 7—16. K. Kodaira, ilolomorphic mappings of polydisk into compact complex manifolds, J. Differential Geometry 6 (1971), 3 3—46. K. Kodaira and T. Spencer, On deformations of complex analytic structures. 1, 11, Ann.

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of Math. 67 (1958), 328—466. S. Lang, Higher dimensional Diophantine problems, Bull. Amer. Math. Soc. 80 (1974),

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159—205. Hyperbolic and Diophantine analysis, BullAnier. Math. Soc. 14( A volume for the Encyclopedia of Mathematics on Diophantine geometry, Yet

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unpublished draft issue. S. Lu, Logarithmic jets and pseudohyperbolicity (to appear).

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S. Lu

, On hyperbolicity and the Green-Gnffizhs conjecture for surfaces (to appear). and S..T. Yau, Holomorphic curves in surfaces of general type, Proc. Nati. Acad.

(Mi)

Sci. U.S.A. 87 (1990), 80—82. M. Maruyama, The theorem of Grauer:-Mulich-Spindler, Math. Ann. 255 (1981), 317— 333. V. Miyaoka, Algebraic surfaces with positive indices, Classification of Algebraic and An-

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alytic Manifolds (Katata Symposium Proc., 1982), Progress Math., vol. 39, Birkhauser, Boston, Basel, Stuttgart, 1983, pp. 281—301. , On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977),

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225—237. 1M21

V. Miyaoka and S. Mon, A numerical criterion for uniruledness, Ann. of Math. 124 (1986), 65—69.

(MM) S. Mori and S. Mukai, The uniruleness of the muduli space of curves of genus 11, Algebraic Geometry Conference (Tokyo-Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer-Verlag, Berlin and New York,, pp. 334-35 3. Mois&on, Resolution theorems for compact complex spaces with a sufficiently large field of meromorphic functions, Izv. Acad. Nauk SSSR Ser. Mat. 31(1967), 1385—1414. [Mum) D. Mumford, Algebraic geometry 1: complex projective varieties, Springer-Verlag, Berlin, Heidelberg, New York, 1976. , Lectures on curves on an algebraic surface, Ann. of Math. Stud., vol. 59, Prince[Mm!) ton Univ. Press, Princeton, 1966. [Ni) .1. Noguchi, Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. 1. 83 (1981), 213—223. , Logarithmic jet spaces and extensions of de Franchis' theorem, Proc. Conference (N2J in Several Complex Variables in Honor of W. S toll, Wiesbaden, 1984. of the family of rational and meromorphic mappings J. Noguchi and 1. Sunada, ENS) into algebraic varieties, Amer. J. Math. 104 (1982), 83—96. T. Ochiai, On holomorphic curves in algebraic varieties with ample irregularity, Invent. (0) Math. 43 (1977), 83—96. [S-Tj M. Schneider and A. Tancredi, Almost-positive vector bundles on projective surfaces, Math. Ann. 280 (1988), 537—547. (Sb.) B. V. Shabat, Distribution of valuer ofholomorphic mappings, Transl. Math. Monographs, vol. 61, Amer. Math. Soc., Providence, 1985. [Sb) G. Stolzenberg, Volumes, limits, and extensions of analytic varieties, Lecture Notes in Math., vol. 19, Springer-Verlag, Berlin and New York, 1976. H. Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology [Tsj) [Mo)

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Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Math., vol. 439, Springer-Verlag, Berlin, Heidelberg and New York, 1975. S-I. Yau, Calabi 's conjecture and some ne'v results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798—1800. , On the Ricci curvature of a compact Kdhler manifold and the complex Monge Ampere equation. I, Comm. Pure Appi. Math. 31(1978), 339—441.

JOHNS HOPKiNS

Proceedinp of Symposia in Pure Mathematks Volume 5211991), Part 2

Aspects of Käbler Geometry on Arithmetic Varieties NGAIMING MOK The unity of the theory of Riemann surfaces is reflected to a large extent by the Uniformization Theorem, which says that any connected Riemann sur, the complex plane C or the face is uniformized by the Riemann sphere upper half-plane X'. These simply-connected Riemann surfaces carry Hermitian metrics of constant positive, zero and negative Gaussian curvature, given by the Fubini-Study metric, the Euclidean metric and the Poincaré metric respectively. Accordingly, we have a trichotomy of Hermitian geometry in one complex variable into the elliptic, the parabolic and the hyperbolic geometries. Most Riemann surfaces are in this sense hyperbolic, the exceptions being given by the Riemann sphere P' , the complex plane C, the punctured

complex plane C and compact complex tori C/L for lattices L. Among the hyperbolic Riemann surfaces a special role is played by those of finite volume with respect to the Poincaré metric. They are isomorphic as complex manifolds to (quasi)projective algebraic curves. In the theory of complex manifolds a lot of work has been done in recent years along lines of thought inspired by the Uniforniization Theorem. Here the model manifolds are the Hermitian (locally) symmetric manifolds. The irreducible ones among them carry Kähler-Einstein metrics of constant positive, zero or negative curvature. Their Riemannian sectional curvatures and hence holomorphic bisectional curvatures are nonnegative, zero and nonpositive respectively. The analogues of P' are given by projective spaces F", the Grassmannians G(p, q), the hyperquadric Q" c and other Hermitian

symmetric manifolds of the compact type. The analogues of C are given by the complex Euclidean space C". The analogues of the upper half-plane, which is conformally equivalent to the unit disc are given by the set of bounded symmetric domains, such as the unit balls B" = { z fi z < 1). The latter are particular examples of bounded symmetric domains defined by 1980 Mathematics Subject Classification (1985 Revision). Primary 53C55, 53C35, 141(99, 32L20.

Research partially supported by NSF grants NSF 8704209, DMS-84-5 1749 (PYIA). This paper is in final form and no version of it will be submitted for publication elsewhere. © 1991 Amencan M.ihetnatical SocIety

0082-0717/91 51.00+5.23 prepage

335

NGAIMING MOK

336

matrix inequalities such as the domain Dptq = {p by q complex matrices Z — Z'Z > O}. There is a duality between Hermitian symmetric manifolds of the compact type and those of the noncompact type X0 (which are equivalently bounded symmetric domains). In such a duality the noncomcan be embedded, by the Borel Embedding Theorem, as pact dual X0 of in such a way that the biholomorphisms of A'0 extend an open subset of is dual to the For instance the unit ball to biholomorphisms of and the domains projective space q are dual to the Grassmannian :

G(p, q).

Unlike the situation of one complex variable, the set of Hermitian (locally) symmetric manifolds of higher dimensions constitute a very special class of

complex manifolds. However, this class is of particular importance partly because of the richness of their geometric/arithmetic structure and partly because they sometimes serve as model manifolds for rather general classes of manifolds. The study of Hermitian (locally) symmetric manifolds subdivides roughly into:

1. Characterization of Hermitian (locally) symmetric manifolds. Riemannian (locally) symmetric manifolds arise in differential geometry as very special manifolds, characterized by the existence of local isometric involutions. For compact Hermitian (locally) symmetric manifolds it is believed that such Hermitian manifolds and/or their underlying complex manifolds can be characterized by topological/curvature conditions. Problems of characterization manifall under the general problem of uniformization of compact folds. 2. Hermitian (locally) symmetric manifolds as model manifolds. Hermitian

(locally) symmetric manifolds can also be regarded as very special examples of a much larger class of Kähler manifolds. For instance, quotients of bounded symmetric domains by torsion-free arithmetic groups of automorphisms can be regarded as special examples of Kähler-Einstein manifolds of finite volume with a negative Einstein constant. In this regard well-known facts of Hermitian (locally) symmetric manifolds motivate the formulation and Sometimes the proofs of structure theorems for wide classes of complete Kähler manifolds. A problem in this direction is to compactify complete Kähler-Einstein manifolds (with negative Einstein constants) of finite volume.

3. Hermitian (locally)symmetric manifolds as individual manifolds. Hermitian (locally) symmetric manifolds are sometimes studied as individual manifolds in their own right. One of the reasons for their importance comes from the fact that they often arise as moduli spaces for certain algebrogeometric problems. Of particular importance are the moduli spaces for polarized abelian varieties possibly with some additional structures. Classical differential-geometric problems of interest include vanishing theorems and

KAHLER GEOMETRY ON ARIThMETIC VARIETIES

337

computation of cohomology groups, holomorphic mappings and representation of fundamental groups in the case of quotients of bounded symmetric domains. For questions related to the characterization of Hermitian (locally) symmetric manifolds, we refer the reader to the survey article of Siu (1] on the uniformization problem in several complex variables. In the 1989 Summer School on Several Complex Variables in Santa Cruz the author delivered a series of lectures on Hermitian (locally) symmetric manifolds with the emphasis on problems related to quotients of bounded symmetric domains of finite volume. The bulk of the lectures was devoted to two topics: a compactiflcation of Kähler-Einstein manifolds of finite volume and Hermitian metric rigidity theorems on arithmetic varieties of rank 2. Since survey articles on the problem of compactification have already appeared [Mok6, Mok7], we will concentrate in the present article on phenomena concerning quotients X of bounded symmetric domains £) of finite volume arising from curvature properties of the canonical Kähler-Einstein metrics, such as 1-lermitian metric rigidity. By the Arithmeticity Theorem of Margulis, for X irreducible

and £ of rank 2, such quotients are obtained from arithmetic lattices. As most of our results will either be for irreducible X of rank 2 or for X of rank one rising from moduli problems, the present article is grosso modo a treatise on the Kithier geometry of arithmetic varieties. Here and henceforth by arithmetic varieties we always mean quotients of bounded symmetric domains by torsion-free arithmetic subgroups, unless otherwise specified. Many interesting moduli problems give rise to noncompact arithmetic varieties. For the study of such arithmetic varieties, it will usually be necessary to use the arithmetic theory of compactifications. The article wll be divided into four chapters. Chapter 1 is on Hermitian metric rigidity theorems on arithmetic varieties of rank 2, which is basic

to the rest of the article. The basic result is that in the locally irreducible case the Kähler-Einstein metric is the unique Hermitian metric of nonpositive curvature in the sense of Griffiths, up to a normalizing constant. Applications

and related results in the rank-one case are also given. In Chapter 2 we study geometric superrigidity theorems of representations of lattices r of rank 2 into isometry groups of Riemannian manifolds of nonpositive sectional curvature in the complexified sense. We prove a generalization of the archimedean case of Margulis' Superrigidity Theorem in case r is

a uniform lattice in a noncompact simple Lie group of rank 2 of the Hermitian type. In Chapter 3 we consider those arithmetic varieties arising as moduli spaces for certain polarized abelian varieties. We prove a finiteness theorem of the Mordell-Weil group (i.e., the group of holomorphic sections) of the associated universal abelian variety. Given any holomorphic family of polarized abelian varieties with the same supplementary structures, we have an associated classifying map. When f is a ramified covering, we also obtain a geometric upper bound for the rank of the Mordell-Weil group in

NGAIMIT4G MOK

terms of the ramification divisor. Chapter 4 deals with algebraic curves on arithmetic varieties. As a somewhat surprising outgrowth of our study of

Mordell-Weil groups we develop a method of proving pinching theorems for modular curves. We give in the case of Siegel modular varieties a very precise pinching theorem by such a method. The underlying theme of all topics discussed in the article is the very rich

geometric structure of arithmetic varieties as reflected by the properties of the canonical Kahler-Einstein metric. The structure of the underlying local symmetric structure is very much captured by the curvature tensor, as the latter gives rise to faithful representations of the maximal compact subgroups of identity components of groups of holomorphic isometrics. There is a dichotomy between the rank-i case and the case of higher rank. In case of rank-i the metric is characterized by the constancy of holomorphic sectional curvatures, giving rise to relations of certain Chern classes in case of complex dimension 2. In case of higher rank the geometry is characterized by the abundance of flats, i.e., totally-geodesic flat submanifolds of real dimension ? 2. A basic observation in the proof of the Hermitian Metric Rigidity Theorem is that this geometric property can also be recaptured by the vanishing of certain characteristic classes. Methodologically, the use of integral formulas arising from topological invariance of Chern numbers permeates the entire discussion. In Chapter 1 on Hermitian metric rigidity the key point is to find vanishing Chern classes through a geometric construction using duality and minimal rational curves on compact duals. Chapter 2 on geometric superrigidity can be regarded as an application of Hermitian metric rigidity. We use the method of harmonic maps and Bochner formulas to obtain pluriharmonic maps into complete Riemannian manifolds nonpositive sectional curvature in the sense. The key fact is that even though the mapping is not holomorphic (the image might not even have a complex structure), the (1,1) part of the pullback of the Riemannian metric is, where nondegenerate, a Kähler metric of nonpositive holomorphic bisectional curvature. Chapter 3 can be divided into two pans, the vanishing theorem for the rank of Mordeil-Weil groups of universal abelian varieties over modular varieties, and the geometric estimate by localization for families obtained from branched coverings. The vanishing theorem is based on some form of infinitesimal Hermitian metric rigidity. On the universal abeian variety, there are canonical locally homogeneous Kahler metrics. By scaling in the vertical direction one obtains then a one-parameter family of such metrics. The existence of a holomorphic section of infinite order then gives rise to a one-parameter family {v1} of Kithier metrics on the base arithmetic variety X, which are of nonpositive bisectional curvature up to an error of order 0(t4). This is the basis of the vanishing theorem for

X of rank> 2. In the rank-i case we use the same set-up but obtain the theorem by the Gauss-Bonnet formula (for curves) and its generalizations (in case of higher dimensions). The geometric estimate is obtained by a

KAHLER GEOMETRY ON ARITHMETIC VARIETIES

339

method of localization analogous to the counting argument in the proof of Siegel's Theorem. Chapter 4 is an outgrowth from an attempt to deal with the general case in the geometric estimate for ranks of Mordell-Weil groups over complex function fields. In the general case one can reduce to classifying maps f: X —. Xi-. which are ramified coverings over the image f(X). In

this case one expects geometric estimates in terms of the ramification of f: X —. f(X) and the second fundamental form of f over the unramified part. We only have a method that deals with small second fundamental forms. This method is the basis of the pinching theorem on Siegel modular varieties. The style of presentation varies from chapter to chapter. Hermitian metric rigidity is treated systematically in the author's monograph (MokSJ. We will hence devote most of Chapter 110 giving a geometric rationale of the meth-

ods of proof, leaving aside most technical details. Many results in Chapters 2—4 are new, a significant part of which was obtained after the lectures in Santa Cruz. In Chapter 2 we will give a proof of geometric archiinedean superrigidity on uniform lattices of the Hermitian type in an illustrative special case. Chapters 3 and 4 can be regarded, on the other hand, as a summary of results obtained by the author on ranks of Mordell-Weil groups over complex function fields, together with an introduction to differential geometry on families of abelian varieties. Ideas of proofs will be indicated, leaving details to future publications. I would like to thank the organizers of the Summer School in Santa Cruz for their invitation and for asking me to write up a survey article on materials related to my lectures. I would also like to thank Corlette, Gromov, Margulis, To, Tsai, Siu and Yeung for discussions related to the contents of the present article, and Silverberg for sending me her reprints on Mordell-Weil groups of generic abelian varieties and for her comments on the first draft of the article. I am particularly thankful to for having raised the question of proving geometric superrigidity in the Hermitian case by the method of harmonic maps, and to Margulis for pointing out to me that the finiteness of the Mordell-Weil groups of universal abelian varieties also follows from vanishing theorems of Raghunathan [RagI] and Margulis [Mar21 in case the base space is irreducible and of rank 2. Chapter I. Hermitian Metric Rigidity

1.1. In this article we are interested in quotients X of bounded symmetric domains fl of finite volume (with respect to Kähler-Einstein metrics). We will always assume implicitly that X = DIr, where F C Aut(fl) is a torsion-free discrete group of holomorphic automorphisms, unless otherwise specified. We study analytic or topological objects associated to such manifolds X, such as holomorphic mappings between them and representations of their fundamental groups. Every result discussed here will in some way be a consequence of the very special properties of the curvature tensor of

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canonical Kahler-Einstein metrics. To start with we treat the subject of metric rigidity, which will be relevant to all later topics. A mairr motivation for Hermitian metric rigidity will be its application to rigidity theorems on holomorphic mappings. To motivate the formulation of Hermitian metric rigidity, consider the very simple situation in the case of compact Riemann surfaces. Let E be an elliptic curve, i.e., analytically C/L for some lattice L, and let R be a compact Riemann surface of genus

2. Then, there is no nonconstant holomorphic mapping from E to B. One way of showing this is to use the Poincaré metric d4 on R (of constant negative Gaussian curvature) and the (flat) Euclidean metric on E. For any nonconstant holomorphic mapping f: E —. R we obtain a possibly degenerate Hermitian metric fdsR on E. Wherever I is f*d4 is of negative Gaussian curvature. When nondegenerate (E, is degenerate one an interpret the curvature as given by negative multiples of the curvature the ô-function. In fact if h is a local expression for form is given by and the degeneracies of h correspond to poles of the subbarmomc function log h. The total Gauss-Bonnet integral of is then a negative number, contradicting with the vanishing of (E, f

the total curvature of (E, d4). To avoid using degenerate Hermitian metrics we may also argue as follows.

Consider the Herinitian metric 0 = 0 can alternatively be +1 x (R, obtained from the Cartesian product (E, which is a Kähler manifold, if one identifles E with the graph of the map (id, I) : E —, E x R. The Gauss equation in the case of Kähler manifolds (S, hi5) -. (X, h) , with curvature tensors R5 and R resp., says that for Q, fi vectors of type (1,0) tangent to the submanifold S. In particular, (E, 0) is of nonpositive Gaussian curvature. At a point where f is unramified, (E, 0) is even of negative curvature due to the negativity of (R, The existence of a nonconstant f: E R yields then immediately a contradiction to the Gauss-Bonnet formula. The simple arguments above can be summarized under the statement that

on E there is, up to scalar multiples, a unique Hermitian metric of nonpositive curvature, namely the flat Euclidean metric. This is a prototype of what we call a metric rigidity theorem. One considers a holomorphic vector bundle V over a (quasi)projective manifold M. Assume that V is endowed with a Hermitian metric h such that the curvature is either nonpositive or

nonnegative. We say that Hermitian metric rigidity holds on (V. h), or that (V, h) is metrically rigid, if h is the only Hermitian metric with the same curvature properties up to normalizing constants. The proper notion of positivity/negativity of the curvature tensor should be one for which one has the curvature nonincreasing property for Hermitian holomorphic subbundles. This notion is provided by positivity/negativity in the sense of Griffiths. The

curvature tensor of (V, h) is given by

where i, f are indexes for

XAHLER GEOMETRY ON ARITHMETIC VARIETIES

fi are indexes for fiber varieties. We say that (V g) is of negative curvature at x M in the sense of Griffiths if base variables and

0). The embedding C M(p, q) is the Harish-Chandra C describes

Embedding. It can be compactified to give G(p, q) in such a way that

C G(p, q) is the Borel Embedding. More precisely, the noncompact semisimple Lie group SU(p, q) acts on with isotropy S(U(p) x U(q)). •q

347


generated by the column vecwith the q-plane in q) on tors of [fl. Then the action of q extends to G(p, q) consisting of those C as SU(p, q) C SL(p + q, C). Let now is invariant under SU(p, q) A fortiori lying over elements of over X = descends to a fiber bundle 9' C so that In the example of the Grassmannian G(p, q) above the procedure for constructing minimal rational curves gives in fact all possible minimal rational curves. It is clear then that SU(p + q, C) acts transitively on the of all minimal rational curves and hence transitively on Douady space Identify Z E

q

In particular, the isotropy group K = S(U(p) x U(q)) at one point It follows then that the noncompact x acts transitively on the fiber over = G0 1K = real form G0 = SU(p, q) acts transitively on is a locally homogeneous holoSU(p, q)/S(U(p) x U(q)). Thus 9' —. morphic fiber bundle. In particular 9' is smooth. The description given generalizesto aboveforthepair /K; of irreducible Hermitian for any pair (Xe, Xe); 10 = G0 /K, =

I

pN is chosen to be the first symmetric manifold, where the embedding canonical isometric embedding according to Nakagawa-Takagi [NT]. This is

characterized by the fact that any minimal rational curve C of X (i.e., [C] is a positive generator of H2(XC, Z) Z) is mapped to a rational line in pN = We will in general call the holomorphic fiber bundle 5° —. thus obtained the characteristic bundle on I. If [a] E 5' we will call a a

I

characteristic vector.

16. We proceed to explain the proof of Theorem 1 by an integral forWrite r : mula on the characteristic bundle 5° for the for the tautological line bundle over canonical projection and (L, endowed with the Hermitian metric derived from a canonical metric g on X. Weequip P(T1) withaKAhierform ii deflnedby ii = —c1(L,

I

I.

U) being the Kähler form of (I, g). Let h be a Hermitian metric on I of nonposilive curvature in the sense of Griffiths. We will show that for any [a] the negative semidefinite Hermitian form = has a q-dimensional null space with q 1. Granted this we will use the integral formula (*)

J[_ci(L,

*)J2

=

f

= o.

proof that h = cg can then be obtained from a modification of the second proof in the case of irreducible quotients of the polydisc given in The

§ 1.3, in which we used Moore's Ergodicty Theorem. To justify (*) we need to show that q 1 and that the integrand [—c1 (L, A vanishes

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identically on Wolf [Wolfl.

For this we will use the Polydisc/Polysphere Theorem (cf.

Letnow pair of irreducible Hermitian symmetric manifold of rank r 2. Embed A'0 into using the Borel Embedding. Then, the Polydisc/Polysphere Theorem asserts that (i) (Xe, contains a complex r-dimensional polysphere (Pt totally geodesic with respect to (ii) all such polyspheres are S equivalent under the action of (iii) the intersection of each S with A'0 gives a complex r-dimensional polydisc D = Ei' totally geodesic with respect to g0; and (iv) all such polydiscs D are equivalent under the action of G0. D (resp. S) will be cafled a distinguished polydisc (resp. polysphere). For c M(p, q) c G(p, q), q p, an example of such a polydisc is given diag(z1,... by D = {Z = [Z' OJ€ M(p,q) < I for I p}. From the description of minimal rational curves on G(p, q) given above, one deduces that at any Z e a tangent vector X is a characteristic vector if and only if X is a matrix of rank I. Thus, in the example the distinguished polydisc D is a product of minimal discs (i.e., 1

intersections of minimal rational curves with This is, in general, true for bounded symmetric domains More precisely, given any x E and given any characteristic vector a at x, there are (many) distinguished polydiscs D such that a is tangent to one of the dlrect factors of D. As D is totally geodesic, Ha has a kernel of dimension r — 1, showing that q I when c is of rank 2. To justify the integral formula (*) it suffices to show that the eigenspaces of c1 (L, are tangent to .9. Fix x E and [a] E passing through Let D be a distinguished polydisc in x such that a is tangent to one direct factor of D. Let be the zeroeigenspace of in = q. By the curvature formula for c1 (L, dime §1.3 it is immediate that c1(L, has a q-dimensional zero-eigenspace We such that projects onto via it: C —. call the inverse mapping the lifting map at [a]. The question is —' is generated as a complex linear space whether Wa is tangent to .S'. by the nullspaces as D varies over all distin:= of guished polydiscs tangent to [a]. In case of distinguished polydiscs D it is clear that the lifting of is tangent to PTD n at [a) (cf. § 1.3). By varying D one sees that the lifting of is tangent to This allows us to assert the integral formula (*) and hence to deduce the Hermitian Metric Ridigity Theorem in case I = is compact. When I is noncompact but (X, g+h) is of finite volume, replacing g by g + h in (*) one can still justify the integral formula to establish Theorem 1.

This is the case when h is Kahler. In fact, if X is a quasiprojective

variety admitting a (possibly singular) compactification X such that ? — I is of codimension > 2 in then any Kähler metric on X is of finite volume (cf. Mok [MokS, p. 169]), as a consequence of current extension


349

theorems. On the other hand if h is only assumed to be a Hermitian metric of nonpositive curvature in the sense of Griffiths, it is not a priori clear that (X, g + h) is of finite volume. In To (To], however, it is shown using toroidal compactifications that h is of at most logarithmic growth at infinity. Using this, one can still justify the integral formula (*) to prove Theorem 1 in case of noncompact quotients I = 1.7.

We state here some applications of Theorem I and its proof, starting

with:

Let I be a quotient of an irreducible bounded symmetric of rank 2 and of finite volume in the canonical metric g. Let (N, h) be any Kähler manifold of nonpositive bisectional curvature and let f: X N be a /zo/omorphic tnapping which is not totally degenerate. Then, f is necessarily a totally geodesic isometric immersion up to a normalizing THEOREM 2.

domain

constant.

The fact that f is an isometry follows immediately from Theorem 1 since (X, g + f h) is a KAhler metric of nonpositive bisectional curvature. (For this part we need not assume that (N, h) is Kähler.) Identify (X, g) locally as a submanifold of (N, h) and write a for the second fundamental form. By comparing the curvature tensors RA' with K" it follows from the Gauss equation that a(a, fi) = 0 whenever = 0. By the polarization argument of [Mok3, p. 130ff.] it follows that a 0, so that (X, g) '-' (N, h) is totally geodesic. From the method of proof we also obtain the following improvement of a theorem of Yang [Yan]. Let be a bounded symmetric domain. Suppose admits a complete Hermizian metric of bounded torsion and of negative bisectional curvature bounded between two negative constants. THEOREM 3 (Mok (Mok6, p. 127]).

Then,

is of rank 1.

In the same way that one proves the vanishing of the first Beth number from Hermitian metric rigidity, one can also prove in this way a vanishing theorem for F(X, V) for certain Hermitian locally homogeneous vector bundles V. We have more precisely a vanishing theorem stronger than classical results in case X is noncompact, as we do not impose any growth conditions on holomorphic sections. THEOREM 4 (Mok (Mok6, p. 212]). Let X be a quotient of an irreducible bounded symmetric domain of rank 2 and of finite volume in the canonical metric. Suppose (V, s) is an irreducible nontrivial locally homogeneous

Hermitian vector bundle on I, and F is a locally flat Hermitian vector bundle on X such that F(X, V ® F) 0. 'Then, (V, s) is of strictly positive curvature in the sense of Griffiths.

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Another application of Hermitian metric rigidity theorems is to study rigidity phenomena for representations of lattices of rank 2 in semisimple Lie groups of Hermitian type into isometry groups of Riemannian manifolds of nonpositive curvature. This will be taken up in Chapter 2. Finally in relation to the rigidity theorem for holomorphic mappings (Theorem 2) for the case of rank 2, in the rank-I case there is a restricted form such a theorem. which we state here. By a complex hyperbolic space form

we mean a quotient B'1/f with F torsion-free.

(X, g) and (Y, Ii) be complex livX — I. perbolic space forms. Suppose X is compact and dime V 2 be a holomorphic immersion. Then, f is necessarily a iota//v Y Let f: X — THEOREM 5 (Cao-Mok [CMI).

Let

geodesic isometric imniersion.

Write n dime X and write N for the normal bundle of the immersion f. By considering differences of Curvatures we obtain a nonnegative (1,1)form p which vanishes if and only if f: A' Y is totally geodesic. From the vanishing of one obtains p" 0 from standard cohomological considerations. We deduced p 0 by studying the foliation on X arising from the nullspaces of a. Chapter II. Geometric Archimedean Superrigidity in the Hermitlan Case

2.1. In this chapter we apply Hermitian metric rigidity to study representations of irreducible lattices of rank 2 of the Hermitian type into isometry groups of complete Riemannian manifolds. Let

be a bounded

symmetric domain of rank 2 and F c Aut(Q) be a lattice. Let (N, h) be a simply-connected complete Riemannian manifold of nonpositive sectional curvature. Let D : r —+ Isom(N, h) be a representation of r into the isometry group of (N. h). A Continuous mapping f: —, N is said

to be P-equivariant if and only if f(yz) =

for all y E F. We say that f :

LI —+

for all z E LI and

N is a (degenerate) totally geodesic

f'

mapping if there is a (proper) direct factor LI' of LI such that f = o ir for it : LI —' LI' the canonical projection and for some f' : LI' N which is a totally geodesic embedding up to normalizing constants. We look for conditions on C1 to guarantee the existence of a D-equivariant totally geodesic mapping f: LI — N. The motivation for proving such a theorem comes from the arch imedean case of Margulis' Superrigidity Theorem, which gives as a consequence: THEOREM (special case of Margulis [Marl]). Let G be a semisimple Lie group with no compact factors, F c G be an irreducible lattice of rank 2. Let G' be a semisimple Lie group without compact factors, and 1: F a group homomorphism such that the image 1(fl is Zariski-dense in G'. Then, .1 extends to a group homomorphism D: G —. G' between Lie groups.

KAHLER GEOMETRY ON ARITHMETiC VARIETIES

As a consequence, for any maxima/ compact subgroups K of G there exists a maximal compact K' of G' for which c K', and a smooth pnap f: G/K G'/K' between the associated Riemannian syminetnc manifolds such that f is a total/v geodesic mapping.

Here G/K and G'/K' are endowed with G- resp. G'-invariant metrics arising from Killing forms. Reducing to the case when 1' is injective we can choose K' to be the fixed point set of an extension of the Cartan involution

0 of (G, K) to G' (cf. Satake [Sa3, p. 15]). As D preserves the Cartan decomposition. f is a totally geodesic mapping. We note however that there exist injective homomorphisms 'F: SL(2, —. SL(n, R) such that 'F maps

thc orthogonal groups SO(2) into SOn) but such that the corresponding mapping f : SL(2, R)/S02) —. SL(n, R)/SOn) is (an isometry but) not totally geodesic (cf. Satake [Sa3, p. 89ff.]).

We look now for a generalization of the theorem above in case G/K = is a bounded symmetric domain, using the method of harmonic maps, by replacing G'/K' by some complete Riemannian manifold. There are two parts to the problem. First, one looks for the existence of harmonic representatives for maps from G/K = Secondly, one looks for curvature conditions on the target manifold (N, h) to guarantee that the harmonic representative is actually a totally geodesic mapping. As in Chapter 1, by a quotient A' = of a bounded symmetric domain we will always assume implicitly that r C is torsion-free. On the other hand a lattice in Aut(Q) may have torsion elements.

2.2. The origin of using harmonic maps for proving rigidity results was the following Strong Rigidity Theorem for Kahler manifolds of Siu [Siul, Siu2]. THEOREM ([Slul. Siu2]).

symmetric domain

Let X be a quotient of an irreducible bounded

of rank 2. Let Al be

a compact Kãhler ntanifold

homotopic to A'. Then, M is biholomorphic or conjugate-biholomorphic to

I.

The Strong Rigidity Theorem of Siu was later generalized to cover the case of irreducible but locally reducible quotients, such as irreducible quotients of the polydisc, by the work of Jost-Yau [JY1, JY2J and Mok [Moki]. Extensions to certain cases where X is only of finite volume were also obtained by [JY3J and Mok [MokS]. In Siu's theorem, the existence of harmonic maps homotopic to a gwen map is guaranteed by the general existence theorem of Eells-Sampson [ES], which asserts that any smooth map from a compact manifold into a compact Riemannian manifold of nonpositive curvature can be deformed by heat flow to a harmonic mapping. The key point of Siu 's theorem was to prove that in the special case considered, the harmonic map is actually either holomorphic or antiholomorphic. The fundamental tool for this is the following formula.

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352

formula. Let (X, g), with Kahler form w, and The manifolds. Let f: X —. N be a smooth map(N, h) be compact on the complexified tangent • ping. Consider the differential df: —, bundles. Decompose the complexified tangent bundles into the (1,0)- and (0,1)-components with respect to the almost complex structures of X and N. Decompose df into Lif + + Of + Of correspondingly by projection, (0, 1)-form on X, etc. Denote by V the is a so that * 0) .0) For induced by the Hermitian connection connection of f (0,1)-form denote by D0f the covariant exterior the f derivative with respect to the Riemannian connection on X and the con-

f

nection v on p

.0)

(N, h) and denote by (

for the Riemannian curvature tensor of inner products induced by h. Then, we have

Write ,

)

OO(g, OfAOf)N = (RN,

DOJADO7).

now f: (X, g) —' (N, h) is a harmonic map. Taking wedge prodwhere m is the complex dimension of X and denoting by norms arising from g and h, we have by Stokes' Theorem

Suppose

uct with

+(RN,

=0.

The integrated identity Is usually referred to as the formula. When N = 1/1' is a quotient of a bounded symmetric domain by a torsion-free group r of automorphisms and h is the (canonical) KählerEinstein metric, the curvature term in the integral formula can be shown to be nonnegative. One concludes hence that the curvature term vanishes identically. The proof of the Strong Rigidity Theorem of Siu is obtained by using the vanishing of the curvature term and the fact that the harmonic map I is of maximal rank at some point to the force the identical vanishing of or Of. One does not actually need the fact that f is a diffeomorphism at some point. It suffices to assume that the real rank of f is at least 2p(Q) + I, where p(c) is an integer depending only on the Lie algebra of Aut(fl). In we have the case when N is a quotient of the complex ball 1, of real rank at least 3 so that any harmonic map f I —' N = is either holomorphic or antiholomorphic. The curvature condition which makes the formula useful is a condition stronger than the nonpositivity of Riemannian sectional curvatures. Let (Roki) be the curvature tensor of (N, h) in terms of holomorphic coordinates. We adopt the convention that for Raji < 0 in caste of quotients of the ball. In tSlul] the complex n-dimensional KAhier manifold (N, h) is said to be of strongly nonpositive curvature if and only if at

z,jk,1

353

KAHLER GEOMETRY ON ARIThMETIC VARIETIES

at every point of N. This for any complex square n-by-n matrix notion is equivalent to saying that the Hermitian holomorphic tangent bundle h) is of nonpositive curvature in the dual sense of Nakano (cf. ESiu3, p. formula is 74 if.]). The curvature term appearing in the positive or zero if (N, h) is of strongly nonpositive curvature. In case (N, h) is Hennitian locally symmetric it follows from expressing the curvature tensor in terms of Lie brackets and the Killing form that (N, h) is always of strongly nonpositive curvature. To be more prccise, what one needs is actually the weaker curvature condition (A'B7

—

CD))

AIBk

—

C!Dk

i,j,k,l for any complex column n-vectors (A1), etc. In ISiul, Siu2] a compact KAhler manifold (N, h) is said to be of adequately nonpositive curvature at x E N if and only if the last curvature condition is satisfied.

formula lies in the absence of Tile strength of the the curvature of the domain manifold (X, g). There is a second method of proving the formula by using the Hodge *-operator which explains the disappearance of the curvature tensor of (X, g) by analogy to the theory of harmonic forms (cf. (Siu3, p. 78ff.]). One regards *Of as a "harmonic" (0, n — 1)-form. The formula can f T,, 0 be regarded as an analogue of the V-Bochner-Kodaira formula in the theory of harmonic forms, which yields the Kodaira-Nakano Vanishing Theorem H"(X, E 0 Kx) = 0 for k > 0 whenever E is a Hermitian holomorphic vector bundle of positive curvature in the sense of Nakano. 2.3. There is a generalization of the formula for harmonic maps from compact Kähler manifolds into Riemannian manifolds, obtained by Sampson [Sam]. Let f : (X, g) —, (N, h) be such a map. We no longer have a decomposition of the complexified tangent bundle can still be decomposed as df = OF + where However df:

OF is a

(1, 0)-form on X, etc. We write deliberately OF

for the first component so as to distinguish it from the notation Of in case (N, h) is KAher. In the latter situation our notation gives OF = Of + Of. Write for the complexification of the curvature tensor on TN and define in a way analogous to the KiIhler case. Sampson applied the divergence

theorem to q

*OF to get

=0. In case (N, h) is Kähler Sampson's formula is the same as the sum of the

formulas applied to N and to the conjugate complex manifold N' (so that Of becomes Of, etc.) since the curvature tensor of (N, h) is of type (1,1). An alternative derivation of Sampson's formula is

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A 0f)N in A ÔF)N, in place of the original derivation of Siu 's formula. Sampson's formula shows that the curvature term is positive or zero provided that (N, h) is of nonpositive sectional curvature in the complexified sense, i.e., RN(V, W;W,V)0,

given by computing

for any complexified tangent vectors V, W at any x E N. This notion agrees with the notion of adequately nonpositive curvature in the Kähler case, where one needs

0, for complex column n-vectors (A'), etc., which is the same as assuming ..j i — I -.1 — I N R 0 for tangent vectors of ® — ® ® — 0 , type (1,0). Writing V = + and W and for the curvature +

operatorof (N,h), RN(V,

in view of the fact that the curvature tensor on a Kähler manifold is of type

2.4. Consider now a complete simply-connected Riemannian manifold (N, h) of nonpositive sectional curvature. Let N be the Martin compactification of N obtained by adjoining a sphere at infinity corresponding to equivalence classes of geodesics on N. Then, an isometry on (N, h) extends to a homeomorphism on N. Let Q be a bounded symmetric domain

of rank 2 and T C

A sufficient condition for the existence of c1-equivariant harmonic maps is given by:

THEOREM (Corlette [CorlJ, Labourie [La], and Jost-Yau [JY]). Let (M, g) be a Riemannian manifold and T c Isom(M, g) be a discrete subgroup such that MIT is compact. Let (N, h) be a complete simply-connected manifold T —p Isom(N, h) induces with nonposizive sectional curvature. Suppose an action at infinity of N which has no fixed points. Then there exists a (N, h). CD-equivariant harmonic mapping f: (M, g)

In the special case when (N, h) is a Riemannian symmetric manifold of the noncompact type, so that Isom(N, h) = G is a seinisimple Lie group with no compact factors, the condition that 1(T) has no common fixed point at infinity is guaranteed if we assume that CD(T) is Zariski-dense in G, as in the archimedean case of Margulis' Superrigidity Theorem. The theorem above was also generalized to the case when (M, g) is complete and MIT is noncompact, provided that there exists some 'D-equivariant : M —, N with finite Dinchiet energy. Since there are still mapping technical difficulties with finding such maps in some cases of quotients of bounded symmetric domains of finite volume, we will only discuss the case of cocompact lattices in this chapter.

KAMLER GEOMETRY ON ARITHMETIC VARIETIES

2.5. We are now ready to state the main theorem of this chapter. Let ci be an irreducible bounded symmetric domain of rank Aut(ci) be a cocompact lattice. Let (N, h) be a Riemannian manifold of nonpositive sectional curvature in the complexified sense and THEOREM 1.

2 and r c

Isom(N, h) be a representation such that the induced r action on the sphere at infinity of N has no fixed point. Then, there exists a totally geodesic mapping F : ci N.

We want to show that a harmonic representative f : ci —. N is a totally geodesic mapping. Unlike most problems related to strong rigidity, the structure of the curvature tensor is now imposed on the domain manifold. An additional ingredient exploiting this special feature comes into play. If (N, h) is Kähler and f is either holomorphic or antiholomorphic, then the Hermitian Metric Rigidity Theorem of Chapter 1 implies readily that f is a totally geodesic mapping. However, the only known method of deriving holomorphicity or antiholomorphicity exploits curvature properties of the target manifold and the real rank of the harmonic map f (cf. There exists however totally geodesic embeddings of ci into some bounded symmetric domain which are neither holomorphic nor antiholomorphic. Thus, we cannot expect to exploit the special curvature properties of the target manifold to get superrigidity. After obtaining the harmonic 4'-equivariant mapping 1: ci —' N for the proof of Theorem 1 we may work on n,r' = X for some torsion-free subgroup r' c of finite index. Extend Ii to a Hermitian metric on Let U be the (1,1)-part of f h. Then 0 is a possibly degenerate Herniitian metric on invariant under r'. We will use the same symbols g and 0 for induced metrics on X. By the Hermitian Metric Rigidity Theorem the proof of Theorem I here can be essentially reduced to showing that on X the Hermitian metric g +0 is of nonpositive curvature in the sense of (iriffiths.

r

More precisely: PROPOsITIoN 1. In the notation of Theorem 1 denote by 0 the (1, 1)part of f h on ci. Then 0 is d-closed and the Kah!er meirw g + 0 is of

nonposilive bisectional curvature.

2.6. We give here a proof in the important special case when (1) (N, h) is Kàhler and (ii) (N, h) is of strongly nonpositive curvature. In this case the proof is most transparent, bringing us closest to the ideas of Hermitian metric rigidity in Chapter 1. The starting point is the idea of [Siu3J that in some cases the pullback of the holomorphic tangent bundle can be given a complex structure even if f: ci N is neither holomorphic nor antiholomorphic. A proof was written for the case when N is the polydisc. The same proof can be modified to give: PROPOSITION 2.

Let (X, g) be a complete Kãhler

a torsion-free discrete group of holomorphic isometries such that

NGAIMING MOK

M = x/r' is compact. Let (N, h) be a Kähler mantfold of strongly nonpositive curvature. Let c1: r Isom(N, h) be a group homomorphism and let f: (X, g) (N, h) be a 4'-equi variant harmonic map. Then, the smooth Hermitian vector bundle (f Tb,, fh), endowed with the connection V arising from the Her,nitian connection of (TN, h), can be endowed holomorphic structures in such a way that V is the (unique) Hermitian connection. Furthermore the f TN-valued (1, 0)-form Of is a holomorphic section of T.ef TN when the second tensor factor is endowed with this holomorphic structure. In

a similar way one can endow f T with a holomorphic structure in such a way that o7 is section of T ® f7.

Given Proposition I, we can consider g + 0 as the Hermitian metric obtained from the holomorphic bundle map id +01 Of PROOF OF PROPOSITION 2. The only point that requires a proof is the

statement on integrability. Fix P E X, Q = f(P) on N. Let (z(k) resp. (wt) be local holomorphic coordinates at P resp. One can assume that

0. Write h.7(Q) = and for the = Riemann-Christoffel symbols of (N, h) and (RI;ki) for its curvature tensor. Write e' = f(8/Ow') near P and write s Es1e1 for a smooth section of f TN in a neighborhood of P. We have by definition (using the Einstein convention) V—s'

=

To verify the integrability by the criterion of Newlander-Nirenberg it suffices

02s'

= — —

As

for any pair of indices (fi, y). We have

=

to check that

02s' —

+

ía

+

(0

f)) \Of' Of' k +(— of 0 \ Of' —-= 0,

= 0 one deduces

P I

= 8787s

k

\0w, J j

J/

I

=

giving at

S - RJlkiOifi (9711k

so that (*)

V7V7s'

- V7V7s'

-

From ihe 0. formula one as (RN Since (N, h) is of strongly nonpositive curvature it follows that for any fi, y the 2-tensor

-

8771) 0

KAHLER GEOMETRY ON ARITHMETIC VARiETIES

lies in the kernel of the curvature operator. In particular, we obtain from (*) V is the Hermitian connection of an that * integrable almost complex structure on f TN. A modification of the proof also yields the analogue of Proposition I for the case when (N, h) is a Riemannian manifold with nonpositive curvature operator. In this case one works only with (See Carleson-Toledo [Cr]). We are going to compute the curvatures of f TN and f We have:

f

PRorosTIoN 3. Let P e X and Q = f(P) E N. We write U' E (JTN)p * for u. Write 6' and 6" for the curvatures TN and f 7 with respect to restrictions h' and h" of fh. Let a E and write 0,!

f

f

Of(8/Oz°),

etc. Then,

and

U'; a,

Since (fTN,

—8(u', ü'; a,

h") are dual to each other as Hermitian holomorphic vector bundles it suffices to establish the first formula. PRooF.

h') and

Let U and W be open neighborhoods of P X and Q E N. Choose local holomorphic coordinates (zn) resp. (w) on U resp. W as before. Let (a) be a holomorphic basis of fTN over U. Write e. = f*(0/Owl) U. We may assume that

j

(a) later. For any a,

Further restrictions will be placed on

= we have

ôs'

(3)

=0,

where

I so that k

k

VkCI —

._

k,!

k

Write

= h'(a,, a). Then

8'-

—

— —

+ v-'

a z0

We have ti,- =

p,q

U

ft

so that k

OS,

—

k

k k

a

NGAIMING MOK

From (1) and (3)

=

+

(4)

0

at P.

so that Os'

andhence

(5)

We may now further assume that 0s/0z0(P) = 0 by making a quadratic holomorphic change of coordinates at P. With this choice of (o,) we see from (5) that in fact (o) is a local basis for [TN adapted to h' at P. We have 02 OzaOYfi

—

— —

(6)

—2 Re

—

k7 k15i5

ç 1,p,q

8

2k S,

2, k

—

'

O2hki Ow P

q

8J I

'

Oza

J

= (7)

It remains to calculate the first term in (7) at P. We have from (4) at P rk vS 8 I2\/

r-'ki

I

k

a

—

i,k,1

so That —

0

9\ —

p.k,I8P80OWP

Putting (9) into (7) and putting i = j,

= R(e1,ë1; 00f,

—

ki —

p,k.IO7POZQ

kp17

= fi, we have:

ë,;

—2R(e1, ë1;

(10)

from which Proposition 3 follows. PRooF OF PROPOSITION 1 IN A SPECIAL CASE. We are now ready to prove

Proposition 1 of §2.4 for the special case when the target manifold (N, h)

359


is Kähler and of strongly nonpositive curvature. Use local holomorphic co-

ordinates (z°) and (w') as above. Recall that 0 is the (1,1)-part of f*h. where

We have

0afi- ——

h

-

'i Oza

+

h

$

-

"Oz a

fi

We have at P Oz

Oza OzvOY

— V

a?

+

—

z

L. Oz aOz y

fi

fi

The last two terms vanish since D8f T58f 0, while the first two terms are symmetric in (a, y). It follows immediately that dO 0 and, hence, that g + 8 is a Kahler metric on X. (We remark in passing that the same proof shows that the (2,0)-part ii of fh is a holomorphic tensor.) To prove Proposition lit remains to show that (X, g+0) is of nonpositive bisectional curvature. Consider the bundle map 'P := TN$ Proposition 2, 'V is a holomorphic bundle map with respect to By f TN. the holomorphic structures of f * TN and f 7 defined there. If we endow with the Hermitian metric gEBh'EBh" =gafh, then g +0 = 'Jt(g fh). Denote by S and T the curvature tensor of (X, g) and (X, g + 0) resp. From the Gauss equation we have (3)

(4)

C=

Write A = Oafs B = (5)

and D = +

We can rewrite (4) as —

In the special case when a = fi we have A = B and C = D so that (6)

= Q(A®1—

C

( 0,

where Q is the curvature operator on (N, h). It follows immediately that (X, g + 0) is of holomorphic sectional curvature. For holomorphic bisectional curvature the inequality (5) is not strong enough. However, from Siu 's formula we have on the other hand that =

lies in the kernel of the curvature operator of (N, h). It follows that (7)

—

+

=0,

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360

and hence

+

—

—

(8)

The proof of Proposition 1 for the special case is completed. We proceed to prove Theorem 1 in the special case when (N, h) is Kähler

and of strongly nonpositive curvature. Let a be the second fundamental

formoftheholomorphicbundlemap 'F: 'F— id ô7. Choose complex geodesic coordinates (f) at P and (w') at Q = f(P) as above. From the proof of Proposition 1 in the special case and from the Hermitian Metric Rigidity Theorem of Chapter 1 we have immediately c(a, P(C)) = 0 for any characteristic vector a and for any C such that (a, C) is a zero of bisectional curvatures. Extend a and to a

neighborhood as a constant vector field with respect to (f). We compute a(a, '1'(C)) as follows. Write V for the Hermitian connection on any of the holomorphic bundles involved. We have at P = V0(C,

(1)

Since

is

01f) =

complex geodesic at P, we have V0C(P)

0. On the other

hand +

=

atP

=

since (zr') is complex geodesic and 15Sf

0. We have a similar formula for For the second fundamental form a of the embedding 'F we know that a(a, 'Y(C))(.P) = 0 if and only if = lies on (0, V0(Ocf), The latter consists of vectors of the form (fi, Oaf, From a(a, 'F(C)) 0 and (1) we see that for

any j, 021j

(2)

05(P) = 55(P) = 0.

From 150f(P)=150f(P)= 0,

52) —

—

for any p, i'. From (2) by polarization (as in the Hermitian Metric Rigidity Theorem) it follows that = 0 for any ji, u. Combin-

ing this with (3) we have immediately V(df)

0 on X (since P

is


arbitrary). This is equivalent to saying that the image of f is totally geodesic in (N, h). From Proposition I and the Hermitian Metric Rigidity Theorem Since we only have g +0 = Const. g, so that 0 is a canonical metric on 8 is only the (1, 1)-part of fh, to prove that f: (Q, g) —+ (N, /i) is an isometry up to a normalizing constant it remains to show that the (2,0)-part

v of f h vanishes. Here zi— —. L.1

0

Oz ft

0 Oz ft

)

From (2) and (3) we deduce that Vt'(P) 0 and hence that ii is a parallel symmetric (2,0)-tensor on g). But such tensors must vanish on proving that f: (11, g) —' (N, h) is an isometry.

2.7. In the general case, where we only assumed that (N, h) is of nonsense, in the absence of integrability of positive curvature in the (as in §2.6, Proposition 2) Theorem 1 can only be obtained by direct computation (cf. tMok9]). The curvature formula must agree with §2.5 in the special case when (N, h) is Kähler and of strongly nonpositive curvature. Recall that in this case writing A = B = 8sf, C = 8sf, and D = we have

+

—

—

(1)

Writing the Kähler case Q is of type (1,1), we may also write (2)

This gives an indication that the latter should be the curvature inequality in the general Riemannian case. In place of the inequality in (2) we can also write an equality by introducing the second fundamental form a of the bundle mapping 'P. Let denote the orthogonal projection on to the orthogonal complement of the smooth subbundie Im('P). Then, we have in the special case of §2.6 the curvature formula (3)

+R(00F,

8,F,

This formula is in fact true in the general case by direct computation. Using

(3) ii is obvious that the proof in §2.6 generalizes to give a proof of Theorem

2.8. We have always been formulating our result for irreducible bounded

symmeinc domains fl of rank 2. It is evident from the discussion that the same applies to irreducible lattices of possibly reducible provided that each irreducible factor of is of rank 2. in case some factors are of rank I, the statement Theorem 1 has to be modified, since there exists

362

NGAIMING MOK

isometric embeddings SL(2, R)/SO(2)

SL(n, ]R)/SO(n) which are not

totally geodesic.

Recently Corlette [Cor] obtained a geometric superrigidity for lattices of certain Riemannian symmetric manifolds (X, g) of the noncompact type. His results apply to those , g) carrying parallel forms satisfying certain properties. He did this by developing Bochner formulas in which plays the role of the Kahler form w in Sampson's formula. A remarkable fact about his result is that it also applies to the case of rank 1 spaces, yielding in particular the archimedean superrigidity of any lattice in Sp(n, I) (corresponding to (corresponding to the Cayley the quaternionic hyperbolic space) and plane). We have: THEOREM (Corlette [Cor, Theorem (3.2)1). Let F c Sp(n, 1), n 2, (resp. be a lattice. Write (X, g) for the corresponding quaternionic hyperbolic space (resp. the Cayley plane). Let (N, h) be a complete Riemannian manifold with nonpositive curvature operator. Let 1': I' —. Isom(N, h)

be a representation such that the induced F-action on the sphere at infinity N has no fixed points. Then, there exists a (D-equivariant totally geodesic

mapping f:X—N. Here the result applies to nonuniform lattices F because in this case one can easily construct harmonic maps with finite Dirichiet energy on the quotient spaces. The theorem applies equally to quaternionic symmetric spaces of higher rank. There is a difference in the curvature assumption on (N, h) from that of our Theorem 1. It is interesting to know if the result of Corlette persists when (N, h) is only assumed to be of nonpositive sectional curvature in the complexified sense. According to Gromov, our technique of proving geometric superngidity can be generalized to many Riemannian symmetric manifolds of the noncompact type and of rank 2 by considering families of totally geodesic Hermitian symmetric submanifolds of rank 2. Such foliation techniques were developed in Gromov [Cr0]. Together with the results of Corlette one might conjecture: CONJECTURE. Let (X, g) be an irreducible Riemanman symmetric manifold other than a real hyperbolic space or a complex hyperbolic space; and F c Isom(X, g) be a lattice. Let (N, h) be a complete Riemannian manifold of nonpositive sectional curvature in the complexified sense. Let F Isom(N, h) be a representation such that the induced F-action on the sphere at infinity of N has no fixed points. Then, there exists a 'D-equivariant totally geodesic mapping f: X —. N. Chapter 3. Ranks of Mordell-Weil Groups over Complex Function Fields

3.1. Arithmetic varieties are important partly because many of them parametrize algebro-geometric objects. Among these, the most important ones are possibly the moduli space of (special) families of polarized abelian van-


363

eties. We start with the case of 1-dimensional compact complex tori C/L for

an arbitrary lattice L in C. Given a lattice L in C one has the Weierstral3 p-function defined by p(z) = E'[(z — w)2 — where the summation is over nonzero lattice points w. p(z) is meromorphic and invariant under translation by lattice points in L. The induced meromorphic mapping (p. p') on C/L extends to an embedding of CJL into P2, realizing C/L as an algebraic plane curve of degree three, given in nonhomogeneous coordinates by the normalized WeierstraB equation y2 = x3 + A(r)x + B(r) for

some constants 4(r) and B(t), as a consequence of functional equations satisfied by p and its derivative p'. Henceforth we will also call C/L an elliptic curve. Denote by W the upper half-plane {r C Im r > O}. Up to a complex linear isomorphism of C every lattice L C C is of the form Z + Zr for some r A'. Write E(r) = C/Z + Zr. Any two such elliptic curves E(r) and E(r') are isomorphic if and only if there is some E SL(2, Z) such that r' = (ar + b)/(cr + d). Thus, the isomorphism classes of elliptic curves are parametrized by K/ SL(2, Z). Here SL(2, Z) acts properly discontinuously on A'. The kernel of the action is {± 1 } and the quotient group P SL(2, Z) contains torsion elements. There is a bob+ 27B2) on A' which is invariant morphic function j = under the action of . Z). It is called the f-invariant for the elliptic curve E = E(r). The mapping j : Z/ SL(2, Z) —. C is a bijection, thus endowing Y/ SL(2, Z) with the structure of the complex plane. SL(2, Z) contains many torsion-free discrete subgroups of finite index. For instance, the principal congruence subgroups T(n) are defined by the extra congruence relation ía bl 11 01 modn. Ic dj [0 3, r(n) is torsion-free. The quotient space Z/F(n) is in one-toFor n one correspondence with the space of elliptic curves with level n-structure,

i.e., an element of A'/f'(n) corresponds to an elliptic curve E together with an explicit isomorphism of the set of n-torsion points with the group (Z/nZ)2. For F C SL(2, Z) torsion-free and of finite index one can construct a 44universal" family of elliptic curves parameterized by the modular curve Xr := A"/F. To achieve this consider the action of ,

on

[;J)

ESL(2,R)XZ2

x C defined by (r, w) —, ((ar + b)/(cr + d), w/(cr + d) + ar' + fi).

Then r x Z2, as a semidirect product, acts property discontinuously without fixed points on A' x C in such a way that the natural projection i = A' x cir x Z2 —' A'/F = Xi.. realizes as a regular family of elliptic curves over the modular curve Xr and such that the fiber E(r)

NGAIM!NG MOK

364

be the unique smooth projective r mod F E Y/F = Xr. Let can be compactified to an algebraic surcompactification of Xr. Then, In this case such that it face —, Xr extends to it '—' is obtained from (cf. by adding rational curves to infinity points of

for x

:

:

and Kodaira (I(o]). The construction of of SL(2, Z) of finite index provided that

also works for subgroups F F (although may have singularities). This requirement is due to the fact that { acts J trivially on Y but induces an involution y —y on elliptic curves. 3.2. In higher dimensions one considers moduli spaces of abelian vail-

eties A, i.e., compact algebraic complex n-dimensional tori T = Unlike the case of n = I, the assumption that A can be embedded into imposes additional conditions. The construction some projective space of the WeierstraB p-function no longer works. Instead one constructs theta functions on C which are holomorphic functions transforming under translation by lattice points co E L according to the rules O(z+o)) where Q(z; a,) are holomorphic quadratic polynomials in z satisfying certain compatibility conditions. Equivalently, they give rise to holomorphic sections of certain holomorphic line bundles A over T. Classically the existence problem for theta functions was solved by Riemann, culminating in the famous Riemann relations. One starts with a Riemann form, i.e., a real alter-

nating bilinear form E(.,.) on C taking integral values on L x L such that E(ix, iy) = E(x, y). Associated to E is a Hermitian form H(.,.) C whose imaginary part is E. The Riemann form E is said to be nondegenerate if it corresponds to a positive definite Hermitian form H. To construct theta functions one requires the positive (semi-)definiteness of H. In case H is positive definite one can construct sufficiently many theta functions to obtain a projective embedding of T = C'/L into some projective space Ph'. Conversely, the existence of some projective embedding of T guarantees the existence of a nondegenerate Riemann form. T is called an abelian variety (or abelian manifold) if and only if there exists some projective embedding. In modern language one can identify a Riemann form E with a closed real (1,1 )-form on T = C" /L which lifts to a form w = E p.1 dz' A on the covering Euclidean space C". The Hermitian form H is then associated to the corresponding Kahler metric 2 Re dz' ® which is invariant under translations and hence can be defined on T. The requirement that E takes integral values on L x L is equivalent to requiring that the cohomology crass [wI E H' ' '(T, R) lies in 112(T, Z). Moreover, H is positive definite

if and only if cv is a positive (1, 1)-form. Thus, the existence of a nondegenerate Riemann form is equivalent to the existence of a Hodge metric on T = C"/L, which guarantees the existence of a projective embedding by a theorem of Kodaira (cf. Kodaira-Morrow [KMJ). For the classical theory of abelian varieties, we refer the reader to Siegel (Sie] and Swinnerton-Dyer (S-D].


365

Let now A = be an n-dimensional abelian variety. The choice of an ample line bundle A over A determines a first Chern class c, (A) E H' '(A, R). A polarization .9 is the choice of such a class, modulo the identification of proportional classes. We will say that A belongs to the polarization .9. Given A with a corresponding nondegenerate Riemann form E, one can always choose a basis for L over Z such that £ is represented by the skew-symmetric matrix , where 0 = diag(0,, ... , for some positive integers 0,, ... , meaning a divides verifying 011011. .. b). We say that (A, .9) is 0-polarized. If by replacing E by some rational multiple, one can obtain a Riemann form with 0, = = 1, we will say that (A , .9) is principally polarized. Equivalently this means that there exists a positive line bundle A belonging to .9 such that is a generator of Z. Any abelian variety A admits a finite covZ) ering A' which can be endowed a principal polarization. (We say that A is isogenous to A'.) Denote by the Siegel upper half-plane {symmetric g x g-matrix: Im(r) > O}. To each r e we associate the lattice L(r) generated by the column vectors of The torus C"/L(r) is an abelian variety equipped with a principa] polarization such that the Riemann form on Cit is given by E(w, w') = Im(Wt(Im t)'w'). Denote by the matrix ,

representing an alternating bilinear form on R2". By the symplectic

group Sp(2n, R) we mean the group of real linear transformations M on i.e., preserving a bilinear form = The symplectic group as a group of automorphisms. More precisely Sp(2n, R) acts on E (AT + B)(Cr + Sp(2n, R) corresponds to r as in (Cr + = t'. Let Sp(2n, Z) C Sp(2n, R) be the subgroup consisting of elements with integral entries. Then X/ Sp(2n, Z) is in one-to-one correspondence with the set of isomorphism classes of principally polarized abelian varieties. As in the case of elliptic curves, one can also introduce level structures to obtain fine moduli spaces corresponding to congruence subgroups of finite index in Sp(2n, Z). In general, given a subgroup r' C Sp(2n, Z) of finite index not containing —Ia, one can construct a "universal" family of principally polarized abelian varieties over Xr = 4*/F. For the construction one uses the action of on x Cit given by [J) Sp(2n, R) x

(t, w)

For r torsion-free ofprin=

we obtain a regular family it : r cipally polarized abelian varieties over Xr. By Satake (SalJ and Baily-Borel EBBI, Xr admits a projective compactification By using theta functions one can also construct a projective compactification r By Borel (Bo2] any such compactification gives rise to an extension of ir : r —, 'r to it With a sligbt modification the above applies also to the moduli problem for 0-polarized abelian varieties (cf., for instance, Swinncrton-Dyer

[S-D, p. 69—73]). Then the set of 0-polarized abelian varieties is in one-

NGAIMINO MOK

to-one correspondence with

Sp(2n, Z, 5) for some arithmetic subgroup

Sp(2n, Z, 5) C Sp(2n, IR). For any choice of S and any Fc Sp(2n, Z, 5) a Siegel modular variety. F will be called of finite index we will call a Siegel modular group. If the polarization is r is of principal type. 3.3. In the sixties Shimura considered special families of polarized abelian varieties. Generically an abelian variety A does not admit nontrivial endornorphisms. We write End(A)Q := End(A) Q, so that generically (satisfying additional a sernisimple algebra F over Q End(A)Q Fix Q.

properties). Shimura considcred the moduii problem for polarized abelian varieties A with a prescribed choice of polarization and level structure such that Efld(A)Q contains F. He formulated and completely solved such modu11 problems in (Shil, Sbi2l. The corresponding moduli spaces X are later called Shirnura varieties of PEL (polarization, endomorphism and level structure) types. A remarkable fact is that X for some bounded symand some arithmetic subgroup F in such a way that Q is metric domain holomorphically embedded in a totally geodesic isometric manner by some is a real quadratic field then there a : fI —+ As an example, if F = (A' xY)/F such that is a countable family of Hubert modular surfaces parametrizes polarized abelian surfaces A for which Efld(A)Q contains If we fix the type S of polarization and consider the problem of Q( classifying such abelian surfaces A with the given type of polarization, we get a finite set of Hubert modular varieties. Let be the ring of C integers. One may also consider the problem of classifying polarized abelian surfaces A together with a faithful ring homomorphism —' End(A). In this context we refer the reader to Hirzebruch-van der Geer [HG, Chapter 7] for details. After refinement by introducing level structures (replacing F by some subgroup of finite index), the solution of the moduli problems implies of course the existence of holomorphic maps (A' x X")/r —i Y2/T' for some Siegel modular group F', say torsion-free. Explicit construction of such maps is given in [HG, Chapter 7], for instance. The point relevant to us is that such a holomorphic map is induced by a A' x A, —+ such that for some biholomorphism of the map a' := o a x A' is given by the diagonal embedding

a(wl,w2)=[1w

01

which is totally geodesic. The fibers of the modular family of 5-polarized abelian varieties ic : r Xr are however not products of elliptic curves since q' E Sp(2n, R) is not in Sp(2n, Z, 5) but takes values instead in As discovered by Mumford [Mum) there exist arithmetic varieties Xr = f)/F and totally geodesic holomorphic embeddings of Xr into some Siegel


367

modular variety such that the corresponding families of polarized abelian Xr do not admit nontrivial endomorphisms. Let variety it r G/K be a bounded symmetric domain and X1- = F \ G/K be an arithmetic variety. In this article, by abuse of language Ar will be called a modular variety if there is a group homomorphism p : G —+ Sr(2n, R) inducing by some choice of maximal subgroup K' of Sp(2n, IR) a bolomorphic totally into the Siegel upper halfSp(2n. R)/K' geodesic embedding a : such that D(T) is contained in some Sp(2n. ö). Fixing such plane does not contain —f,, a p and such a choice of K' and provided that we havc an associated family of ô-polarized abehan varieties it r

: d r —+ a pnodular family of (ô-) polarized abelian varieties. (With this terminology we assume implicitly that —I,, cD(fl.) X1 of polarThe problem of classifying modular families it : d ized abelian varieties (also called Kuga varieties, ci. Kuga [Kul, Ku2)) was in principle solved by Satake [Sa21. One considers an even-dimensional QWe will simply call it

Q2fl,

an underlying linear complex structure I on the rcai vector space V = R and a real bilinear form E on V such that E takes rational values on VQ and such that E(x, Jj') is symmetric and positive definite on V. Denote by Sp( V, E) the subgroup of GL( V, R) the subgroup of Sp( I', E) preserving the compreserving E, and by Sp(2n, R), and, fixing the choice plex structure J. Then, Sp( V, E) of an Sp( V. E)-invariant complex structure on Sp( V, E)/K1, the latter Given a semisimcan be identified with the Siegel upper half-plane pie algebraic R-group G of the Hermitian type defined over Q (i.e., G is identified as a real Lie group to an algebraic subgroup of some GL(N, R) defined by polynomials with rational coefficients), a choice of a maximal compact subgroup K of G (so that G/K is given the choice of a structure of a bounded symmetric domain), one looks for Q-homomorphisms vector space VQ

p : G —, Sp(V, E) such that p(K) C K, C Sp(V, E) and such that the

induced map a : G/K —' Sp( V. E)/K1 is totally geodesic. The problem is to find all such quadruples (V. p. E, I). Given such a quadruple, for z E G we denote by the linear complex structure on V obtained by conjugating I with p(y). Fixing a lattice L C C V one can construct compact complex tori (V/L, which can be endowed nondegenerate Riemann forms given by some positive multiple of E. Let r C G be a discrete subgroup commensurable with GnGL(N, Z). Suppose furthermore that p(y)(L) = L for any y E F. (Given p and L, such F always exists; cf. [Sa3, Remark 2, P. 1981). In this way one obtains a holomorphic map Xi-. = F \ G/K F' \ Sp(V, E)/K1 for some Siegel modular group F'. Assuming that p(G) one obtains a modular family of polarized abelian varieties it : d r Xr over the arithmetic variety Xr. To determine all quadruples (V, p. E, J) Satake LSa2] first considered the same problem over R, i.e., obtained by forgetting about the rational struc-

NGAIMJNG MOK

368

ture of V. This is the same question as determining all homomorphisms p: G —. Sp(2n, lIt) inducing holomorphic totally geodesic embeddings into the Siege! upper half-plane The question can be reduced to an algebraic

problem on the infinitesimal map dp : Lie(G) —' ofi(2n, R). The complete classification of irreducible solution over R can be read from Satake [Sa.3, p. 1881. It is found that every irreducible classical bounded symmetric domain admits a totally geodesic holomorphic embedding into some On the contrary the two exceptional bounded symmetric domains do not admit such embeddings. The description of solutions over Q can be obtained from the solutions over lIt (which are necessarily defined over some number field) by the method of restriction of scalars: For details see Satake [Sa3, pp. 189—195].

3.4. Let X and d be normal quasiprojective varieties and it : A —.. X be a surjective holomorphic map whose fibers are abelian varieties. We will assume implicitly that the zero section s : X varies holomorphically and that the fiberwise group operations are holomorphic on Let X c—. and d '—. 7 be normal projective compactifications such that it extends to a holomorphic map Suppose furthermore that the zero section extends meromorphically to and that the fiberwise group s X

d

7

d

operations over X extends to meroniorphic maps on d. Then, 31: d —* can be regarded as a geometric model of a polarized abelian variety A defined over a function field C(X), the polarization being determined by the choice of a projective embedding 7 '-' pN• From the latter perspective a rational point a of A over C(:Y) is equivalently a meromorphic section of In the sequel we maintain the convention that it : d —. X has abelian fibers unless otherwise specified. (We will say that 71 : d — I is regular over X.) a set of complex codimension 2 on X. On the other hand, over a small simplyconnected open set U on X, it - '(U) is uniformized by U x C d It follows

:7

by lifting

U —.

i"(U)

to UxCd and the Hartogs Extension Theorem

for holomorphic functions that in fact a is everywhere holomorphic in X (cf. Igusa Thus, the set A(C(I)) of rational points of A over the complex function field C(X) agrees with the set of meromorphic sections of 7 over which are holomorphic over X. In the modular case it —

I

will always take 1r to be the Satake-Baily-Borel compactification. In case Xi-. is a curve and noncompact the family it : d1 Xv is isomorphic to a modular family of elliptic curves, for which one has compactifications as elliptic modular surfaces, in all other irreducible cases Xr are of we

codimension 2 in 1r so that the choice of a completion 31 :

—,

Xr

is unimportant. With respect to any such compactification, the zero section s and the fiberwise group operations extend meromorphically. On the other hand the existence of 71: 7r —.1 can be obtained using theta functions or

xAHLER GEOMETRY ON ARITHMETIC VARIETIES

369

by differential-geometric methods (as given in Mok-Zhong [MZJ). Thus, associated to each modular family r Xr we have a well-defined modular polarized abelian variety Ar over the modular function field C(Yr). We say that an abelian variety A is simple if and only if there are no proper abelIan subvariety. Suppose now A = is not simple.

Then there is a proper complex vector subspace V of C" such that V fl V. Write A' = V/(V fl L) '—' C"IL = A. Choose a L is a lattice nondegenerate Riemann form E with associated Hermitian form F-f. Let W be the orthogonal complement of V with respect to H. Then, it is easy to see that WnL is a lattice in W and, that writing A" = Wf(WnL), the natural map A' x A" —' A is an isogeny of abelian varieties. Thus, every abelian variety A is isogenous to a direct sum of simple abelian varieties, written A A x.. x Ak. This gives the existence part of the Irreducibility Theorem for the complex field. Returning to the family we say that it has constant parts if and only if there is a nontrivial abelian variety A0 which occurs as a simple factor of each fiber over x E X. This happens if and only if there exists a finite branched covering ii : Z —. Y such that the family x A0 Z is isogenous to Z as an abelian variety over the complex function field C(Z). (To construct ii Z I one may use an argument on Douady spaces very much like Riebesehl ERie].) Let now r : SI be a polarized family of n-dimensional abelian varieties over a normal projective variety. Each fiber = 2(x) is of the form for some lattice of C". Over a small open set U on X

I

we may fix the choice of L(x), x E U, in such a way that the generators of L(x) vary holomorphically on U. Putting the uniformizing complex Euclidean spaces together we obtain a holomorphic vector bundle V over X. Denote by the corresponding locally free sheaf. Putting the lattices L(x) together we obtain a locally constant sheaf L of lattices over X. Thus, L is isomorphic over small open sets U to Z2" x U—. U. The local sections of L X are integral combinations of lattice points. As the lattice points vary holoxnorphically, L is a subsheaf of By abuse of notation we will write L C V. The quotient spaces V/L X then gives the polarized family SI —. X of abelian varieties. We call the short exact sequence V o —+ L SI —.0 the exponential sequence for 7t sI -. X. (The exponential sequence for stable elliptic fibrations over a compact curve is defined by Kodaira [Ko]. In this case the subtle part is the extension of the sequence to points of the base curve with singular fibers. For polarized stable fibrations of abelian varieties with singular fibers over a compact curve this extension is given by Néron models [Ne]). If I is compact the Mordell-Weil group A(C(X)) can be identified with H °(X, SI). (The same is true if for some compactification I of X, I — I is of codimension 2 in I, cf. [Sill, Proposition 1].) From the associated long exact sequence we have the

I

NGAIMING MOK

370

exact sequence of abelian groups (*)

O—.H°(X, L)-. Ff°(X, V)-. Jf°(X,sl)-. H' (X, L)—.H'(X, V). Write p:H°(X, V)—.H°(X,d) fortheprojection map. If H°(X, for the 0 write X(s) c d for the graph of p(s) e H°(X, d). Write is irreducible component of the Douady space of d containing [X(s)]. compact. From easy algebraic arguments one can prove that H°(X, V) as an abelian variety. For every x X the is surjective, realizing evaluation map s —. s(x) e gives then the abelian subvariety C with Thus, if R : d —. X is a regular family without constant part then the natural map H°(X.I d) —' H'(X, L) is injective. (In the modular case this corresponds to Silverberg [Si14, Proposition 6], which applies also to noncompact Xr of dimension 2.) in particular, if H' (X, L) is finite then = 0. The mapping H°(X, d) —, H'(X, L) can be interpreted as follows. Reisregularover X. Overanysmallopenset U of X calithat Ux we have the trivialization For x E U, moving along a closed is modified by a symplectic transformation loop the isomorphism by some element of r = Sp(2n, Z, 6). This way one obtains a monodromy mapping 0: (I) — T and a holomorphic mapping F : X —. from (depending on the choice of F(z0) at a the universal covering I into base point z0 e X). The monodromy mapping 4): ir, (X) -. defines an by symplectic transformations. Write for action of ,r1(X) on endowed with this ir1(X)-module structure. For z E X denote F(z) by t. For ye ir1(X) write

r

]€r. Then, a holornorphic section of V over X corresponds to a holomorphic mapping g : X —. C" satisfying the compatibility condition

g(yz) = (C7r +

for any z

I.

A holomorphic section of d over X is, on the other hand, given by those g satisfying the weaker compatibility condition

g(yz) = (C7r+ D7)'g(r)

for some 1a7fl7]' E

in case of principal polarization. Given a E H°(X, d), the mapping c,,: thus defined is a 1 -cocycle in Z (I), (cf. Brown (Br] for relevant background on group cohomology.) Passing to cohomology we

get a map H°(X,d) —.

L).

3.5. Given an abelian variety A defined over a number field k the Theorem of Mordell-Weil asserts that the abelian group of rational points A(k)

KAHLER GEOMETRY ON ARITHMETiC VARIETIES

371

over k is finitely generated. The statement remains true if one replaces k by a complex function field C(X), provided that A has no constant parts. (See Lang [Lan, p. 137—149] for a proof as in the case of number fields.) In case X is compact this follows from the injectivity of the map A(C(?)) H°(X, ii) H'(X, L) in §3.4, since H°(X, V) = 0 and H' (X, L) is finitely generated. In § 1.7 we will sketch a differential-geometric proof of the Mordell-Weil theorem over complex function fields for the gen-

eral case (with X quasiprojective). Our concern in this chapter is to give estimates for the rank of the MorddllWeil group over complex function fields C(X). Suppose for instance it : is of fiber dimension n, has a polarization of type o and a level N-structure for some N 3. Then, there is a congruence subgroup F = F(O, N) c Sp(2n, R, 0) and a unique classifying map f: X — Xr such that ,tld : .ii —, X is isomorphic to the pullback of the modular family it : d r Xr by f. Since r is torsion-free applying the extension theorem of Bore! [Bo2), the classifying map f: X —' I,..

always extends to a holomorphic mapping f : X

—.

where

de-

notes the Satake-Baily-Borel compactification of Xr. If A(C(Y)) has additional structure, such as having extra endomorphisms, Xr should be replaced by some modular varieties of smaller dimensions. Our objective is to estimate in terms of the classifying map f: X —. Xr, where iv :4 A',. stands for any modular family of polarized abeian varieties. The first step in this direction is obviously to study : itself, which defines the modular polarized abelian variety over C(Xr). We assume now that : —, I,. has no constant parts (in particular r,. is positive-dimensional). It was generally believed that in this case is always finite, i.e., rankz(Ar(C(Yr)) = 0. Many partial results were known

before. For the proof of the conjecture we may always assume that F is torsion-free. Shioda [Shio, 1972] proved that = 0 for the case of elliptic modular surfaces : = Z/F and conjectured the same for the Siegel modular case. This was first proved by Silverberg [Sill, 1985 and S112, 1988] which also proved the conjecture for :

certain Shitnura varieties arising from fixed PEL-types. Her results include the cases of Shimura curves (which are compact), Hilbert modular varieties, Siegel modular varieties and certain Sbimura varieties whose covering spaces are products of Siegel upper half-planes. Silverberg made the conjecture that = 0 for the general case (cf. [Sil4, §2.6)). Write p: r Sp(n, R) for the representation defining the family it: 4 As shown in Silverberg [SIL3J, the conjecture for will follow if H' (F', p) = 0. In case A',.. is compact, irreducible and of rank 2 this follows from a vanishing theorem of Raghunathan (Ragi]. In case A',. is noncompact, irreducible and of rank 2, certain special cases were also known to Raghunathan [Rag2) for which H' (A',-., LR) = 0. Very recently, Margulis [Mar21

372

NGAIMING MOK

proved that for any such Xr and for any finite-dimensional locaily constant sheaf E over a field of characteristic zero, one has H' (Xr, E) = 0. We have succeeded to affirm the conjecture in the full generality by a completely different method. We state this as: Let ir : d r —p be a polarized modular family of abe/ian varieties without constant part, : be a projective compactification which is a geometric mode/for the associated modular polarized abe/ian Then, there are at most a finite number of merovarieties Ar over over In other words, morphic sections of Ar(C(Xr)) = 0 for THEOREM 1.

the Mordell- Well group

In Theorem 1, F does not need to be torsion-free, although the proof can be reduced to the torsion-free case. Suppose now that F is torsion-free and —. X is obtained from a classifying map f: X that ir : Xc into some defines a compactmodular variety Xc. Then, the completion 7: : ification X, which is a geometric model for a polarized abclian variety A over the complex function field C(V). Because of Theorem I it in terms of the geomis desirable to give an upper bound of etry of the classifying map. An extreme case occurs when f(X) is totally geodesic and the mapping f: X f(X) is unramified. (This includes the X case of taking finite unramified coverings of Xr.) In this case ir : d is itself a modular family and we have 0. In general one expects therefore that the latter quantity can be estimated in terms of the second fundamental form and/or the ramification locus. We are not able to do this in general. Nonetheless when f: I — 1r is a ramified covering we have an estimate in terms of the ramification divisor. This estimate applies then in principle also to the case of surjective classifying maps f: since the latter can be factored through a ramified covering. In this direction we prove:

Xc be a polarized modular family of abe/ian THEOREM 2. Let hr : d r varieties without constant parts with F torsion-free. Fix a canonical KdhlerEinstein metric w on Xr. Then, there exists a constant Cr depending only on F for which the following holds. Let ir : .ii be polarized family of abe/ian varieties over a normal projective variety X obtainedfrom a class map f: I —. Xr. Denote by R1 the ramification divisor of f counted with the branching divisor on I, i.e., the image of R1 with multiplicities, B1 c projective compactificasion multiplicities counted. Then (denoting by I

I

of I) we have

CrVolume(Bj, ü). We note that by Borel [Bo2] f: 1 —. Xr extends to a holomorphic map 7: 1 so that C 1r is a subvariety. It follows readily that Volume(B1, w) — i + We are going to show that for

1 /4n any foliated section of d over K is horizontal. Denote by R' the curvature tensor of K as a subinanifold of X. By the Gauss equation we

forany a of type (1,0) tangent to K atsome xEK, so that from the assumption on RX we have Let a be —(1 + a foliated section of : d —. K and E C C°°(K, T. 0 V) have

be the corresponding tensor. To prove the proposition we have to show that 0. Denote by the 1-parameter family of Kihier forms ?co +

on ir

: r Xr degenerating to co as in §3.6. Fixing a and restricting to Graph(a) we have a i-parameter family of Hermitian metrics on K degeierating to for the curvature tensor of coLt. Write (K, so that R"(O) = From §3.8 we have

(*)

—

+ 0(t4).

NGA1MING MOK

390

Take now a to be a unit vector with respect to w. Write M(a) for the term in parenthesis. We remark that the bundle (V, h) is Hermitian-Einstein. It follows hence from the curvature formulas above that for any unit tangent vector a of type (1,0) and any orthonormal basis we have } of

in particular we see that M(a) < 1/2. Equality occurs precisely when a is conjugate under U(n) to + + as can be seen from the curvature formulas given above. On the other hand the latter happens if and only it'

=

—I

This reflects the fact that for a totally geodesic

.

curve K' of conslant curvauue —1, VIE. splits into the direct sum of ii square roots of the tangent bundle. Here we have instead the hypothesis —(1 + f) < —1). For a a1 E fl, the curvature hypothesis implies that

(E(4) = _! Let be such that for We have

1, —(1 + e)
1—

(f).

—c5 and

is attained when all the 2 n, are equal. It follows that one may take for ô(e) the solution to the equation .+

=n(1+E), i.e.,

ô()2 = We

return now to the Gauss-Bonnet formula. We have the estimate

(*) so

— 1).

—

412

+ 0(t4),

that for a of unit length with respect to co, we have

(*)'

R_(t)

+ Should the answer be affirmative, can we choose the constant ô(n) to be independent of n?

The last question is a formulation of an L2-analogue of the fact that the pinching constant in Theorem 1 can be chosen to decay like 1/n. Given

a compact algebraic curve K in some X =

constructed algebrogeometrically (for instance, corresponding to the base space of a Kodaira surface), it is in principle easier to compute degrees of than to compute curvatures explicitly. In other words, should an L2-analogue of the pinching theorem (Theorem 1) exist, the question of finding sharp pinching constants might be more accessible. While we consider only quotients of

commensurable with Sp(n, Z),

NGAIMING MOK

394

the interpretation of X as a moduli space of polarized abelian varieties was just a motivation. In the proof of Theorem 1 for compact algebraic curves, we do not need the locally constant sheaf of lattices. Instead, we need the locally constant sheaf LR of real vector spaces for our argument, so that it applies in fact to any quotient of by a torsion-free discrete group r. Furthermore, the setting applies equally to quotients of bounded symmetric domains provided that admits a totally geodesic holomorphic embedding into some

does not contain an By Satake [Sa2] this is the case provided that exceptional domain as an irreducible factor. Details of pinching theorems for curves in arithmetic varieties will appear in a future article. BIBLIOGRAPHY W. L. Baily and A. Bore!, Compact ;fl cation of arithmetic quotients of baunded symmetric domains, Ann. of Math. 84(1966). 442—528. [Bol) A. Borel, Density properties for central subgroups of semisimple groups without compact components, Ann. of Math. '72 (1960), 179—188. (Bo21 —, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, 3. Differential Geom. 6 (1972), 543—560. [Br] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin-Heidelberg, 1982. [CMI H.-D. Cao and N. Mok, Holonwrphic immersion between complex space forms, Invent. Math. 100 (1990), 49—62. EBB]

[Con] K. Corlette, Flat G-bundies with canonical metrics, 3. Differential Geom. 28 (1988), 361—382.

(Cor2J _, Archimedean superrigidity and hyperbolic geometry, (preprint). [Cr] (DWJ

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.1. A. Canlson and D. Toledo, Harmonic mappings of Kahler manifolds to locally symmetric spaces, Inst. Hautes Etudes Sci. Pubi. Math. 69 (1989), 173—201. D. L. DeGeorge and N. R. Wallach, Limit form ulas for multiplicities in L2(r\ 6) Ann. of Math. 107 133—ISO. 3. EclIs and H. Sampson, Harmonic maps of Riemannian manifolds, Amer. J. Math. 86 (1964), 109—160.

(Gd]

[IguJ

P. Oriffiths, Hermitian differential geometry, Chern classes and positive vector bundles, Global Analysis (articles dedicated to K. Kodaira) Univ. of Tokyo Press, Tokyo, Princeton Univ. Press, Princeton, 1969, pp. 185—251. M. Gromov, Foliated Plateau problem (preprint). E. P. Hirzebruch and G. van der Geer, Lectures on HUbert modular surfaces, Séminaire de Mathêmatiques Superieures., les Presses de l'Université de Montréal, 1981. J. Igusa, On the structure of a certain class of Kdhler varieties, Amer. J. Math. 76 (1954),

[JY1]

J. Josi and S.-T. Yau, Harmonic mappings and Kdh/er manifolds, Math. Ann. 262 (1983),

(Gin] [HGI

669—678. 145—166.

(JY2J —, The strong rigidity of locally symmetric complex manifolds of rank one and finite volume, Math. Ann. 275 (1986), 291—304.

[JY3] ., On

the rigidity of certain discrete groups of algebraic varieties, Math. Ann. 278

(1987), 481—496. Harmonic maps and group representations, (prepnnt).

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3. Kodaira and 3. Morrow, Complex mantfolds, Holt, Reinhart & Winston, New York,

(Ko)

K. Kodaira, On compact analytic surfaces. 11—111, Ann. of Math. 77 (1963), 563—626; 78 (1963), 1—40.

[Kul]

M. Kuga, Fibered varieties over a symmetric space whose fibers are abelian varieties. I, II. Lecture Notes, Univ. of Chicago, 1963/64.

1962.

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[Ku2J —, Fibered variety over symmetric space whose fibers are abe/ian varieties,

395

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Japan Scm. in Differential Geometry (Kyoto, 1965), Nippon Hyoronsha, 1966, pp. 72— 81.

(Ui

F. Labouric, Existence d'applicazion harmoniques tordues a valeurs dans variê:és a cour-

ILani

bw-e negative, (preprint). S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York-BerlinHeidelberg-Tokyo,

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Y. Matsushima, On Beth numbers of compact locally symmetric Riemannian manifolds, Osaka Math. 3. 14 (1962), 1—20. (Mar11 G. A. Margulis, Discrete groups of motion of manifolds of nonpositive curvature, Amer. Math. Soc. Transl. (2) 109 (1977), 33—45. EMar2) _, Oral communication. [MaJ

[MoklJ N. Mok, The holomorphic or antiholomorphic character of harmonic maps into irreducible quotients of polydiscs, Math. Ann. 272 (1985), 197—216. Metric rigidity theorems on locally symmetric Hermitian spaces, Proc. Nat. Acad. Sd. U.S.A. 83 (1986), 2288—2290. Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients (Mok3) of bounded symmetric domains, Ann. of Math. 125 (1987), 105—152. (Mok4] —, Uniqueness theorems of Kdhler metrics of semipositive bisectional curvature on compact Hermitian symmetric spaces, Math. Ann. 2% (1987), 177—204. (MokS) —, Strong rigidity of irreducible quotients of polydiscs of finite valume, Math. Ann.

(Mok2I —, ,

282 (1988),

555—477.

[M.k6J _, Compacufication of Kdhler-Einstein man(fo!dr offinite volume, Contemp. Vol.

Math.,

101, Amer. Math. Soc., Providence, k. 1., 1989, pp. 287—301.

(Mok7) ___,

Topics

in complex d(ferential geometry, Adv. Stud. Pure Math. 1*4 (1990), 1-

141. , Metric rigidity theorems on Hermitian locally symmetric manifolds, Series in Pure Math., Vol. 6, World Scientific, Singapore-New Jersey-London-Hong Kong, 1989. (Mok9) —, Geometric arcizimedean superrigidity in the Herniitian case, in preparation. [MR3 Min-Oo and E. A. Rub, Vanishing theorems and almost symmetric spaces ofnoncompact type, Math. Ann. 257 (1981), 419—433. (MS) Y. Matsushima and G. Shimura, On the cohomology groups attached to certain vectorvalued forms on products of the upper /zalf.plane, Ann. of Math. 78 (1963),

(MokS)

417—449. (Marn)

D. Mumford, A note of Shimura's paper "Discontinuous groups and abelian varieties", Math. Ann. 181 (1969), 345—351.

[MZ) (Ne]

(NT)

of complete Kdhler-Einstein manifolds of N. Mok and J.-Q. Zhong, finite topological type and bounded curvature, Ann. of Math. 129 (1989), 427—270. A. Neron, ModE/es minimaux des vaiWlEs abeliennes sur les corps locaux et globaux, Pubi. 1.H.E.S., No. 21, 1964. H. Nakagawa and R. Takagi, On locally symmetric Kdhler manifolds in a complex projective space, J. Math. Soc. Japan 28 (1976), 638—667.

[R.glJ M. S. Raghunathan, On the first cohomology of discrete subgroups of semisimple Lie groups, Amer. J. Math. 78 (1965), 103—239.

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(1967), 408—424; U, 87 (1968), 279—304.

[S2)

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COLUMBIA UNIVEaSITY AND UNWERSFrE DE PARIS-SUD, Fwicu

Proccedinp of Symposia in Pure Mathematics Volume 52 (1991), Part 2

Moduli Spaces of Holomorphic Mappings into Hyperbolic Spaces and its Applications J.NOGUCHI 1. Model cases and motivations. For the model case we recall the following two theorems, which state the finiteness and the rigidity of holomorphic

mappings into a compact Riemann surface (curve) S with genus 2. Let M be a complex projective manifold. DE

THEOREM.

There are only finitely many nonconslant holo-

rnorphic mappings from M into S. HURWITZ'S THEOREM.

Let f, g: M —

holomorphic

S be

mappings that are mutually homotopic; Then f

g.

As we extend these theorems to a higher-dimensional case, we see that they are related to Diophantine geometry, such as (A)

Mordell's conjecture (Faltings' theorem) over function fields due to Manin and (3rauert (Cf. [N1] and its references);

Parshin—Arakelov theorems for curves, principally polarized Abelian varieties, and polarized algebraic K3 surfaces. To state (B) we fix a curve R and a finite subset £ C .R (the higher-

(B)

dimensional case is similarly dealt with), and a positive integer g. Let

ir:X—.R bealocauynontrivialfamilyof (a) curves X1 with genus g (2), (b) principally polarized Abelian variety X1 of dimension g, or (c) polarized algebraic K3 surfaces X1 such that X —. R is smooth outside I. Then, in the case of (a) there are only finitely many such X (Parshin—Arakelov [PA]), and in the cases of (b) and (c) the respective X 's fom1 an algebraic variety over C (scheme of finite type over C) (Faltings (F] for (1,)), of which connected components 1980 Mathematics Subject Classification (1985 Revision). Primary 32H20. The detailed version of this paper has been submitted for publication elsewhere. © 1991 Mnencsn MathemMiosl 0082.0717/91 pcrpa5e

397

1. NOGUCHI

398

have structure of quotients of bounded symmetric domains (Noguchi [N,]). Let D be the TeichmUller space of genus g for the case (a) (resp. SiegeFs of rank g for (b), a bounded symmetric domain of upper-half space type IV for (c)), and r the TeictiinUller modular group (resp. an arithmetic discrete subgroup U c Aui(D)). For a given X R. we have a holomorphic mapping f: R — —. T\D, and the problem is reduced to the study of the F\D) of all holomorphic mappings from R — moduli space Ho)(R — into F'D. We recall the following two facts: (i) F\D is complete hyperbolic (in the sense of Kobayashi), (ii) F\D has a compactthcation F\D (Satake-Baily-Borci's compactifica-

iion) such that f\D is hyperho!ically imbedded into F\D: that is. for any of P in F, E D(F\D), i = 1, 2 with P, P2, there are neighborhoods T\D such that Q2); Q. E U, fl (U\D)} > 0,

whcre dr\D denotes the Kobayashi hyperbolic distance on U\D.

2. Structure theorem for Hol( M, N). Let M (resp. N) be a Zariski open (resp. N). subset of a compact complex space THEOREM 1 [N2].

Assume 1/tat N is complete hyperbolic and hyperbol-

ically i'nbedded into N. Then Hol(M, N) carries a structure of a Zariski open subset of a compact complex space with ihe natural universal properly as moduli space.

The proof depends on the following Extension-Convergence Theorem:

with the unit disc Ac C, and N be —. N, 1, 2, ... be holomorphic mappings that as above. Let converge uniformly on compact subsets to f0: N. Then ) ) N and 7o: N of there are unique holomorphic extensions converge uniformly on compact subsets of respectively, and and THEOREM 2 [N2J.

Let

'=

REMARK. The extension of holomorphic mappings from due to Kierman [Xl, but here we use a different method.

into N is

We deduce Theorem 1 from Theorem 2, the Douady theory and Hironaka's flattening theorem. APPLICATION 1. Let N = IT\D be the quotient of the TeichmOller space (cf. (a) in §1), and M = R — L as in §1. Then Theorem 1 implies that there

are onJy finitely many homotopy types of f Hol(M, N); specilically, we have to deal with nonconstant holomorphic mappings from M into PtD that are liftable to equivariant holomorphic mappings from the universal

covering space M of M into D. Combined with a local rigidity of f E Hol(M, N), this yields the Parshin—Arakelov theorem for curves [P, A, IS].

MODULI SPACES OF HOLOMORPHIC MAPPINGS

399

APPLICATION 2. By making use of Theorem 1, we have a higher-dimen-

sional analogue of Faltings' theorem (Mordell's conjecture) over function fields for hyperbolic fiber space, which implies that the original ManinGrauerts' theorem as a special 1-dimensional case [N1]. There is another application: THEOREM 3 (Miyano-Noguchi [MN]). Let N be as in Theorem 1. Then the group Aut(N) of all holomorphic transformations of N is finite.

In the case of compact N, this is due to Kobayashi (Ko). 3. Arithsi.etic quotients of bounded symmetric domains. In this section let

= F\D be the quotient of a bounded symmetric domain D by an arithmetic discrete subgroup r C Aut(D). For the sake of simplicity, we assume

that r is torsion free, but note that the case where f contains torsion can be dealt with similarly. We also assume in this section that M Kähler manifold. Put

is

a compact

(f, P) Hol(M, f\D) x M —' f(P) I'\D, cDi,: fE Hol(M, V\D)

f(P) E IT\D(P EM).

THEOREM 4 [N2]. (i) For any fixed P E M, 1,, is a proper holomorphic isometric I miners ion onto a complex totally geodesic submanifold of V\D. (ii) Let f E Hol(M, T\D) be nonconstant. Then dim1 Hol(M, fD)

ftV)(T\D))

0, (hen dim1Hol(M, f\D) < 1(f), where 1(D) (resp. 1(f)) denotes the maximum dimension of proper (resp. f-rational) boundary components of D. 1(D); moreover, if

(iii) Let 0.

f

Hol(M, f\D). If rank f> 1(D), then dim1 HoI(Mf\D) =

If rankf> /(f) and

then

r'\D)=O.

The proof depends on Theorem 1, Schoen-Yau's result on harmonic mappings [SYJ, and a finiteness theorem due to [NSI. We call such a mapping as in Theorem 4(1) a Kuga-Satake immersion. APPLICATION TO CASE (b) IN § I. Let A(g, R. Z) denote the moduli space

of g-dimensional principally polarized Abelian variety A R over R that degençrate at most on Then we see by Theorem 4 that A(g, R, E) is a finite union of quotients of bounded symmetric domains that are mapped into Sp(2g, Z)\Hg by Kuga-Stake immersions. Let

E Hol(M, f\D) be a nonconstant mapping such that

dim1 Hol(M, f\D) >0. Let X be the connected component of Hol(M, F'\D) containing Then X = F1 \D1 , where D1 is a bounded symmetric c omain and F1 c Aut(D1). We have a holomorphic mapping

J. NOGUCHI

Let Y = f2\D2 be the component of Hol(X, f\D) containing we have the natural mapping

D: (x,y) E Xx Y THEOREM 5 (Miyano-Noguchi [MNJ).

,(M).

Then

—'y(x)E

The above 1: X x Y —. 1\D is a

Kuga-Satake immersion.

This theorem may have an application to the study of A(g, R,

We

can also use this to study case (c) in § 1. APPLICATiON TO CASE (c) IN § 1.

It is known that the moduli space of

polarized algebraic K3 surfaces is given by where is a nineteendimensional bounded symmetric domain of type IV. In this case we have

1(f) = 1(D) =

Therefore, X and Y

1.

Theorem 5 are quotients of the upper half plane H1 C C. One way to construct such a '1 is the following. Let E(r) denote be the elliptic curve corresponding to r E PSL(2, Z)\H1. Let X(r1, the desingularization of (E(r1) x E(z2))/(—id) at sixteen fixed points of the surface and determines involution — id. Then X(r1, 12) is an algebraic K3 a point

in

12)1

(r1, 12)

Thus we get a nontrivial holomorphic mapping

(PSL(2, Z)\H1)2 —. (X(r1, 12)1

admits no more deformation. Lately, Masahiko By Theorem 4 this Saito and S. Zucker gave a complete description of such 4k); that is, they classified all nonrigid families of polarized K3 surfaces over R with degenerations at most over E [SZ]. One of their interesting results is the following: There are only finitely many birationally equivalent classes of locally nontrivial fiber spaces of polarized K3 surfaces of THEOREM 6 (Saito-Zucker [SZ]).

degree 2k over .R with degenerations at most over £ whose general fibers have the

Picard number

2, 6, 10, 14, 18. REFERENCES

(Al

S. lu. Arakeiov Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 1277—1302.

(Fl

G. Failings, Arakelov

ILSI

Y. imayoshi and H. Shigs, A finiteness theorem for holomorphic families of Riemann surfaces, and moduli. II, Springer-Verlag, New Yak, Berlin, 1988,

theorem for Abelian varieties, Invent. Math. 73(1983), 337—347.

pp. 207—2 19. (KJ

P. Kiernan, Extensions of holomorphic maps, Trans. Amer. Math. Soc. 172 (1972), 347— 355.

S. Kobayashi, Hyperbolic mansfolds and holomorphic mappings, Marvci Dekker, Ncw York, 1970. (MN) T. Miyano and J. Noguchi, Moduli spaces of harmonic and holomorphic mappings and Diophantine geometry, Proc. Taniguchi Intemat. Conf. 1989, Katata-Kyoto, Lecture Notes in Math., Springer-Verlag (to appear). (N1) .1. Noguchi, Hyperbolic fibre spaces and Mordell's conjecture over function fields, Pubi. Res. Inst. Math. Sci. 21 (1985), 27—46. [KnJ

MODULI SPACES OF HOLOMORPHIC MAPPINGS

[N2] —, [NSI IF]

Modul: spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces, Invent. Math. 93 (1988), 15—34. J. Noguchi and T. Sunada, Finiteness of the family of rational and ineromorphic mappings into algebraic varieties, Amer. J. Math. 104 (1982), 887—900. A. N. Faishin, Algebraic curves over function fields. I, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1145—1170.

[SZ]

[SY]

M.-H. Saito and S. Zucker, Classification of non-rigid families of K3 surfaces and a finiteness theorem ofArakdov type, Math. Ann. (to appear). R. Schoen and S.-T. Yau, Compact group actions and the topology of manifolds with

non-positive curvature, Topology 18 (1979), 36 1—380. TOKYO INSTITUTE OF TECHNOLOGY, JAPAN

Proceedinp of Symposia in Pure Mathematics

Volume 52(1991), Patti

The Analytic Radon Transform SALOMON OFMAN

0. Introduction. The purpose of this article is the construction of an analytic transformation we can see from two different points of view on one hand, we get a holomorphic form on a cycles space; on the other hand, we get a section of a holomorphic line bundle on a space of flags. This second point of view is suggested by the work of Gindikin and Henkin [C-H) on the Radon transformation. Our construction generalizes the definition given in [C-H] and studied in (G-H, H-P] in the particular case of linear subspaces of the complex projective space To simplify, we study here the case of smooth effective divisors and we indicate how to generalize this construction to cycles of any dimension.

NotatIons. Let Y be an analytic complex connected n-manifold, the sheaf on Y of germs of holomorphic functions (resp. differential (r, s)-forms), the space of the analytic compact cycles of dimension n — 1; for c E Cn_i(Y), we denote by id the support of C (i.e., the set of points in c). Let W be a smooth open set of a subspace of we denote by the space defined by G1 = {(x, C) E Y x x icI}, and a, fi, respectively, the natural applications from G1 on Y 1.

and W. Let M(Y) be the set of meromorphic functions in Y; for f M(Y) — 9(Y), we denote by (1) the divisor of f, and (resp. the positive (resp. negative) part of (1). We denote by 4' (resp. J) the set of meromorphic functions f such that M(Y)

(resp.

(resp.

(f)f) is compact and

J) the set defined by M(Y) = M(Y)

(resp.

A'=A—(Anf(Y))).

DEFINITION. An analytic space is pseudoconcave if there is a compact set

K and an exhaustion function such that the Hessian of nonpositive proper values at any point of Y — K.

has at least 2

1980 Mathrmaticr Subject Classification (1985 Revision). Primary 32L10. This paper is in final form and no version of it will be submitted for publication eliewhere. Ameñciz 0082.0717/91 $).0O+ 8.25

403

SALOMON OFMAN

404

EXAMPLE. Let Z be an analytic complex manifold, X a submanifold of codimension not less than 2; then Y = Z — X is pseudoconcave. REMARK. If Y is a connected pseudoconcave space, then a holomorphic function on Y is constant (cf. [AJ); this property is essential in the rest of this article. DEFINITION. Let

v in

denote

current defined on W C

by

=

restriction of p to (resp. p°). LEMMA

For

1.

in

(resp.

a

form (resp.

is a holomorphic

(resp.

ferential form, pt

is noted p'

(resp. to

The

function) on 0'. PROOF. Let v' be a (n — r, n — 1)-differential form with r in {O, l}; a'yi isa (n—r,n-- 1)-differential formon W. Theapplications a and ft

analytic and the differential commutates with their direct and inverse images. The fiber above any point c E W is of pure dimension n — 1 (for c mW); then fi,a' is a (1 — r, 0)-current on W. Since w = fl,a 0 i,v = 0, $0a is a holomorphic Current on W and then, in fact, a differential form on Y of degree (1 — r). From Dolbeault isomorphism, we see that the transformations p° and p' define applications on the groups of Y (resp. (Y, and £1"fl; they are still noted p° et are

where 0 0. pç'(c)—p

.

n—I

(Y,fl n—i ),

e ci,

There is a very simple relation between these transformations: LEMMA

2.

dp°ço =

p' dç.

PROOF. The lemma results immediately from the commutativity of the differential d with the direct and inverse images of applications. In the following, we suppose that Y is pseudoconcave. 2. Meromorphic functions and the space of analytic cycles. To any function in £' (resp. A'), we associate the following families of functions and of cycles:

f

=

{i;

=

—

t,

t e

,

}

=

= c,

t

4(e)},

THE ANALYTIC RADON TRANSFORM

{z C; Jzf <e}. is the disk of C defined by: By construction, we must choose c small enough so that for any t in

where

the function f (resp.

(resp. .4(e))

belongs to

(i.e., c (resp.

c) is smooth, we deduce from Bertini's theorem that we may choose e such that for any t in De the cycle c (resp. —, c,) is still smooth. Under these hypothesis, we denote the map by J:

C. LEMMA 3 [04). c 4' (resp.

For f in 9' and

a disk of C small enough so that

c .1), the family

(resp.

is analytic.

PRoPosITIoN I. For g in 4', the following statements are equivalent

(i) (g)=c' (ii)

and

g=Af+/z,AEC',pEC.

(i) is easy; so we may assume (i) is true. PRooF. The implication (ii) By Bertini's theorem, we may also assume that c has no multiplicity. Set and g' = the function g' is meromorphic on c and moreover c= g' is in fact holomorphic; c is a compact variety so we have g' p E C. We note = g — 1u; it is easy to show that the divisors of f and of g" are the same so that we have g"/f = (g — =A E and from the remark

3. Relation between p° and p'. In this section, the a-closed differential form

is fixed.

(a) Let f be in 4, and set c' = We fix e small enough so that we can define and we note / (resp. instead of f (resp. The differential form f) is d-closed in Y — C'; from the Gelfand-Leray theorem we know there are forms 0 in Y — c' such that 0 A df = 'u/f; moreover, 0 can be extended by zero in Y. Let be such a form. PRoPosITIoN 2.

For f and p'

as above, we have

=

.

d/.

PROOF. (a) Let 8 be another form such that 8 A df = w; outside c' the form — 8) is divisible by df and, for any t in De, the integrals of and 0 are equal on so we have = p°8(c) for any in The application ft is analytic and defines locally a trivial fibration above W and, by restriction, above let v be a form on

SALOMON OFMAN

406

we

I=

set:

v); by Fubini's theorem, we easily see that

*APS

j

=

(j

Aa'

A

fi'v

J

=

(j

f

cx'

A fi'v;

J

d)= cx' df = /3'df, we deduce

since

'

fi'(dJAv) = fl'(dfnv)

L.

(dfAv))—(cx

=(cx

ç'Afl df,fl v);

finally, we obtain the equality

A fi' df.

=

(b)For fin 4',wedefine

by

iffE4", I,(f)=O where Rest. y is the residue-class of y in the sense of Leray. The form represents the residue-class (cf. (LI) and Proposition 1 is equivalent to the equality

p'

=

dJ'

(f

E

We showed PROPOSITION 21.

For any

f A' and c E

p'

1, (3:) df(c).

4. The Radon bundle.

A flag-cycle in an analytic manifold Y, is a sequence of analytic cycles of Y such that

DEFINITION.

...

,

=

DF

1(Y) the subset of denote by negative parts of principal divisors of Y,

whose members are the the projection of D on W, and p1 the projection of 4 on D defined by pL(h) = (c', ë) where (h) = We

and ë=(h)1fl(s4). We denote by 9'

4

=

the subset

in 4'

{f= (h, €4' xD; if h

x D defined =

by

((h),

THE ANALYflC RADON TRANSFORM

407

and q be the projections from 4" on D and 9", respectively. We may write 4" = U 4" where 9" = q'(4') and 9" = 4" — Let

have 4"

C x D and the restriction of q to 4' isomorphism on 4", so we may identify f = (h, E We

hE4'.

is

clearly an

and q(f) =

Forany fin 4",we set REMARK, if f E 9", then w'f is a-closed in Y and we have

= 0.

DEFINITION. Let E be the quotient-space of 4" defined by the following relation of equivalence: f g if (i)

(ii) q(g) —

q(f) = constant.

We denote by p2 the projection 4" —' E = f E 4" in E (i.e., f = p2(f)).

4"/

and

I the class of

The following lemma is easy to prove. induces a projection r: E —, D whose fiber LEMMA 4. The application above any point of D is a complex line. Let TW be the tangent bundle of W; we define an application a1: 4" TW by: (i) For h in 4" and a disk in C, we associated in §2 the family

We define an application

—,

W by

= (h1) with i(O) = (h0) = c'. defines a complex analytic curve in W parameterized

From Lemma 3,

thus a tangent vector r(h) E 7,W. Let f be in 4" such that

by

€4" x D = (c', E W x we define an application f= from 4" to TW (still noted) r by t(f) = r(h); the image of f by r is (h,

also denoted

(ii) Assume f is in 4"; we set (c', where

=

E D (i.e., f =

AEC and

Hence we have an application PRoPosITIoN 3.

Let

4"

TW defined by

= r1.

f and g be in 4".

(1) 1fp2(f)=p2(g) then a1(f)=a1(g). a1(f) iszerozn

(2)

ifffbelongsto

9" (i.e.,

f=

with

= (c',

(h,

4' x D and g =

(1,

E 4' x D

EWx (if h is nonconstant, C' = (h) Since p2(f)=p2(g),we obtain l—h=5EC and

and

SALOMON OFMAN

408

(a)

If I is in .4" (i.e., h

then

E

1 is

in .4"

too,

thus a1(f) =

a1(g)=O€

We may therefore assume f (and g) are in (i.e., h in Y). We set c = (h,) and k = (i,) (h1 and 1,, respectively, in (b)

and I

nonconstant

and

; we define

We denote by D(e)

(resp.

and

as in §2.

(resp. lim

the germ of curve defined by Let

D(t)

eterizations of the curves

(resp.

= be the param-

(resp.

(e)) defined by

=

(resp. "2e the

c

= v(O) = C'; we denote by r: = k), where application defined by r(t) = t/(l + ôt). (i) , thus E C and by symmetry we obtain = (ii) We set and ii = wehave = = rID, = thus = v'(O). The application is analytic and clearly De injective, so it is an analytic diffeomorphism; hence, for any t in we obtain 0, then a1(f) = 0. ZIE(t)

Let it1 be the projection from TW on W; then Proposition 3 is equivalent to PROPOSITION 3'. (i) The application

(ii) For any

f in E,

.4" —. TW induces an application a: E TW. we have a(j) = 0 = 0 (identity of ir' (ir(f)).

/

NOTATION. We will also write

for a(f).

5. Construction and study of the Radon transformation. We can put on (E, D, it) a structure of holomorphic line bundles so that (it', a) is a morphism of holomorphic vector bundles. THEOREM 1.

PROOF. We will show that we may consider E as a subbundle of the vector

bundle E' = ir'(TW). (1) Let i be the natural application (of sets) E

E' defined in the

following manner: For

we

set

= (c', ë) in D and / in E E, we denote by E' defined by = a(f) (since = i(f) = We obtain the following diagram:

the point in

W.

(2)

be in

Let

p,(f)=f

'(c)

with 1(f) = i(g)

(resp.

By definition, there is (A, 4u)

and f (resp. g) E 9"

(resp.

C2 such that 1

=A

h

+ a.

THE ANALYTIC RADON TRANSFORM

if 2 =

409

then / is constant (i.e., g = 0 in and (Proposition 3'). We obtain i(j) = = i(t) = = 0, then f = (b) We may therefore assume A 0 and I = 2- h; we have (a)

=

0

I

I

0

1(1

1

We set c = (h,) and k = (l,) ; we obtain k' = 4,. As in the demonstration of Proposition 3, let and ii be the parameterizations induced resp. by the families et since v = 1 o where I is the multiplicationby A in C, weget and Hence, if C* xC), ((A, 1u) E we obtain from Proposition g' =(I+p, (2-h+4u,

3, o1(g)=o1(g')=2a1(f), thus Are, =T1 We have proved the injectivity of i and the linearity on the fibers. We may

then considerE as a of r'(TW) and put on E the structure induced by ir (TW). (ii) Set f (h, 1' (h', in 9" such that p2(f) = p2(f'); then = We may therefore set the (h — h) is constant in Y and following DEFINITION. Let w be a s-closed (n, n — 1)-differential form on Y; the

Radon transform of E defined by

denoted by

/

is the section of the dual bundle

where I f

The residue-class on c of a d-exact differential form on Y — c' is zero [L), so the following definition makes sense. DEFINITION. The Radon transformation is the application

-. H°(D,

R:

defined by:

where WE

For convenience, we will write THEOREM 2.

for II(çv).

Forany f€E, we have

where

(c', e) =ir(f). PROOF. Let f be in the equivalence class of 1.

(1) If f 4", the equality above is trivially true. (2) For f = (I,, 4" , we set = (C', and / = I/h (then / E With the usual notations, it is easy to see that the applications C, t, and c i—s 1, can be identified. Let be the parameterization of the curve in W induced by (i.e., q(t) = c). We

SALOMON OFMAN

have r1.

and

!o q = Id0, thus

(0))

(dl(c'), r1) = = = (d(loii)(O),

= (dt(0),

From Proposition 2',

we

= 1.

get p' çv(c')

('W(1/1)

theorem results from the above equalities.

d!(c'),

and the

As an application of the foregoing, we have THEOREM 3. Let 0 be a point of a complex aigebraicprojective n-manifold Z, Y = Z — {O}. The Radon transformation

-, H°(D, t9(E))

R: is infective.

PRooF. The image of the kernel of p° by the differential d is zero (cf. [01) and [02]). The theorem then results easily from Lemma 2 and Theorem 2.

6. GeneralizatIons. (i) When c' has singularities, we use the residuecurrent of [C-H) instead of the residue-class of [U; more precisely, we set

j), p)

=

function with compact f) is the residue-current and p a support in Y such that p 1 in a neighborhood of Ic'I. We may then extend the construction of the Radon transformation of and 5 to singular divisors and the results of these sections are still true for these cycles. (ii) If we consider (n, q)-differential forms fl-closed in Y, and in Cq(Y), we set G1 = {(x, c) E Y x W; x PCI) and denote by and where RCSCt

.

fi applications from G1 on Y and W, respectively. We define p, an application from a") in 110(W,,, LY) (with p + q = n) by where ii'

We define 4; and =

= ...

((F),

r,

E H4(Y,

= {F = (fe,

4; (f)+}; 4'* = 4V fl =

then

'/1

(F) and Dq =

Qfl)•

... f,) For F in ,

fl ... fl

if i, J we note the flag-cycle

F

If we assume the intersection of p compact principal divisors is not empty, we obtain the following proposition similar to Proposition I.

ThE ANALYIIC RADON TRANSFORM

'if

=

bein

and

Let

PROPOSITION.

=

g.

+

+

+

where x Dq and a quotient set Eq We can then construct a set C as in §4. We obtain a commutative diagram similar to the one in §5:

of

Eq

Using the We denote by a the application a = a' o i from Eq to injection i, we can put a structure of holomorphic line bundles on Eq•

For F= ((f1,

.

..

,

f,,)

,

in

,

we set

and p is a

where c = such that p depends

with compact support in Y f. As in §5, we can prove

1 in a neighborhood of only on the s-class of cohomology of w and on the class of

equivalence F defined by F in Eq• We may then set the following DEFINITION. The Radon Transformation (in dimension q) in Y is the application (1") JI°(D,

PE

is the section of the dual bundle Eq defined by

where

E

E

E

in §5, we write RqW for Rq&/); we obtain a theorem similar to Theorem 2:

THEOREM. For any F E, we have (C',

=

.

F where

= ic(F). REFERENCES

(A)

[A-N)

IC-H)

[C-H]

A. Andreotti, Théorème de dt!pendance algebrique sur les espaces complexes pseu4oconcaves, Bull. Soc. Math. France 91(1963), 1—38. A. Andreotti and F. Norguet, Cycles of algebraic manifolds and Ann. Sd. Norm. Sup. Nsa 25 (1971), 59—114. N. Colefi and M. Herrera, Les courants rêsiduth associEs a sine fonne mEromorphe, Lecture Notes in Math., voL 633, Springer-Verlag, Berlin and New York, 1978.

G. Gindikin and M. Henkin, Integral geometry for 8.cohomology in q-linear concave domains

[H-P)

in CF , Funktsional.

Anal. i Prilozhen. 12 (1978), 6—23.

M. Henkin and P. Polyakov, Homotopy formulas for the on CF' and the Radon-Penrose transformation, Izv. Akad. Nauk. SSSR Ser. Mat. 50 (1986), 556—597.

SALOMON OFMAN

412

[U

[01] [021

J. Leray, Le ca/cu! differentiel el inlegra! sur une variëtE anaiylique complexe, probleme de Cauchy. HI, Bull. Soc. Math. France. 87 (1959), 81—180. S. Ofman, integration des classes de d'd"-cohomologie sur les cycles analytiques, Soludu problème de I'Injectivité, .1. Math. Pures AppI. 68 (1989) 73—94. , d'd", d"-cohomologies et integration sur les cycles anaiyziques, invent. Math. 92 (1988), 389—402.

[03]

[04]

—, La transformation de Radon analytique, C.R. Acad. Sd. Sér. I Math. 307 (1988), 443—446. —, La transformation de Radon analytique en codimension I, Univcrsité Paris 7, Preprint, 1989.

UN!

Pistis 7, FRANCE

E-mail: OFMAN@FRCIRP7 I .BITNET

Proceethnp of Sympos3a in Pure Maihemaucs

Volume 52(1991), Pan 2

Technique in L2 lodge Theory on Complete KAhIer Manifolds Applications of the

TAKEO OHSAWA

1. We shall survey recent progress in the L2 theory for the s-operator on complete manifolds with emphasis on the extensions of Hodge theory to noncompact manifolds.

2. Let X be a (connected) complex manifold of dimension n, and let I1'(X) and be respectively the r-th de Rham cohomology group /C and the Dolbeault cohomotogy group of type (p, q) of X. First of all we recall two fundamental facts:

THEOREM I (Hodge-de Rham). If X is compact, H'(X) and are finite dimensional vector spaces /C. THEOREM 2 (Hedge). If X is compact and admits a Kähler metric, then there exist canonIcal C-linear isomorphisms 1': —, H"(X)

such that

=

In case X is noncompact, although neither H"(X) nor arc finite dimensional in general, there exist some significant cases where the finite dimensionality is observed. For instance, let V be a compkx analytic space

of dimension n, and let x E V be an isolated singulai point such that the germ of V at x is irreducible. By Hironaka's theorem there exist a Stein neighbourhood U p, a locally closed complex submAnifold U C pM for some N E N, and a proper holomorphic map x: U U such that is a biholomorphism. Let X = U. Then we have: PRoPOSITION 3.

Under the above situation,

1980 Subject (1985 Revision). Primary 32C35. The author was supported by Max-Planck-insthutc für Mathematik. This paper is in final form and no version of it will be submitted for publication elsewhere ©1991

00*24117/91 *1.00+8.23

413

TAKEO OHSAWA

414

Moreover !-I""(X) (q 1) depend only on the germ (X, sense that the restriction maps —.

in the

urn '(x)

are isomorphisins

for q 2

where

ft' runs

through

i/ic neighbourhoods qf

(x). In fact. Proposition 3 is a special case of a theorem of Andreotti-Grauert IlL which we shall recall briefly below. A complex manifold, denoted by X again, is said to be k-convex (respectively k-concave) if there exists a .1' — [0, (respectively —* (a, 0] for some C2 exhaustion has at least n — k + I a [—ec, 0)) such that the con!plex Hessian is positive cigenvalues outside a compact subset of X. The function called then a k-convex (resp. k-concave) exhaustion function of X, by an

abuse of language.

Let X be a k-convex (resj'. k-concave) manifold. Then

TIIE0R EM 4.

If

—

—

3fdç2f2),

q) form u. Here co denotes the funfor any compactly supported damental form of ds2. The proof is an obvious combination of the Bochner trick and Jacobiidentity technique. Remark 1 In case dco = 0, we recover the classical L2 estimate of 0. (The generalization to the bundle-valued Kodaira—Nakano by letting case is straightforward.) REMARK 2. The Jacobi identity technique is a generalization of the Bochner trick, since Kodaira—Nakano identity is nothing but a modification

of the Jacobi identity for the operators 5, A, and the holomorphic conneclion by the Kähler identity ([7]). We note that there exist real analogue of the above estimates, although they appear only partially in the literature ([12, 30]).

REMARK 3. Suitable modifications of Kodaira-Nakano identity of different kind sometimes yield the sharpest control of the growth of holomorphic differential forms and harmonic maps ([9, 10, 29, 33, 38]).

4. In what follows we suppose that X is embedded as a Zariski open subset of a complex space .?. First we shall describe the extension of Theorem 2 to the k-concave cases. The notion of k-concavity is related to the

embedding I

.Y

in the following way.

TAKEO OHSAWA

420

If

= k, then X is (k + 1)-concave. For the proof see [1] or [28]. In particular, we note that Theorem 9 can be directly applied to X. To extend Theorem 2, we need a proper generalization of Kahlerianity to complex spaces. DEnNITION. A Hermitian (respectively Kkhler) metric on a (reduced) complex space X is a Hermitian (respectively Kähler) metric ds2 on there exist a neighis smooth at x} such that, for any x {x E with U —. R bourhood U x and a C°° function PROPOSITION 10.

2ds2

ds2

(respectively ds2 =

on U n

The following was first discovered by H. Grauert 113]. Our harmonic theory will be based on it.

PROPOSITION Ii. Let ? be a compact complex space with a Hermitian (respectively Kähler) metric ds2, and let X c be any ZarEski open subset.

Then, for a =

—oo

and a =

0, there exists a C°° exhaustion function

X —. (a, I) such that ds2 + is a complete metric on X. immediately if Y\X is discrete. In fact, for each One can find such x E V\X there exist a neighbourhood U x and a holomorphic embedding (13N (U, x) 0), where 3N denotes the complex unit ball of dimension N centered at the origin 0. Letting z be the coordinate around 0, we have OOlogIJzIJ +

Therefore we obtain our X —. (—oo, I] by patching the restrictions of — log(— log to U\(x) for all x and multiplying a small positive number if necessary. For a = 0, one may use the function (log(— log llzII)Y1 instead of — log( — log liz and obtain also a complete metric. In order to generalize this construction to the case > 0, we need to use the property of ds2 that it is locally equivalent to the restriction of a metric on some nonsingular ambient space. This property of ds2 is also crucial in the proof of the following refinement of Proposition 11. PRoPosrrloN 12 (cf. Proposition 1.1 in [28J). Let the notations be as above,

and suppose that V admits a Kahler metric. Then, for any e > 0, there

on I, a C°° exhaustion function '1': (—oo, 0] and a neighbourhood W r\X such that (*) <e (**) 0, letting = A(c — 'P)' and tation we have r',1÷62,pq — —

for p+q>

=

:= {x; 'I'(x) n + dim(?\X), and

forp + q > n +

+ 1.

Here the isomorphism is induced by the natural inclusion homomorphism. COROLLARY.

Using the situation, there exist canonical

(H'(X) p+q=r

for p+q

OPEN QUESTION. Is Theorem 13 also valid for non-Kähler REMAaK. In the above proof we heavily use the Kahlerianity to absorb the term rnto ,p.q' and the author does not know how to generalize our argument to the non-Kähler case. Of course the generalization is trivial if dim(X\X) = 0.

The L2 cohomology group H(2)(X) with respect to a C°° Hermitian metric on ? obviously does not depend on the choice of the metric, so that it deserves to be studied in detail, too. The following was proved in [27] by regarding the metric on as a limit of complete metrics on X for which the L2 estimates are uniform. THEOREM 14.

If dim()\X) = 0,

then

f for p + q > n + 1, and

f 1

for r> n +

1,

with respect to any Hermitian on

NOTE. In [27] the above result is stated only for the Kähler case, but the same proof applies to the general case. REMARK 1. Little is known about the structure of

'1(X) and (X) outside the above ranges, althougb there seems to exist a relation between H(2)(X) and the intersection cohomology groups of ([6, 17, 21, 36, 37, 39]). See [22] and [35] for the Hodge structure of the intersection cohomology groups.

REMARK 2. As for the corollary to Theorem 13, a completely different proof was recently given by Arapura [3]. In case X is projective algebraic, its algebraic version exists ([4, 5]).

5. Now we turn to discuss the opposite case where X is k-convex. Compared to the k-concave case, the situation seems to be more delicate. pN EXAMPLE ([5], [14]). Let be a nonsingular projective surface, and let E Y be a rank two vector bundle defined as the kernel of a homomorphism Let X be any (algebraic) compactification of the total space of the dual

bundle E and let X be the complement of the zero section of E in Obviously X is 2-convex, but a computation shows ([5]) that dim 116(X) = I

THE 0 TECHNIQUE IN L2 HODGE THEORY

and Ep+q6dimH"(X) 2. Moreover H4'2(X)

0 and

423

= 0,

so that the Hodge symmetry does not hold either. Therefore, in order to extend Theorem 2 to k-convex Zariski open subsets we need to impose an additional condition on the boundary of X. Xc Bauer.-Kosarew [4, 5] have shown the following: THEOREM 15. Let Y he a projective algebraic variety of dimension n, is invertible, and let let Y c .V bc an algebraic subset whose ideal sheaf Y is k-a,;zple in the Suppose that the line bundle X= *

sense of Sommese.

Then

dim HT(X),

>

p+q=r

and

=

for r,p±q>n+k. NOTE. It seems to be difficult to see whether there exist canonical isomorphisms

such that in the above range.

The notion of k-ampleness has a differential geometric counterpart: Let

L be a holomorphic line bundle over a reduced complex space Y. L is said to be semipositive of rank I if there exist a fiber metric h of L is semipositive and, for any function such that the curvature of h/ p: Y 11 with supp q' cc Y, there exists e > 0 such that the curvature of the modified fiber metric h?' has at least I positive eigenvalues on THEOREM 16. Let X be a Zariski open subset ofa compact n-dimensional Kãhler manifold V. If V\X is a divisor whose normal bundle is semiposilive of rank n — k — 1, then

:fr>n+k, p+q=r

and

ifp+q>n+k+l. The proof is given in [31]. It is true that the isomorphisms are canonical in the range r, p+q > n+k+1 ([32]). We do not know whether

incasep+q=n+k+l.

* A holomorphic line bundle L —. Y is said to be k-ample if there exist positive integen, 1, N and holomorphic sections of such that the ratio (se: S . s,) , defines a morphism whose fibers have dimension k.

TAKEO OHSAWA

424

NOTE. Suppose a holomorphic line bundle L over an n-dimensional manifold Y is k-ample, but not (k — 1)-ample. Then L is semipositive of rank

n—k ifandonlyifrank 0). But clearly

d(sO:...:sN)_n—k forsomesO,...,sNE

{'semipositive of rank n — k '-bundles)

if Y is a compact

manifold with H'(Y)

{

k-ample bundles),

0.

REFERENCES

1. A. Andreotti and H. Graucrt, ThdorEnse de Jinhtude pour Ia cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193—259. 2. A. Andrec,tti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Etudes Sd. PubI. Math. 25 (1965), 81—130.

3. D. Arapura, Local cohomology of sheaves of d(fferential forms and Hodge theory, (prepnnt; Purdue University 1989). 4. 1. Bauer and S. Kosarew, On the Hodge spectral sequence for some classes of noncomplete

algebraic manifolds, Math. Ann. 284 (1989), 577—593. Some aspects of Hodge theory on non-complete algebraic manifolds, (preprint; Mathematica Goitingcnsis 1989). 6. 1. Cheeger, M. Goresky and R. MacPherson, L2-cohomology and intersection homology of singular varieties, Seminar on Differential Geometry, Ann. of Math. Stud. 102 (1982),

5. —,

303—340.

7. 1.-P. Demailly, Une preuve simple de Ia conjecture de Grauert-Rienu'nschn eider, Séminairc d'Analysc P. Lelong-P. Dolbeault—H. Skoda, Lecture Notes in Math., vol. 1295, Springer-Verlag, Berlin and New York, 1987, pp. 24—47.

8. —, Cohomology of q-convex spaces in lop degrees, Math. Z. 204 (1990). 283—295. 9. K. Diederich, G. Herbort, and 1. Ohsawa, The Bergman kernel on uniformly exiendable pseudoconvex domains, Math. Ann. 273 (1986), 471-478. 10. K. Diederich and G. Herbort, Extension of holomorphic functions with growth conditions, manuscript. 11. H. Donnelly and C. Fefferman, and index theorem for the Bergman metric, Ann. of Math. 118 (1983), 593—619. 12. H. Donnelly and F. Xavier, On the d(fferenzial form spectrum of negatively curved Riemannian manifolds, Amer. J. Math. 106 (1984), 169—185. 13. H. Graucrt, Characteriricrung der Holomorphiegebi etc durch die vollstdndlge Metrik, Math. Ann. 131, (1956), 38—75.

14. H. Grauert and 0. Riemenscbneider, KOJileroche Mannigfalligkeiten mit hyperqkonvexem Rand, Problems in Analysis, Symp. in Honor of S. Bochner, Princeton University Press, Princeton, NJ., 1970, pp. 61—79. IS. R. E. Greene and H. Wu, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier (Grenoble) 25 (1975), 215—235. 16. L HOrmander, L2-estimasesandexistencetheorenisfortlie Acta Math. 113 (1965), 89—152.

17. W. C. Hsiang and V. Pail, L2-cohomology of normal algebraic suifaces, Invent. Math. 81 (1985), 395—412.

18. H.

cohomology of (H, C)-gmupt, Pubi. ken. Inst. Math. Sd. 20, (1984),

297—317.

19. E. Looijenp, L2-coliomology of locally symmetric varieties, Comp. Math. 67 (1988), 3—20.

20. B. is

La cohomologie d'une vat Wti analytique complexe a bonipseudo-convexe par nt1cessairement (English summary) C.R. Acad. Sd. Paris Sft. A-B 208

(1975), A93—A95.

THE

TECHNIQUE IN L2 HODGE THEORY

425

21. M. Nagase, Remarks on the L2-cohomology of singular algebraic surfaces, J. Math. Soc. Japan 41 (1980), 97—116. 22. V. Navaro-Aznar, Sur Ia de i-lodge des variflés algebriques a singularizes isolées, Asténsque 130 (1985), 272—307. 23. T. Ohsawa, A reduction theorem for cohomology groups of very strongly q-convex Kãh!cr manifolds, Invcnt. Math. 63 (1981), 335—354. Addendum to: A reduction theorem for cohomology groups of very strongly qconvex Kdhler manifolds, Invent. Math. 66 (1982), 391—393. Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, 25. PubI. Ret. Inst. Math. Sd. 18 (1982), 191—232. 26. Completeness of noncompact analytic spaces, PubI. Res. inst. Math. Sd. 20 (1984),

24. _,

—, —,

683—692.

27.

28.

Hodge spectral sequence on compact Kdhler spaces, Publ. Ret. Inst. Math. Sci. 23 (1987), 265—274.

—, Hodge spectral sequence and symmetry on compact Kdhler spaces, Pubi. Res. Inst. Math. Sd. (1987), 613—625.

29. —,

On

the extension of L2 Iiolomorphic functions II, PubL Ret, Inst. Math. Sd. 24

(1988), 265—275. A generalization of the 30.

formula and an analytic approach to Morse

theory, J. Ramanujan Math. Soc. 4 (1989), 121—144. 31. —, Hodge spectral sequence on pseudoconvex domains II, (to appear Revue-Rouinaine

de Math. Pure et Appi). (in preparation). 33. T. Ohsawa and K. Takegoshi, On the extension of L2 holomorphic functions, Math. Z.

32. —,

195 (1987), 197—204.

34. —,

lodge spectral sequence on pseudoconvex domains, Math. Z. 197 (1988). 1—12. 35. M. Saito, Modules dellodge polarizables, Pubi. Ret. Inst. Math. Sd. 24(1988), 849-995. 36. L. Saper, L2cohomology and intersection homology of certain algebraic varieties with isolated singularities, Invent. Math. 82 (1985), 207—255. 37. L. Saper and M. Stern, L2.cohomology of arithmetic varieties, Proc. Nat. Acad. Sd. U.S.A. 84(1987), 5516—5519, Ann. of Math. 132 (1990), 1—69. 38. K. Takegoshi, Energy estimates and Liouville theorems for harmonic maps, (to appear Ann. E. Norm. Sup.). 39. S. Zucker, of warped products and arithmetic group,, Invent. Math. 79 (1982), 169—218.

Kyoro

JAPAN

of Symposli m Pwe Mathematics

Volume 52(1991),Pszll

Bounds on the Derivatives of Holomorphic Endomorphisms MARLUS OVERHOLT volAbstract. We use the Kobayashi metric and the ume to find points in certain bounded domains where the derivatives of all holomorphic endomorphisms of the domain simultaneously satisfy a good bound. The resulting inequalities resemble those that occur in Cartan's Lemma, but do not involve fix-points.

0. IntroductIon. Let Q c C' be a domain. Unless otherwise stated, al is the set of domains in this paper are bounded. The set End(fl) = to be the We define holomorphic endomorphisms of the domain to be the supremum and distance from z to the boundary ÔQ of

of

as

z rangesover

The Kobayashi metric on Q x C" is given by

where the supremum is taken over all with

f'(O) =

and 1(0) =

> 0

for which there exists

z. Since

an I e it

=

is sometimes convenient to consider as a function on Q x C!'"'. It is well known that bolomorphic mappings are distance-decreasing with respect to the Kobayashi metric. In particular

0. Since f0(D) ç and again to see that we may assume —' istaut, f0(D)çQ or But f0(0)=p€OQ,so a = 0, we have a nondegeneratç analytic disk n Since contradiction. The usefulness of Proposition I is enhanced by a simple geometric condition that is sufficient for a domain to be taut and its boundary of analytic disks. A domain fl c C is said to (locally) strictly convexifiable if for

any p E OQ, there exists a neighborhood U of p in C, anda biholomoris geometrically phic mapping h: U h(U) C, such that h(U n strictly convex with respect to the side on which h(U Cl Q) lies. In particular, a strongly pseudoconvex domain is strictly convexiflable by Narasimhan's lemma. PROPOSiTION 2.

If

c C" is strictly

then £) is taut and

OQ does not contain any analytic disks. PROOF. The problem is local, for if the boundary contains an analytic disk, then it contains a small analytic disk, and similarly with the tautness. Let g: D —, be an analytic disk with g(D) £ and g(D) and so small that there is a neighborhood U of g(D) in C" and a biholomorphism h: U —. h(U) c C so that h(U fl c9Q) is strictly convex with respect to

the side on which h(U Cl

h(U ii Ofl) ç OD, and if domain, a contradiction.

lies. Then there is a convex domain D with were not taut, D would be a nontaut convex

429

DERIVATIVES OF HOLOMORPHIC ENDOMORPHISMS

Let g: D —, M be a nonconstant analytic disk so small that a U and h exist as before. Let j: C C' be a complex affine mapping with (j o hog)(D) ç {Re(z1) O} and (johog)(0) = 0. Then Re(irojohog) is

harmonic on D. Here ir(z)

= z1 is the projection onto the first coordinate.

We have Re(iroj ohog)(z) 0 there exists some

PRoPosITIoN 3.

Qx

point (a, is taut and

such that 1+e for all f€End(Q). does not contain analytic disks, we may take e = 0.

has a positive infimum Q on Q x Foreach e >0 we can findapoint (a, forwhich We may obviously assume that (1 + e)Q. Let f otherwise there is nothing to prove. We have

If

is bounded,

PROOF. Since

0,

,

= F(f(a),

q) (I + e)Q.

By canceling the common factor Q, we obtain the first claim. Now assume taut and free of analytic disks. For each m By Proposition 1 we can extract a convergent subsequence

1 + l/m(j), for all f E End(Q). But f, so we obtain

is continuous in (z, for any single fE It Is natural to ask whether one can take 8 = 0 in Proposition 3 if is only assumed bounded. By Proposition 2 we can take e = 0 if fl is a bounded strictly convexiflable domain. It would be nice if one could replace the particular direction Sition 3 with an arbitrary direction 5. But the example 1)

shows

in

Propo-

f(z1, z2)=(z2/2, 2z1)

and

this to be impossible. We have

'

rn

I

"

I

'UoJ)L2 0

0—

2

forall ZE(1. At a fix-point, we have the following PROPOSITION 4.

vector

C' with R(C)

1

m)

431


In particular, there exists a compact set K ç Q such that Zm K. We can then extract a convergent subsequence Zm(j) —. a E K, and clearly a satisfies the claimed boundary distance estimate. Now for any f E End(Q), we have

(1 + and thus

Jdet(f'(zm(J)))! I + l/m(j). det(f'(z))I is a continuous function of z for any fixed f, and so End(Q). det(f'(a))I < 1 for all

But I

I

f

It follows from a result in [2) that when n 1 the dependence on y(fl) in the boundary distance estimate of Theorem 1 can be removed. Can it be removed in general? Can the dependence on n be removed? 3. Convex domains. For convex domains it is possible to make fairly detailed statements about the set of minimum points of the Kobayashi metric. In particular we can show that for a bounded strictly convex domain there is a unique minimum point that depends continuously on the tangent direction. We also show that for a bounded convex domain that is symmetric about the origin, the origin is a point where all the eigenvalues of the derivatives of all the holomorphic endomorph isms are bounded by one in absolute value. x (C\{O}) An extremal mapping for the Kobayashi metric at (z0, with f'(O) and f(O) = is an f Q(D) such that = z0. Lempert 111 has geometrically characterized the extremal mappings of sufficiently smooth strictly convex domains. We will require a weaker, but For this we need the concept of a more general result due to Poletsky

p-pseudoconvex domain. It is a domain C C for which there exists a plurisubharmonic function u E and a constant c > 0 such that u = 0 cju(z)I. Note that any convex domain on u 0 for which there exists an E Q(D) such that

=

and

(if1 i- (1 — t)f2)'(O) (if1 + (1 — t)f2)(0)

=

Now

(t.Z1 + (I — + (1 — t)z2 = 23,

so I

= supA > t supA + (I — I) sup). = —

I

1

+

I—I

where we have used that if1 + (1 — t)f, is a competitor in the extremal problem sup23. Now assume that is strictly convex. Since fl is taut, sup j= 1,2,3, is attained, say by j 1,2,3. If we have equality in the inequality that asserts that l/F is concave, then if1 + (1 — must be an extremal mapping. Put E {e0: f1(e'°) f2(e°)}. Then + (1 — z)f2)(e°) E (1 for e0 E E. By Poletsky's results, E must have zero linear measure, and by Fatou's theorem it follows that f1 But then z1 = f1 (0) = 12(0) = 22, a contradiction. Since f is taut, is continuous. Furthermore, by —, +00 at Propositions 1 and 2. Thus has a minimum point , which is unique by the strct concavity of By the continuity of

is Continuous as a function of

Thus —

But since that

is the unique minimum point of Ff(z,

we conclude

a must be continuous.

Proposition 5 is a generalization of a result about the that is due to Minda and Wright (3].

mewic

if 1) ç C" is convex añ'd symmetric about the origin, 1 for any eigenvalue A of f'(O) for all f E

PROPOSITION 6.

then IAI

E put g(z) (1(z) — f(—z))/2 Then g E End((l) by the convexity and symmetry of Since g(0) = 0, Cartati's Lemma applies, and since f'(O) = g'(O), the claim follows.

For a bounded convex domain• symmetric about the origin, we have for any by Proposition 5. Proposition 6 can be proved by using minimum points instead of appealing to Cartan's Lemma.


433

4. Concluding remarks. Related results in the context of functions of one complex variable appear in a joint paper with David Minda [2]. I would like to acknowledge a suggestion by Ian Graham that enabled me to improve my original version of Theorem 1. BInLIooaJ1PHY 1. L. Lempert, La meirique de Kobayasht ella representation des dotnaines sur Ia boule, Bull Soc. Math. France 109 (1981), 427—474. 2. D. Minda and M. Overholt, The minimum points of the hyperbolic metric (to appear). 3. D. Minda and D. J. Wright, Univalence criteria and the hyperbolic metric, Rocky Mountain 3. Math. 12 (1982), 471—479. 4. D. A. Pelles, 1-folomorphic maps which preserve intrinsic measure, Amer. J. Math. 97 (1975), 1—15. 5. E. A. Poletsky, The Euler-Lagrange equations for extremal holomorphic mappings of the unit disk, Michigan Math. 3. 30 (1983), 317—333. UNIVERSITY OF TENNESSEE, KNOXVILLE

Pmcaedinp of Symposia in Pure Mathematics Volume 52(1991). Part 2

Monge-Ampére Functions

with Large Center GIORGIO PATRIZIO AND PIT-MANN WONG

Introduction. The complex homogeneous equation provides a very useful tool in the classification of Stein manifolds. The first work in this direction is the well-known theorem of Stoll [SI (see, also, Burns [BJ and Wong [W]), classifying C and the unit ball in C" by the existence of a nonnegative smooth strictly plurisubharmonic exhaustion r (bounded

for the ball, unbounded for C) such that the condition (ddc log r)" = 0 holds on the set {r > 0). Under these assumptions, the center S, which by definition is the set on which the exhaustion r attains its minimum, namely {r = 0}, consists of exactly one point. Recall that the minimal set of a nonnegative strictly plurisubbarnionic exhaustion is totally real, hence its dimension is at most half the (real) ditnension of the ambient manifold. Stoll's theorem is the case where the dimension of the minimal set is zero. We consider in this paper the class of complex manifolds (of complex dimension n) admitting certain nonnegative Monge-Ampère functions such that the minimal set (or center) is of maximum possible dimension, namely n. In § 1 we present a class of examples of this kind, namely the tangent bundles of compact symmetric spaces of rank one. These manifolds have been extensively studied and characterized from the point of view of group theory (cf. [MN] or [115]). In §2 it is shown that geometric informations (of the foliation and of the center) can be derived directly from the Monge-Ampère function. These results provide a function theoretic characterization of the tangent bundles of compact symmetric spaces of rank one in the spirit of Stoll's theorem. In §3 examples of manifolds 1980 Mathematics Subject Classification (1985 Revision). Primary 32F99. The research of the first author was partially supported by MN. The research of the second author was supported by a Sloan Fellowship and NSF Grant #87-02144.

This paper is in final form and no version of it will be submitted for publication elsewhere. American Mathematical Society 0082.0717/91 51.00+8.25

435

PATRIZIO AND PIT-MANN WONG

436

with bounded Monge-Ampère function are presented and hyperbolic metrics are constructed on these manifolds along the line of ideas developed in the article {LPWJ. 1. Examples. The list of compact symmetric spaces of rank 1 is well known. It consists of

(I) the n-sphere, S", for n 2; (H) the real projective space, P'(R), n 2; (III) the complex projective space, P"(C), n 1; (IV) the quaternionic projective space, n 1; (V) the Caley projective plane P2(Cayley). With the exception of the real projective spaces, the spaces on the list are simply-connected. The tangent bundle of a compact symmetric space admits a natural complex structure with the symmetric space embedded as a totally real submanifold. Standard affine algebraic models of these manifolds are well known (cf. Morimoto-Nagano [MN], Huckleberry-Snow [HW], and Azad (AZ]); they will be denoted by

M' = Qfl = complex affine hyperquadric, n 2, M1 = ?(C) — = compact quadric, it 2, (1.1)

n

1,

N=n(2n—1)—l,

is a 16-dimensional Stein manifold, where '(C) is a hyperplane at infinity in the complex projective space pN(C) the upper index indicates the complex dimension of the manifold,

and the lower index indicates the rank one symmetric space embedded, as a totally real submanifold, in the complex manifold. The existence of special Stein exhaustions on these manifolds is given by the following theorem. For each of the Stein manifolds defined in (1.1), there exists a real analytic strictly plurisubharmonic exhaustion .p: M —+ [1, oo) with the property that the center S = {p = 1 }, is the given symmetric space and dimk S = dime M. Furthermore, the plurisubharmonic function u = cosh 'p satisfies the homogeneous Monge-Ampere equation on M — S. The closure of a leaf of the Monge-A foliation on M — S is biholomorphic to a one-dimensional complex quadric, and its intersection with the center S is a closed geodesic (in the natural invariant metric of the symmetric space S). Converse'v, each closed geodesic of S corresponds to exactly one leaf of the foliation. In particular the leafspace of the foliation is the space of oriented geodesics (every geodesic is closed) of S. THEOREM 1.1.

The case of a hyperquadric Q" = {z E

hard to see. The function u =

+

+1z,j2

1} is not

p, where p is the restriction of the

MONGE-AMPtRE FUI4CFIONS

437

standard exhaustion IIziI2 of to Q', is plurisubharmonic on Qfl and strictly pseudoconvex in the complex directions tangential to the level sets of p. The minimal set is easily seen to be the unit sphere S' in The vector field V— a

tangential to Q' and ddcu( V, F) = 0, hence (ddcu)n 0. The leaves of the Monge-Ampère foliation are precisely the complexification of the great circles (hence are the one-dimensional quadrics) of the sphere 5". is

The real projective space P' (R) is naturally embedded in the complex projective space P"(C) as a totally real submanifold: P'2(R) = {[z] P"(C)I[z) be the compact hyperquadric in P"(C) given by the equa= [21}. Let n = 0. Thus P"(R) is contained, as a tion = 0. Then = P"(C)—Q totally real submanifold, in the affine algebraic manifold —, = defined by The map it:

[z0:

:

—, tz3/z0: ••:

:

a two-fold unramified covering. Mapping (2 to I) the center of Q" onto Qfl exhaustion on descends to the center P"(R) of The satisfying the homogeneous Mongea plurisubharmomc exhaustion on is

Ampere equation outside the center P' (R). By construction, the leaves of the foliation are exactly the images of the Monge-Ampbre foliation on Q' under it. Thus the closure of each leafmeçts P"(R) in the image of a great circle on the center S' of Q" and therefore in a geodesic, in the induced metric, of P'(R). Coilversely, for any geodesic of P"(R) there exists a unique leaf passing through it. Furthermore the restriction of p to the closure of any leaf L is an exhaustion with no critical point outside the minimal set L n Since L n P"(R) is a simple closed curve, it follows that L has the same homotopy type of a one-dimensional quadric and therefore it is biholomorphic to one. Denote by the underlying real space of the complex projective space P'(C) with the Fubini-Study metric. The space is naturally embedded

as a totally real submanifold in the complex manifold P"(C) x P"(C) via [z) —. ((2], [TJ). The product space P"(C) x P"(C) is embedded in PN(C), where N=(n+l)2—l,viatheSegreembedding Oa, P n, = where denotes the homogeneous coordinates on P'1'(C). The (image of the) totally real subspace P(C) is given by = 0 a, ft < 1*, which is easily seen lobe disjointed from the hyperplane at infinity defined by the equation = 0. Thus P(C) is a totally real subspace of the complex affine algebraic manifold

L

= P"(C) x P"(C) -

c cN

pN(C)

—

GIORGIO PATRIZIO AND PIT-MANN WONG

438

Define a plurisubharmonic exhaustion on PN(C)

by

—

i. I

On

the function V

>11

0,

if (I, A) is

PRooF. The growth condition in Definition 2 gives

L,

tim

L, nf)=0.

lim

Now let — A, L) denote the (algebraic) cohomology of the line bundle L. Then we have

PRoPosrnoN3. where Xis projective and is a differentiable function belonging to (X, A).

Before going into the proof let us see how this proves Theorem 1. By — A, L) = 0 for each line bundle L on Propositions 2 and 3 we get X. Now I isprojective, so Y) = 0 for each coherent sheaf and

q>0. ByacriterionofSerre, I—A isafilnewhichimptiesthat (X,A) is

I is a divisor on X. If ç'

affine. The proposition has been proved in [M] for the smooth case. —' be a modification such that

PRooF OF PROPOSITION 3. Let -. is biholomorphic and

a function belonging to (X, A), the function

(I,

q' o ir

is

need not belong to

But by the Nullstellensatz we get

a < fi, where

belongs to (X, A). Hence llmJf(2)(X — A,

,r'(A), ,rL, n(çooir))

L, =

Of course, H q(1

— A, L)

H

—

—

'(A),

i('(A), ,rL, np). ir

L), so we may assume that

AcX isadivisor. Let D bealinebundlewithasection s suchthat A isthezerosetofs. For m> n, multiplication with gives a map L®D") L®Dm), and there is a natural isomorphism — A, L) lim So we have to prove L 0 D") = — A, L,

L®

where we can

AFFINE VARIETIES

L ® JY) as an analytic cohomology group because X is interpret projective. But this isomorphism can be constructed by means of a fine resolution

L,

L, nço)—..

are the L2-integrable (0, q)-forms on X — (X0uA), L, The cohomology of the (X0 U A) with respect to the weight function above complex is just the L2-cohomology. where

—

5. An application. THEOREM 3. Let X c P,, be a branched covering, X being locally 1,-reducible with branching locus V c X. Then X — V is an affine variety.

We only give the idea of the proof. Let d be the lifting of the Fubini metric on to X which degenerates on V. Let = — log where is the distance to V in this metric. This function is not differentiable, but by smoothing we obtain a function with the required properties in Definition 2. We get that (X, V) is algebraically 1-complete, hence (X, V) is weakly affine by Theorem 1.

If X is smooth, Em (E] has proved that V is an ample divisor. Our theorem follows from his result in this case, It is open if Theorem 3 is valid over arbitrary algebraically closed fields. REFERENCES L. Em, The ramification divisors for branched coverings of P, Math. Ann. 261 (1982),

[EJ

4*3—485.

[MJ

fOl

F. Deligne), Asttnsquc 17 (1974), 141—160. T. Ohsawa, Vanishing theorems on complete Kdhler manifolds, Pubi. Res. Inst. Math. Sd. 20 (1984), 21—38. 0. Maltsinoitis, CIA GA affine

KIRCHaERO 59, D-4006

2, GERMANY

Proceedinp o( SyiEposta ie Pure 2 Vohame 52(1991).

Compactifications of C'1: A Survey THOMAS PETERNELL AND MICHAEL SCHNEIDER

Introduction. Nowadays it is "obvious" that is the natural compactification of This was not so clear from the beginning when Osgood x as the TMrigbt" compactification of C2. proposed By a compactification (1, 1) of C we understand an irreducible compact complex space X (mostly smooth), a closed analytic subspace Y C X

such that I \ Y is biholomorphic to C'. Y then is automatically a hypersurface. Of course the only smooth cornpactification of C is P1. For n = 2 there are infinitely many: just take the Hirzebruch surfaces I = = P(t9etf(—k)) and set Y = FUC0, where F is the fibre of Ek and C0 is the exceptional section of If one assumes Y to be irreducible (and I to be smooth) then (I, Y) (P2, P1) by a result of Remmert and van de Yen [RV]. Note that the smoothness of X is important for this result; just take as above the Hirzebruch surfaces Zk and set

Y=F. In his famous 1954 problem list, Hirzebruch [H] proposed the problem of

classifying all compactifications (I, Y) of C' with I smooth and Y irreducible. The object of this survey paper is to report on the progress made on this question recently. A concluding section is devoted to open problems, hopefully serving as a stimulus for future research on these matters. 1. PrelimInaries. Here we collect some basic facts about compactifications which are used frequently in the sequel. 1.1. DEFINITION. A pair (X, Y) consisting of a connected compact com-

plex manifold I and a closed analytic subset Y is called a (smooth) cornpactification of C' 1ff I \ Y C" (biholomorphically). Two compactifications (X, Y) and (I', Y') are isomorphic ill there is a biholomorphism 1980 Mathematics Subject Classification (1985 Revision). Primary 32J05, 14i40. This paper is in final form and no version of it will be submitted for publication elsewhere. © 1991 American Mathentaitcal Society 0082.0717/91 per page

455

THOMAS PETERNELL AND MICHAEL SCHNEIDER

456

It is clear that Y is purely 1-codimensional i.e. a hypersurface. Since C" is contractible, the standard exact sequences for the pair (I, Y), the universal coefficient theorem, and Poincare duality give the following topological information.

Let (X, Y) be a (smooth) compactification of C".

1.2. PROPOSiTION.

Then

H'(Y, Z) for

(a) H'(X, Z)

2n —2,

for i2n—2,

H1(Y,Z)

H'(X, Z) = 0,

H1(X, Z) 0, (c) b2(X) = dime H2(X, R) is the number of irreducible components of Y.

(b)

We will be concerned mostly with compactiflcations (I, Y) of C" with Y irreducible, i.e. b2(X) = 1. Using 1.2, the exponential sequence and Hodge-decomposition, one easily obtains:

Let (I, Y) be a (smooth) compacujication of C" Then with Y irreducible, and let L = 1.3.

PROPOSITION.

(a) c1(L) generates (b) If X is Moishezon, then

and Pic(X)

is generated by L. Much deeper information is given by the following result of Kodaira [K]. 1.4. PRoposiTioN.

Let (X, Y) be a (smooth) compactification of C".

Then

H°(X,o1)_—O form1, where denotes the canonical line bundle of X. In other wods the Kodaira dimension ,c(X) = —00.

In case b2(X) =

1

this has the following important corollary.

Let (I, Y) be a (smooth) projective compactification of C" with irreducible Y. Then X is a Fano manifold. 1.5. COROLLARY.

Since Pic(X)

Z and X has ample line bundles it follows from 1.4 that is ample i.e. X is a Fano manifold. If b2(X) 2 Corollary 1.5 is no longer true c.f. §5. Fano manifolds of dimension 3 are classified [I], (IS], [MM]. This is one of the reasons that one has a rather good understanding of compactifications of C3 with b2 = 1.

COMPACTIFICATIONS OF C": A SURVEY

457

2. Semiclassical results. In this section we will report on the progress made on the compactification problem in the sixties and seventies. As mentioned in the introduction, the two-dimensional case with irreducible curve at infinity was completely settled in 1960 by Remmert and van de Yen [RV].

2.1. THEOREM. Any smooth compactification (X, Y) of C2 with irreducible Y is isomorphic to (p2' Ps). Of course the situation is completely different for b2(X)> 1. A solution in this case has been given by Morrow [Mo]. His results show in particular that X is always rational. Prior to this van de Yen [V] investigated the higher dimensional case and proved among other things: 2.2. THEOREM.

(i) Any compactification of C2 is algebraic. (ii) Any projective compactification (X, Y) of

n5.

phicto

Y smooth, is isomor-

One compactification of C3 with irreducible but singular surface at infinity different from P3 was well known: X Q3 C P4 is the smooth 3-dimensional quadric and Y = Q0 is the quadric cone obtained by intersecting Q3 with a hyperplane tangent to Q3. In [V} it was speculated that these were the only compactifications of C3 with b2 I. We will see later on that this is not the case. In the seventies Brenton and Morrow investigated compactifications of C" and related questions. Using Brenton's work (Bi, 2] on singular surfaces they IBM] were able to show that a smooth compactification X of C3 with normal irreducible surface Y at infinity is automatically projective provided b3(X) I. This latter condition is automatic (c.f. (PS]). 3. Fano manifolds. In this section we collect the facts on Fano manifolds which are relevant in the subsequent sections. 3.1. DEFINITION. A projective manifold X is a Fano manifold if is ample.

The index of a Fano variety is the largest positive integer r such that = Hr for some H E Pic(X). r n + 1, n dimX. This can be seen easily by considREMARK. I ering the polynomial P(t) = co'). Kodaira vanishing implies that P

haszeroesat 1=—I, —2,..., —r+1 and P(0)= 1. Since degPn we get r — I

n.

A first important result is due to Kobayashi and Ochiai [KO],

3.2. THEOREM. Let X be an n-dimensional Fano variety of index r. Then (i)

r

n

I

a smooth hyperquadric.

ThOMAS PETERNELL AND MICHAEL SCHNEIDER

458

As an easy consequence of this one gets back the theorem of Remmert and van de Yen. Just use the fact that the Fano surfaces of index 1 are the Del Pezzo surfaces with b2 > 1. Fano and later Roth have given an (incomplete) classification of Fano 3-folds. Iskovskih [II provided a modern and almost-complete classification of Fano 3-folds of index 2 and of index one provided b2 = I His work was done under the additional auumptions that Fano 3-folds contain lines and that the anticanonical system has smooth members. This was established later by Shokurov (Shi, 21. The classification

of Fano 3-folds with index one and b2 2 has been done by Mon and Mukai [MM). Two invariants of Fano 3-folds are important:

g = g(X) = d = d(X) =

+ c1

(H)3.

g is called the genus and d the degree of X (H is uniquely determined by

(4 = H').

As an example we state the classification of Fano 3-folds of index 2 and second Betti number one d

g

b3

type

1

5

2

9 13 17 21

42 20

cone double cover of the ', along a quartic duubie cubic 3-fold in £4 complete intersection of two quadrics in section of Gr(2, 5) C P9 by a linear subspace of codimension 3

3

4 5

10

4 0

We will see in the next section that only the last 3-fold occurs as a cornpactification of C3 of index 2. ft is usually denoted by V3. Fano 3-folds of index one and b2 = I satisfy the estimate g < 12. In the next section we will see that the compactifications of C3 of index 1 and b2 = necessarily have genus 12. 1

4. Compactlficatfona of C3 with b2 1. In this section we describe an (almost) complete classification of all smooth projective compactifications

(X, 1) of C3 with b2(X)= 1. 4.1. EXAMPLE (Furushima [Full). Let X be the Fano 3-fold of index 2 b2 = I and deg = 5 i.e. X = Gr(2, 5)nS, where (3r(2, 5) is embedded (via Plucker) in P9 and S is a generic linear subspace of coditnension 3. X is a compactification of C3 in two ways, more precisely there exists with

COMPACrIFICATIONS OF

A SURVEY

459

Y c X, a normal surface and Z c X a nonnormal surface, such that

in the 3dimensional smooth quadric Q3 C P. Choose points p0, p E I and tangent hyperplane sections H0, H at the points p0 respectively p. Let C be a twisted cubic curve contained in H0. Then necessarily p0 E C. Let it the strict transform Q3 be the blow-up of C c Q3 and denote by of in 2 . Then

To see this we need another description of X. Take a line

I

I

H0

—'

—. P1 to get a smooth 32 can be blown down along the ruling fold X, c I X. X is nothing else than our Fano 3-fold of index 2,

and

b2(X)= 1, degX=5. Themap o:X—'X istheblowupofaline /0cX with normal bundle

—1)

—1).

Now let H be the strict transform of H in X and let Y = a(H). It can be checked that V is a rational normal surface with exactly one A4singularity and X \ Y C3. To obtain the compactification with nonnormal surface at infinity let Z = ai('(C). Z is a nonnormal surface (isomorphic by identifying the excep= to the surface obtained from C3 we get the desire tional section and a fibre). Since X \ Z Q3 \ compactification.

For another construction of (X, Y) we refer to [FNI). Now we return to the classification of compactifications X of C3 with b2(X) = 1. Since I is a Fano-manifold we may assume by 3.2 that the index r of X is either one or two. We fix the ample generator L of Pic(X) Z. 4.2. PRopositioN.

Let (X, Y) be a

= I and index r = of C3 with (i) c1(L)3=5.

smooth projective compactification 2. Then

(ii) if V is nonnormal, the nonnormal locus E of V is a line i.e. E

and LE= 1.

The proof of 4.2 can be found in [PS] and we make only some remarks about (i). If Y is normal, then Y is a Gorenstein surface with b2(Y) = I and ample These surfaces have been classified by Brenton [B2J. Going through his list it is easy to get c1(L)3 5. If V is nonnormal we use Iskovski's list (cf. §3) and check all four cases

with d 0 and ca1 is

In case of dim Z = I one has Z P1 and for dim Z = 2 one gets For dim Z = 3, Z is a—possibly singular—Fano variety. In this Z case Z has only isolated terminal singularities of very restricted type (see [M] for a list). The contraction (p is given by an extremal rational curve I i.e. I is a rational curve with 0 < I 4 dimX + and R÷[/] is extremal in the cone of numerical equivalence classes of effective curves. 1

Since 1's, 1'2 form a basis of Pic(X) Z2 it is natural to consider the = .1. Up to permutation there are 3 cases: (a) a1> 0, cr2 > 0,

numbers

(b) a1>O,a2 This metric was also studied in the physical literature. See [4, 5, 6, 7]. In §2 we show how to find a potential of this metric on Dir(S')/PSU1 because we can interpret as all possible C°° complex structures on D = {'I' E C and < l}. Each point of Dir(S')/PSU11 defines an elliptic operator that acts on the space of all quadratic differentials on S'. The operator is defined by — where (1

—

{t E Cftf > 1} such that

.

is a univalent function defined in D := = on S', and S(f(4)) is the

1980 Mathematics Subject CIas.sification (1985 Revision). Primary 32013; Secondary 58B20, 30F20. This first author was partially supported by NSF grant DMS-8922699. This paper is in final form and no version of it will be submitted for publication elsewhere. 1991

Ameiican Mathematical Society

0082.0711/91

467

51.00+5.25

T. RATIU AND A. TODOROV

468

Schwarzian derivative. Then using Quillen's construction we get the so-called determinant line bundle This line bundle has a natural section deQuillen's metric on is defined in the following noted by !1, is the conjugate where way: := :=

with respect to the Poincarë metric. Let Cs(s) = , where A, are of the eigenvalues of "Almost" copying the proof of a theorem of Quillen in [21] we prove:

=

THEOREM.

Petersson metric on .7 :=

is just the Well-

Difr (S' )/IFSU1

This turns out to be part of a more general principle, which was also observed by Fujiki and Schumacher. See [10]. Namely, we prove that the Quillen metric is the potential for the Weil-Petersson metric, whenever it is defined. This principle is formulated precisely at the end of the article. In this paper we are concerned with a 1. The space very particular example of a coadjoint orbit, defined by a central extension. Let us denote by G := the orientation preserving diffeomorphisms of the circle relative to the length form di The Lie algebra of

Dir(S') consists of Vect(S'), the space of vector fields on S', endowed with minus the usual Lie bracket on vector fields. Relative to the L2-pairing, the "dual" of Vect(51) consists of quadratic differentials. Gelfand and Fuks

proved that up to an isomorphism there exists a unique central extension of Vect(S') by R. (See [12].) This central extension vir is called Virasoro algebra. Bott proved that up to an isomorphism there exists a unique Lie group, which is a central extension of by R/Z. See [3]. This group is called the Bott-Virasoro group. 1.1. THEOREM extension (1.1)

Thus the

[151 and Witten [24]). The corresponding central Vir is the coadjoint isotropy of

(Kirillov

of PS U11

in

p0(2(dt)

,—l)Evir

foranyp0ER.

right quotient Dir(S' )/PSU11 is diffeomorphic

orbit of Vir in vir through the point p0(2(dt)®2 ,

to the coadjoini

— I).

We will use this theorem in order to define up to a constant a unique left invariant metric on 1.2. DEFINITION. A complex structure on a manifold M is a global splittingof the complexified tangent bundle TM ® C = T1'0 Ta" such that = T°" plus the integrability condition, i.e. the bracket of two (1, 0) vector fields is again (1, 0) vector field.

Let Vect(St) ® C be all complex-valued vector fields on S1. It is easy to

THE WEIL-PETERSSON METRiC

469

see that (1.2)

Vect(S') 0 C =

L

,

is naturally a subalgebra in Vect(S' ) ® C spanned by {L..1, L0, L1}. Thus the complexified tangent space at id(modPSU1 in Dilr(S')/PSU, can be identified with The Lie algebra of PSU1

1.3 DEFINITION. Let

(1.3) Use

the right action of Diff'(S') on S to define T' .0y and T°'

Then it is easy to check that for every pair of vector fields u and v of type (1, 0), Eu, v] is again of type (1, 0). So the above defined complex structure on S is integrable. Moreover, if I denotes the complex structure operator defined by the splitting in (1.3), it is easy to check that (1.4)

Iv =

Clearly

where v =

—

VmLm•

we defined a left invariant complex structure on 5.

1.4. DEFINITION. Let w be the Kirillov-Kostant..Souriau form on 5 (S' )/PS U1,1 thought of as the coadjoint orbit of the Bat:- Virasoro group that passes through = (2(dt)®2 , — 1) and let I be the complex structure op-

erator on 5 defined by the splitting in 1.3. The metric (u, v) := co(u, Iv) is called the Weil-Petersson metric on 5. 1.5. THEOREM.

(A) Let

v,, exp(2r

v = I

and w =

n>1

exp(2ir

1

n>1

of type (1, 0). Then

two left invariant vector fields on the Weil-Petersson metric is given by be

(1.5)

(v,w)=

wherea>0.

(B) Let H1(S') := {ai/ H3 vectorfields v on 5' of type (0, 1) for s> Then H1 (5') is a complete Hubert space with respect to the metric defined in (A).

PROOF.

Part (A) of this theorem follows directly from the definition of

the two cocycles that define the Virasoro algebra, i.e., ELmS

= (m — n)Lm+n +

—

T. RATIU AND A. TODOROV

470

From this formu!a, the definition of the orbit symplectic form w and by direct computation we get that (1.7)

Lm) =

From (1.4) and (1.7) we get that C

co(u, Iv)

.8)

>(m

3 —

m)UmVm.

This series is absolutely convergent since u and v are of class s> 4. The fact So (1.8) shows that Iv) is a metric on Difr(S' )/PSU1 that this metric is a Kähler metric follows from the fact that w is a closed form on Y. So part (A) of Theorem 1.5 is proved. i.e. Part (B). Let v bea it vector field of type (1,0) on S' for s> 2 and (1+m2s ) 2

Then u =: p1(p) =: The function

G —, Z÷ is

—

p)

uniformly for all z E U.

is said to be the zero multiplicity of f at p. called the zero-divisor of f.

A function v: G Z is said to be a divisor of G, if every point of G is contained in an open, connected neighborhood U in G carrying two holomorphic functions g 0 and h 0 such that (1.7)

z/IU=Pg—/1h.

of m—2 canbe required. The set 0 functioning as the null-divisor. The closure S in G of {z E 0} is called the support of G. Also the divisor 1 can be taken in (1.7). If v 0, the is nonnegative, if and only if ,'z support S of v is a pure (m — 1)-dimensional analytic subset of G. if A is is the regular a pure (m — 1)-dimensional analytic subset of G and if set of A and if 1(A) is the singular set of A, then there exists one and only = A and L14(z) = 1 for all z one divisor 11A such that supp Then 0. If is a locally finite family of analytic sets of pure dimension m — and if is a family of integers then In addition divisors is a module with v

1

(1.8)

>2bAzJA 2E/t

is a divisor, which is nonnegative if and only if bA 0 for all A E A. If ii is a divisor on G with support S 34 0, then v is locally constant on SR(S). Let be the set of branches of S. For each B E there is a unique integer a8 such that (1.9)

zJ=>2a8z18.

If v1 and are divisors on G, and if E is an analytic subset of G with dimE < m—2, then v1 0, the counting function is defined by (1.10)

= r2_2mf 5[r]

0m—1

=

f

S[rJ

+

CARTAN CONJECTURE FOR MOVING TARGETS

479

if m>l. Ifm=1,then v(z). ZES[rl

If

for some polynomial F, if 0 increases. Also v = 0, then ii is bounded. For 0 < s < r, the valence function and only if ii 0 and of a divisor ii on M is defined by (1.12)

s) =

f

be a meromorphic function on the open, connected subset G of M. Take p G, then there is an open, connected neighborhood U of p in G carrying holomorphic functions g and h 0, such that hi = g and such m — 2. Take a C with f a. Then the athat fl multiplicity = /'g-h.a(P) and the pole multiplicity do not depend on the choice of g and h, and define the nonnegative aon G. If I 0, then and nonnegative pole divisor divisor 0 and k 0 are is called the divisor of f. If f Pj.o — 0 is meromorphic meromorphic functions on G, then If = +/Sk. on M, then Jensen's formula states

f

(1.13)

/

M(')

M(s)

Let V be a hermitian vector space of dimension n + 1 > 1. Naturally the previous notations apply and are distinguished by an index V where is an nconfusion may be possible. The projective space P(V) = P(V) be dimensional, connected compact complex manifold. Let F: V V abbreviate F(S) := n The dual the residue class map. If S c P(S vector space V of V consists of all C-linear functions a: V —. C and we call (a, := a(s) be the inner product of a E V and E V. If a 0, the n-dimensional linear subspace E[a] = ker a depends on a = P(a) E P(V) only and E[al = P(E[a]) is a hyperplane in P(V). Thus P(V)

bijectively parametrizes the hyperplanes in P( V). The dual vector space V is hermitian such that the dual base of each orthonormal base is orthonormal. Schwarz inequality holds V. The distance from a)I s II;II hail for x = P(s) to E[a) with a = P(a) P(V) is defined by (1.14)

0 < Ox; aD

I.

= The Fubini-Study form on P( V) is the unique positive form of bidegree (1, 1) on P(V) such that = wi,. V is said to be meromorphic, A holomorphic vector function m: M — I

if I is analytic with I M and if there are a holomorphic vector function u: M —' V and a meromorphic function h 0 such that m = hi. Here we can assume that 1(m) := I is minimal. If m 0 is meromorphic, then

M. RU AND W. STOLL

480

m —2, we say ti is reduced. If so D can be taken such that := 1th does not depend on the choice of h and o, and is called the divisor

of to.

A holomorphic map f: M—1 —. P(V) is said to be meromorphic on M, if I is analytic with I M and if there is a meromorphic vector function ni o on M suchthat f(z)=P(ro(z)) forall zEM—(IUI(ro)) with is called a meromorphic representation; if m is holomorphic, it If so, is called a representation, if to is holomorphic and reduced, it is called a reduced representation. If to and t' are meromorphic representations of f, there is a meromorphic function h 0 such that to = hi,. if t' is reduced and if to is holomorphic, h is holomorphic. If both u and to are reduced, h is holomorphic and without zeroes. A reduced representation exists if f is meromorphic. We can assume that 1(f) := I is minimal, then 1(f) is called the indeterminacy of f and dim 1(f) ( m — 2. Also 1(f) = o '(0) for each reduced representation of f. The holomorphic map 1: M —1 —. V E M — 1} is meromorphic on M if and only if the closure of {(z, is analytic.

For r > 0, the spherical image function A1 of a meromorphic map f: M —. P(V) is defined by A1(r) = r2_2mf

(1.15)

f(Q) A em_I 0.

MI,]

Then A1 increases. Thus the following limits exist. (1.16)

0 A1(0) = urn A1(r) 0 N the compensation function mjg is defined by .

(1.21)

If (f, g) is free and if 0 < s (1.22) If

1 = m, the defect relation was proved by Cartan [2]. He conjectured the defect relation for n > k I = m in [2], and this case was proved by Nochka [16, 17 and 18) and for m> 1 by Chen are constant, but if n k = m = 1, the defect relation [3]. If not all was conjectured by Nevanlinna [151 and proved by Steinmetz [29]. The case n = k> 1 = m was proved by the authors [22], see also [33] and [231. The following theorem is a trivial consequence of Theorem I. THEOREM H.

M —' P( V)

Let

=

be a finite family of meromorphic maps

in u-subgeneral position with n u and 2u —

#5 < cc. Let f: M

n

+I

P( V)

be a transcendental, meromorphic map which grows slower than f for is linearly nondegeneraze over Assume that each jES. Then wehave

2u—n+

(1.29)

1.

fEd

We will prove Theorem II first and derive Theorem I from Theorem II. However this derivation is much more complicated than in the case of constant targets, Chen [3]. 2. The First Main Theorem for an operation: applications. The reference for this section is [33]. Let V, and W be hermitian vector spaces. Let (2.1)

be a p-linear map over C. If ; =

...

,

;,,)

x

V1 x

we write

(2.2)

and say that; is free for the operation 0 if 0(;)

0. If

and (2.3)

is well defined. Given ;, free or not, also the "nonn" 0,,ii (2.4)

0x10x20•..Ox D=

II

ll;2 II

It

= P(;3) we also

M. RU AND W. STOLL

484

is well defined. This norm is positive if and only if (x1, ... , x,,) is free for normalize the hermitian metric on W such that Dx1è.. . Ax,,D 0. M —+ 1. Take meromorphic maps for j = 1,... 'p. Then f, are said to be free for (1 if and only if DJ O. .. 0. If so, the compensation function

mjb...èj(r)

(2.5)

1

0.

Df

=

M —' V be a reduced representation, then b = o1 0. 0 is a representation of f10. . Of,,. The nonnegative divisor := depends on f1, ... , J, only and its valence function is denoted by Nfeef. Now (1.19) and (1.20) imply Let

.

T1901(r, s) + N18 T1(r, s)

=

s)

= fMfr> log

JM(r) log

—

—JM(s)

log

Thus (2.4) and (2.5) yield the: FIRST MAIN THEOREM.

E T1(r, s) =

(2.6)

If dim W = 1,

s) +

—

then P( W) is a point and

0.

An example istheexteriorproduct O=A:

= = —'A V with are free f,, A if and only if n + 1. In this case .1,, are linearly independent, which implies p ?(V) for 1, ... , p. Dx1A Let be a finite family of meromorphic maps M —' P(V) = with T = 0; see (1.26). Assume that n + 1 #d' < óo. For A E T abbreviate A = and V = V

an W = A,, V. Then f, ...

hf

.

(2.7)

N,(r, s) =

s),

m0(r) =

lET

lET

For 0 < s < r we obtain the First Main Theorem for a family (2.8)

E T1(r, s).

N0(r, s) + m0(r) — m,(s) =

AET(e) j—O

The gauge

(2.9)

is defined for all z E M —

= inf{Dg1(0)(z)A. .

.

with A

= €

1(g1)


Then 0

F(s)

1. For r> 0, the gauge function

re(r)

(2.10)

f

=

1

log £

M(r)

48S

is defined by

a 0.

Define

(2.11)

=

fi

Then m(e)n. For ).E 7',,, theset S2 is analytic. Then S = is analytic and N = M — S in open and dense in M. Obviously N n = 0. For each z E M —1(e), the orbit family e(z) = is defined. Since our theory applies to constant families, we may ask when e(z) is in u-subgeneral position. Then

LEMMA 3.1.

If z E N, then e(z) is in u-subgeneral position.

PRoOF. Take P ç S with #P =

u

+ I. Since 0 is in u-subgeneral

position 2 E T(0) exists with 1m2 ç P. Then toA 0 and A e Thus 0 by the construction of N. Thus A T(0(z)) with ImA P. Hence 0(z) is in u-subgeneral position. Q.E.D. For 0 P ç S and z M, let H(z, P) be the linear subspace of V spanned by LEMMA 3.2.

E P}. Define d(z, P)=dimH(z,p). then d(p)=d(z, F) isconstanifor ZEN. N and z2 E N. Put p + 1 = d(z1, P). Then 0 p

PRooF. Take n. There is a bijective map 2:Z[0, P1

—.. P

, ... ,

such that

a base of H(z, F) - Then m2(z1) 0. Thus m1 0 and A e 7,. Because z2 E N, also m2(z2) 0. Hence m2(0)(z2). ... , are linearlyindependent. Hence p+l d(z2,P). Thus is

By symmetry we obtain equality.

Q.E.D.

Obviously if P ç S and #P u + 1, then d(P) =

n

+ 1. The basis

for the product to sum estimates in the case of u-subgeneral position are the

M. RU AND W. STOLL

486

weights introduced by Nochka for constant families. Let be = and a family of points P( V*) in u-subgeneral position with u For each oc > q = with = > 2u—n+ 1. Take

0

P ç & let 11(P) be the linear subspace generated by E P} in V'. Then there exists a function w: — R(O, 1) called a Nochka weight and a real number 0> 1 called Nochka constant satisfying these properties: (1) (2)

O<w(j)0jEPw(i) dimH(P). (3) If 0 P (4) 1 0.

=

.

0 by the construc-

II

2(0)

II

> 0 by (2.9). Q.E.D. T(0) is finite, For x E P( V), for 0 < r E P and for z E M —1(0) define

Since

(3.4) PROPOSITION

then

g1(z)D < r}. 0(x, r, z) = {j 3.7. If x E P(V), if ZEN and if 0< r(n+1)

#0(x, r, z) u.

PROOF. Assume that #0(x, r, z) u+ 1. Then 2 E T(0) exists such that for j=0, ... , n. Definition 1m2 ç 0(x, r, z). Hence Ox, 2.9 and Lemma 3.5 imply 0g2(0)(z)A. .

0

Ox, which is impossible.

Q.E.D.

THEOREM 3.8 (product to sum estimate).

Take x E P(V) and z

N

suchthat Dx;g1(z)O>Oforallj€d. Then (3.5)

(

i

PRooF. Take r =

u. Thus A

p. Then there is

m=>wjbjEP, where J0 E Z[0, k] exists such that w1

0= (u, m)

0. Then

b.)=

by),

which contradicts the assumption that (o, b0), ..., (o, bk) are linearly inThus dim dependent over = p and f is k = n — p flat. Q.E.D. REMARK. In the previous theorem we can take b0, ... , b, as an orthonormal system in

M. RU AND W. STOLL

$00

7. Proof of Theorem 1.

First

we need some preparations.

M —' V be a holomorphic vector function with Let ,n — 2. Let h 0 be a meromorphic function on M. Let Then be a base of V. Define it = hD w0e0 + +

LEMMA 7.1.

dim u

(0)

-

e,

IZh.0Uw,,O (7.2)

AUh,oo

PROOF. W.Lo.g. we can assume that w. 0 for all j E Z[0, n). Then 0 is holomorphic. We have w1 o = v0e0 where each + for all f Z[0, n] which implies .0

The

=

.

=

+

=

,0

+ Isv, .0

set

A

=o'(0)UI(h)UUJ(w1) 1=0

is analytic with dimA

m

—

2. It suffices to prove (7.1) and (7.2) for every

= 0, then (7.1) is trivial. Assume that /4h,O(Z) >0. z EM—A. = 00(z) = 0 = Then h is bolomorphic at z and so is Hence Thus (7.3) implies auh,0(z)

which

+

=

proves (7.1).

If

fore

that vk(z)

= 0, then (7.2) is trivial. Assume that > 0. There= 0. Also we have o(z) 0. Hence k Z[0, n] exists such 0(z) = 0. Thus (7.3) implies 0. Thus = '4Wfr.OO —

which

proves (7.2).

Q.E.D.

0 and w0, ... , w,, be meromorphic functions on M. Let A be an analytic subset ofM with dimAm—2. Let c>0 bea LEMMA 7.2. Let h

continuous function on M — A. Assume that (7.4)

Then we have (7.5)

onM—A.


m = (w0, PROOF. 0 function. There exists a

...

is a meromorphic vector

w,,): M —i

,

vector function D = (v0,

... ,

v,,): M —

0 on M

with dim m — 2 and a meromorphic function u such that m = ut'. By Lemma 7.1 we have •

00

• 00

on M — A outside a thin g = h/u. Then (7.4) implies (g(2 analytic set. Hence g is holomorphic on M. Therefore Define

It suffices to prove (7.5) for z M—(AuI(h)) with /1h00(z) Mh,O(Z)—O. We obtain

=

>0.

Then

—

—

Q.E.D.

14g,00(Z) S

Now, we shall make these general assumptions:

(BI) Let M and V be finite dimensional berrnitian vector spaces with and

(82) Let 1: M -.. P(V) be a transcendental meromorphic map. be a finite family of meromorphic maps g1: M (B3) Let = P(V) in general position. Put and q #8.

(B4) Assumethat f is k-flatovcr with 0k

—

+

= + 11w1,O — Therefore Lemma 7.5 and Lemma 7.9 imply s)

—

N11(r,

s) =

s) ÷ o(T1(r, s))

LEMMA 7.11.

s)

for

—

s)

r

Q.E.D.

Assume that (B 1 )—(B6) hold. If j E Q, then

o(J,

(7.23)

= ô(f, g3).

PROOF. We have 1

—

N1 g

J(f, g1) =

N1

(r, s)

T1(r, .c) + Tg(r, s)

= limsup

=limsup

=limsup N1

s)

T1(r, s)

N11(r,s) Tj(r,s)

(r, s) Q.E.D.

THEOREM

7.12.

(the defect relation: Theorem I).

hold. Then we have (7.24) JEQ

Assume that (B 1 )-.(B6)


PROOF. We apply Theorem 5.4 with the aid of the translation table: Theorem 5.4 Theorem 7.12 Theorem 5.4 Theorem 7.12

M M g

V

m

W

m

n +1 k+I

V

u

q

i f

Q Q j

vS(f,

q

(Al) is trivially satisfied. Lemma 7.8 implies (A2). The map I is linearly nondegenerate over as shown after (7.8). Thus (A3) holds. (A4) is proved in Lemma 7.7. Therefore we have (7.25) JEQ

Now (7.25) and Lemma 7.11 yield (7.24).

Q.E.D.

REFERENCES

1. L Ahlfors, The theory of meromorphic curves, Ada. Soc. Sci. Fenn, Nova Ser. A3 4 (1941), 171—183.

2. H. Cartan, Sur lea zeros des combinazsons lin#aires de p fonctions holomo,phes données, Mathematica (cluj) 7 (1933), 80—103. 3. Wanxi Chen, Cartan conjecture defect relation for meromorphic maps from manifold to projective space, Notre Dame Thesis, 1987. 4. Chi-Tai Chung, Uné d'une inégalitC de Nevanlinna, Sd. Sinica 13(1964), 887—895. On the zeros of some differential polynomials of meromorphicfitncrions, Science

5. —.

Report 89-002, Inst. Math., Peking University, 1989, 1—29. 6. M. J. Cowen and P. A. Griffiths, Holomorphic curves and metrics of negative curvature, J. Analyse Math. 29 (1976), 93—153. 7. J. Dufresnoy, Sur les valeurs exceptionnelles des fonctions méromorphes voisines d'une fonclion mEromorphe donnée, C. R. Acad. Sci. Paris Set. I. Math. 208(1939), 255—257. 8. G. Frank and 0. Weissenborn, Rational deficient functions of meromorphic functions, Bull. London Math. Soc. 18 (1986), 29—33.

9. H. Fujimoto, On the Gauss map of a complete minimal surface in Rm, J. Math. Soc. Japan 35 (1983), 279—288. 10.

On

the number of exceptional values of the Gauss map of minimal

3.

Math. Soc. Japan 49 (1988), 235—247. 11. —, Modified defect relations for the Gauss map of minimal surflzces. II, J. Differential Geom. 31 (1990), 365—385. 12 W. H. Greub, Multi/meat algebra, Grundiehren Math. Wiss., voL 136, Springer-Verlag, Berlin-Heidelberg-New York, 1967. 13. S. Mori, Remarks on holomorphic mappings, Contemp. Math. 25 (1983), 101—114. 14. R. Nevanlinna, Zur Theorie der meromorphen Funkzionen, Ada Math. 46(1925), 1—99. 15. —, Le théorème de Picard-Borel et la théorie des fonctions meromorphes, Gauthier Villars, Paris, 1929; Reprint, Chelsea, New York, 1974. 16. E. 1. Nochka, Defect relations for meromorphic curves, Izv. Akad. Nauk. Moldav. SSR Ser. Fiz.-Teklam. Mat. Nauk 1 (1982), 41—47. 17. —, On a theorem from linear algebra. Izv. Akad. Nauk. Moldav. SSR. Ser. Fiz.-Teklam Mat. Nauk 3 (1982), 29—33, 18. —, On the theory of meromorphic curves, Doki. Akad. Nauk SSSR 269 (1983), 377— 381. 19. Ch. F. Osgood, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds or better, 3. Number Theory 21(1985), 347—389.

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508

20. Mitt Ru, On the Gauss map of minimal surfaces immersed in R' , J. Differential Geom. (to appear). 21. Miii Ru and W. Stoll, Courbes holomorphes ëvitant des hyperplans mobiles, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 45—48. The second main theorem for moving targets, I. Georn. Anal. (to appear). The Nevanlinna conjecture for moving targets, prepnnt 16 pp. 24. Mitt Ru and Pit-Mann Wong, Integral points of P' — {2n + 1 hyperplanes in general posiiion}, preprint 26 pp. 25. B. V. Schabat, Distribution of values of holomorphic mappings, Transi. Math. Monographs vol. 61, Amer. Math. Soc., Providence, 1985. 26. B. Shiffman, New defect relations for mero,norphic functions on C', Bull. Amer. Math. Soc. 7(1982), 599—601. 27. —, A general second main theorem for meromorphic functions on C' , Amer. 1. Math.

22. _, 23. _,

106 (1984), 509—531.

28. M. Shirosaki, On defect relations of moving hyperplanes, Preprint 17 pp. 29. N. Steinmetz, Eine Verailgemeinerung des zweiten Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math. 368 (1986), 134—141. 30. W.Stoll, Introduction to value distribution theory of meromorphic maps, Lecture Notes in Math., vol. 950, Springer-Verlag, Berlin and New York, 1982, pp. 210—359. 31. —, The Ahifors- Weyl theory of meromorphic maps on parabolic man4folds, Lecture Notes in Math., vol. 981, Springer-Verlag, Berlin and New York, 1983, pp. 101—219. , Value distribution theory for meromorphic maps, Aspects of Math. El (1985), p. 32. 347.

33. _, An extension of the theorem of Steinmets-NevanJinna to holomorphic curves, Math. Ann. 282 (1988), 185—222. 34. A. ViUct, A lemma of the logarithmic derivative in several complex variables, Duke Math. .1. 44 (1977), 90—104.

35. Pit-Mann Wong, Defect relations for maps on parabolic spaces and Kobayashi metric on projec:i.te spaces omitting hyperplanes, Noire Dame thesis, 1976. 36. H. Weyl and J. Weyl, functions and analytic curves, Ann. of Math. Stud. 12 (1943). UNiVERSITY OPSINOAPORE, SINGAPORE

Current address: University of Noire Dame

Procecdinji of Symposia in Purt Mathematics 'Ioltjmc 32 (19911. Pin 2

On Kollair's Conjecture MORIHIKO SAITO

1. IntroductIon. In [12] Kollár studied the generalization of his vanishing theorem and torsionfreeness of higher direct images of dualizing sheaves, and obtained some conjecture about abstract (i.e. not necessarily geometric) variations of Hodge structures. In this note we show that his conjecture naturally follows from the general theory of Hodge Modules [16, 17). Let I be an irreducible algebraic variety over C (assumed always separated and reduced), and H a polarizable variation of Hodge structure on a smooth dense Zariski open subset of I. Then it is shown in [171 (cf. also

[19]) that H is uniquely extended to a polarizable Hodge Module 4 on X whose underlying perverse sheaf is an intersection complex of GoreskyMacPherson-Deligne [1, 5] with support X. For simplicity, assume X is a closed subvariety of a smooth variety I'. By definition 4' is represented by a filtered (M, F) with rational structure given by an isomorphism DR(M) = K®QC for a perverse sheaf K with rational coefficient, and they satisfy very special kind of conditions (16]. Let

Then p' dependsonlyon H,and F,M on H and X (i.e. independent of embedding of I) as

and we denote them

by p(H) and

respectively. More generally, DR(M) the graded pieces of the filtered de Rham complex of M (see (2.2.4) below) are also well defined as complexes of and DR(M) = we use right We show that

because

=

behaves very much like the dualizing sheaf = in the smooth case (and DR(M) like other differential

forms 12" shifted by diml +p to the left), and the results of Kollár hold with replaced by S1(H). For example, if I is smooth so that X 1980 Mathematics Subject Clarnjication (1985 Revision). Primary I 4C30, 32C35. Key and phrases. Variation of Hodge structure, Hodge Kodaisa

decomposition theorem. This paper is in final form and no version of it will be submitted for publication elsewhere. Mathematiosi Soâaty

0032.0717/91 $10095.23 peipsje

509

MORIHIKO SAITO

510

-

and the singularity of the variation of Hodge structure is a normal crossing divisor, then is obtained by taking the intersection of the direct image of Hodge filtration with Deligne's extension of the integrable connection [4]

up to tensor of

(see (2.2.6), (3.1.1)) just as in the case of higher direct images of dualizing sheaves [12]. The main theorem is the stability of Sx(H) by direct image under a proper morphism f: X Y (with the into a direct sum of cohomology sheaves):

(1. 1) THEOREM.

®p(H')-=p(H)

in particular = if p(H') = p(H) and 0 otherwise. Here H' is the variation of Hodge structure whose fibers are the intersection cohomology of the restriction of H to the generic fibers of f, see (2.4)(2.6), (3.2). If f is birational then we have 0 for i > 0, and = where H is viewed as a variation of Hodge structure on a smooth Zariski open subset of Y. As to the vanishing theorem, let L be an ample line bundle on X. Then we proved in [17, 2.33]: (1.2) THEOREM.

H'(X,

DR(M) ®

(See (2.2.4) for DR(M).) DR(M) replaced by Y,

=

0

for

I

0.

For direct image this holds with X, DR(M)) :=

In

the case where the singularity of M is a normal crossing divisor, DR(M) can be expressed more explicitly using Deligne's extension (4], see (2.2.7), and this can be extended to the mixed case as in [71, see (3.5)(ii). Note that

DR(M) is the dual of the self-duality of 4' (where n assertion on

DR(M) as a complex of by is the weight of 4'), and this gives the dual corresponding to (the direct images of) ef. §3.

2. Polarlzable Hodge modules. In this note we use right 9-Modules, which are more convenient for our purpose (for example, is naturally a right

if X smooth). (2.1) Let X be a complex algebraic variety (always assumed separated and reduced). In [16, 18] we define MH(X, n)" the category of polariz-

able Hedge Modules of weight n on X. If X is smooth, this is a full subcategory of the category of regular bolonomic filtered 9-Modules with Q-structures

Q) whose

objects are (M, F, K) with (M, F) a

filtered 9x.Module and K a perverse sheaf with rational coefficient such that M is regular holonomic [2, 10, 14], Gr'M is coherent over and M, K are given isomorphism DR(M) = K C. Here the filtration F of is defined by the order of differential operators, and the de Rham functor DR induces an equivalence of categories between regular holonomic and perverse sheaves with complex coefficient, cf. [bc. cit]. (One may use either algebraic or analytic 9-Modules as preferred by the cxtendability of Hedge Modules and GAGA, but the stratifications of perverse

CONJECTURE

ON

sheaves are always assumed algebraic.) The difference between MH(X, n)" and Q) is very large, and we use the vanishing cycle functors [5] and the filtration V of Kashiwara-Maigrange [9, 13] inductively to define the condition of MH(X, n)". In general MH(X, n)" is defined by using locally defined closed embeddings of X into smooth varieties. For example. if X is a closed subvariety of a smooth variety Y, then MH(X, n)P is defined by the full subcategory of MH( Y, n)" whose objects are supported in X, and this definition is independent of embeddings, cf. [16, 17]. By definition we have a strict support decomposition

MH(X, n)" =

(2.1.1)

n)"

indexcd by closed irreducible subvarieties Z of X, where 4 E MH(X, nY' belongs to it has strict support Z, i.e. its underlying perverse sheaf is an intersection complex (with twisted coeffIcient) supported on Z. The above independence of embedding means an equivalence of

if

categories (2.1.2)

n)"

Mz(Y,

for a closed embedding of smooth varieties i: X

Y, and (2.1.2) holds F) of a F) F), and locally we have

for X, Y singular. Here the direct image (M', F) = filtered (right) s-Module (M, F) is defined by (M, F) with

(2.1.3)

F)

hI' =M®C[81, ...

,

8w],

C

if X = {y, = 0; 1

VENd

i d} with (y1, ...

'Ym) a local coordinate system

of Y, where —_ii,a7 for ii = = 8/ay, and If X is embeddable into a smooth variety Y so that J E MH(X, n)t' is represented by (M, F, K) E Q), then p' := min{p: O} and the Module hI are independent of closed embedding by definition, cf. (2.1.3), and we define (2.1.4)

p(4r) = min{p:

0},

Then p(4) and

are globally well defined even if I is not globally embeddable into a smooth variety, because is locally well defined, and glues together globally. For an irreducible variety I we denote by w)t' the category of polarizable variations of Q-Hodge structures of weight w defined on smooth Zariski open subsets of I, whose local monodromies at infinity are quasi-unipotent. Then (2.2) THEOREM 117] (see also 119]).

(2.2.1)

We have

an equivalence of categories

n—

MORIHIKO SAITO

512

This means that a polarizable Hodge Modules is generically a polarizable variation of Hodge structure, and any polarizable variation of Hodge structure can be uniquely extended to a polarizable Hodge Module with strict

support X. For simplicity assume X is a closed subvanety of a smooth variety X'. Then, by the theory of intermediate direct image [I) (or minimal extension in the terminology of Kashiwara), the underlying local system L of a variation of Hodge structure is uniquely extended to an intersection

complex K with support X, and M is obtained as the regular holonomic corresponding to K ® C. The Hodge filtration F on M is determined by its restriction to any open dense subset using the filtration V of Kashiwara-Malgrange [9, 131 as follows. Let g be a function defined on an open subset U of X' such that denote the embedding 0, and by graph of g. Put (M, F) = F)Iu, cf. (2.1.3). Then F) is determined by its restriction to U x C (and (M, by that to U\f' (0)) using the formula F,,M = E( V 0. Let D := f3(F). Now for s <s0,

DctT.

COMPACTWZCAT1ON PROBLEMS FOR STEIN 3-FOLDS

537

Now assume that D C U; hence s0 E W and it readily follows that W is an open neighborhood of s0 in [0, 1]. In particular s0 < I, since for W one must have C U but Jj(r') f1(z) = f(z) = w U. s Consequently, there exists an e > 0 such that (0, s0 + e] c W, which contradicts the choice of s0. Therefore D U and hence D F and our claim is proved. Now it follows readily from this argument that there exist infinitely many compact hypersurfaces {F,} in M; but, in view of (111 this is impossible since a(M) =0. Q.E.D. (b) Now, let T be a C-analytic torus with a(T) = 0 and let G be a finite group of analytic automorphisms of T such that M is a nonsingular model

of the quotient T/G. Then a(M)

0 and one can check that M cannot

be the compactification of any Stein manifold. M is the so-called Kummer manifold, see e.g. [15]. in order to measure the algebraic structure of X we need the following DEFiNiTION 2 [14]. Let M be a compact C-analytic manifold, let F C M be a divisor that is assumed to have only normal crossings and let X := M\r.

Now let KM be the canonical bundle of M and let us consider a basis for the vector space V:= H°(M, O(mKM+(m—l)l)), which gives rise to a well-defined meromorphic map:

'sm: M-PPN :=

w where

:

N:=dimV—1.

Let

N(X)

' 'l—oo

#0,

ifN(X)=0

is called the analytic Kodaira dimension of X. Notice that ic(X) could take any of the following values

—00, 0, 1, ...

,

DEFINITION 3. In Definition 2, if one replaces V by W := H°(M, O(mF)),

then c(i') := ,c(X) is the so-called Iltaka D-dimension. Furthermore, F is called big if REMARK 3. In Definition 2, if one replaces V by U := H°(M, O(mKM + F)), then one obtains the notion of logarithmic Kodaira dimension, Which is birationaily invariant and yet is not even biholomorphically invariant [8); meanwhile, the analytic Kodaira dimension is bimeromorphically invariant, hence it fits well in our framework. Furthermore, when F = 0, sc(X) coincides with the notion of Kodaira dimension for compact C-analytic manifolds. Also notice that < ic(X) ( a(M) always.

VOVAN TAN

538

=

EXAMPLES. (a)

where

:= C\{O}.

(b) Let I' be a divisor with only normal crossings in P,, and let X Then, one can check [14] that K(X) —cc (resp. = n) if degT = —00. More generally, one has: THEOREM 2. Let M, r', and X be as in Definition 2 and let us assume

n + I (resp. > n + 1). In particular,

that r is big. Then ic(X) is either —cc or n. PROOF. Let us use the notations as in Definition 2. If N(X) =0, for any m, then certainly ic(X) = —cc. Otherwise, let us assume that there exists an

such that, for m (*)

2mKM+(2m—l)r= D+r where 1) := 2mKM+2(m-l)F. Hence in view of hypothesis (*) with m m0, D is linearly equivalent to some effective divisor, i.e. D+F F. Since F is big, Ic(2mKM+(2m— l)f') K(fl = n. Consequently ic(X) = n. Q.E.D. COROLLARY 1. Let X be a nonsingular affine variety (hence X is a cornpactijiable Stein manifold). Then K(X) = —cc or n.

PRooF. In view of Hironaka's resolution of singularities, one can assume

that there exist a nonsingular complete variety M and a divisor F with M\F. A main result in (6] tells us only normal crossings such that X that there exists some monoidal transformation M —' M with smooth

center c r such that e :=

is the support of some effective ample (and hence big) divisor. Now Theorem 2 applies and one gets the desired conclusion. Q.E.D. 4. This result is well known; however, we could not find an ex-

plicit proof for it in the existing literature; Our original plan was to use Corollary I to detect compactifiable Stein manifolds that do not carry any affine structure.

DEFINITION 4. A compactifiable Stein manifold X is said to carry some affine structure if there exists a nonsingular affine variety A such that (the associated C-analytic manifold) is biholomorphic to X. As far as the one-dimensional case is of concern, a complete answer is well known. PROPOSITION 2.

Any compact ifiable noncompaci Riemann surface is affine.

On the other hand, one has THEOREM 3 (16]. —cc

Let X be a compacuflable Stein surface. Then ic(X) =

or 2. In particular,

X admits a

M, which is not a

(birationally) ruled surfizce, then necessarily ic(X) = 2.

Naturally, one would like to ask: PROBLEM 2. Do there exist compactifiable Stein manifolds Xwith dime X

3 such that —cc < ic(X) max(p, q) are acycic sheaves. For example, its term of degree p + q — I —The portion

(g.P , q

•

,q+t

q

is the single complex associated to the double complex n shiftedbyonedegree indegree p+q+1),withdifferential d. An obvious involution maps onto Lq ,. q DEFINITION. A generalized d-c!osed (p, q)-form on a complex space X is an element of

The tail morphism sends X9"(X) to := Its image is by definition the space of good d-closed (p, q)-forrns on X. We also define

:=

So X"

is exactly is

Lp,q)• C

—'

,

the cokernel of the canonical morphism from

and to

A word of warning must be said here: The space of "generalized d-closed rn-forms" may also be defined. It is Gm of the de Rham complex. There is a canonical morphism taking generalized d-closed (p, q)-forms to generalized d-closed (p + q)-forms, obtained by considering a convenient morphism of complexes of sheaves. But it is not injective general. So the term "generalized d-closed forms" is ambiguous. We will call "generalized q doubly closed forms" the elements of (X).

According to the above considerations, elements of .2" (I) are represented by systems of (tech cochains in

• C°(u,

C)

which may be written in a matrix form

a

with

a E C"'4(u, C),

g0

gk

... ...

flk)

ThE

h

ON COMPLEX SPACES

(U

E

E q

means that whenever three entries of

The cocycle condition the above matrix are in position

B

CA' then the (tech differential öA is equal to

x

(DB + DC) where D

denotes the sheaf differential. So finally

= ôg° = = o7i' —

for

1 k

for

I

for

I

1,

for 1 k

1,

I

p — 1, q

—

1,

g°) =

for

+84k_1.I)

J#k.!

I

Two such matrices define the same hypercohomology class if, after taking a common refinement of their corresponding coverings, we can write their difference as the is-differential of an element of

C)s The tail of

C°(U, *

q

will be the d-closed (p, q)-form co"

q

such that

,q—1

(—

It is very important to notice that the standard definition of a Káhler metric on a complex space is exactly that of an element of X' "(X) satisfying a positivity condition. It is required to have a covering

it = (U0),

of X and P0 — Pp

strongly plurisubharmonic functions p0 on U0 such that are pluriharmonic. The Kähler form w is determined by

e(w) = Write

forsome

tial of is holomorphic and self-conjugate, hence constant-valued; we call it a. Denote by 4) the 0-cochain (—ip0). Now, under the above conventions, the matrix

01,1._IC

f\

4)j, represents an element of X' ' 1(X).

Its tail

is o.

J. VAROUCHAS

We emphasize the fact that, if the space X is smooth, then = fl Kerd, :=

=

,q—I

(X),

since the tail morphism is then an isomorphism.

The prodnct law. This will be the first nontrivial operation we shall use (nontrivial=not obtained by diagram-chasing). It amounts to show that "good doubly closed forms" are closed under the wedge product. To do this, we will show the existence of a bilinear law such that

tail: defines a morphism of bigraded algebras of presheaves with involution. To

do this, we denote by it an arbitrary open covering of I and consider tech cochains of U with coefficients in sheaves of differential forms (holomorphic,

antiholomorphic or r°). If

4)

a q-cochain of k-forms and w

is

an r-

cochain of I-forms, we define (*)

(4)

w)0

q+r

:= 4)

00

A q

g" q+'

if dealing with ordinary cochains, and we antisymmetrize the right-hand side of (*) if dealing with alternating cochains. We always have 6(4). p') = (64)). + Moreover, in the alternating case, we also have

_J. Let us define

We will omit the letter it from

= to be the direct sum of all groups

...

Cm(C)

cm_2(go.o)

... ...

1

.QI)

where every group involving some negative superscript is understood to be zero. Notice that this is exactly The differential

in

Apq

in

m+l

p.q —'Jè'p,q

ON COMPLEX SPACES

THE

will be ô + (—1 )mD, where D is the sheaf differential induced by the double

complex L. and each in matrix form, their product be the element of represented by a matrix decomposed into blocks of respective sizes (rows x columns) Given x

E

will

ixp sxp qxp

lxi sx] qxl

Ixr sxr qxr

The corresponding entries will be a1

a2

.

T

.

a2

g2 .

f

•

.1-s

T

•

the unspecified signs being, respectively, (1)fl(k+I)

+1

(1)n(k+I)+1

+1

(_1)(fl_I)P J

(1)fl(IS+I)

J

(

J

1

This choice of signs makes the following true. The proof is by direct computation and will be omitted. PRoposiTioN 3.

(i) The product law x

(ii) For

is

associative.

and

+

=

X It follows that if we set

,q (it)

_,

:=

and p,q•

p,q p.'!

then the direct sums of

q

(it) and

p,q

p.'!

'!(tt) respectively are two bigraded

algebras. The tail morphism

—.

is defined by taking

of the C°O1, which is a 0-cocycle O(CIOSed (p, q)forms. A direct computation shows that tail(t1)1 X c1)2) = tail((11) A

tail(D2).

574

J. VAROUCHAS

The "generalized

,"

at the to the matrix having C°(U, ')-component and 0 everywhere else. These morphisms induce, via the canonical epimorphism, a commutative diagram

takes

q

j tail which, after taking the direct limit over £1, is the one contained in Proposition

1. The direct limit over U of X' q (U) cannot be taken because when U, the induced morphism are two coverings with a refinement from U to to to (unlike the one from from depends on the choice of the refinement. Let us now prove Theorem 1. We may use alternatively

In a complex space, any compact rn-dimensional complex-analytic subset admits a fundamental system of (m + 1)-complete BARLET'S LEMMA [2J.

neighborhoods. (Stein =1 -complete);

or REIFFEN'S LEMMA (9). If is a coherent analytic sheaf on a complex space and Y any rn-dimensional complex-analytic subset, then

denotes cohomology with compact supports and for all q > Pn. Here is the topological restriction of .? to Y.

Using Barlet's Lemma, let Y be a compact complex rn-dimensional complex-analytic subset of a complex space. X. Take neighborhoods U C V of Y in X such that V has the homotopy type of Y (hence H2m+2(V, C) = of the 0) and U is (m + 1)-complete. Take the (m + I )th power Xm41 .m+l(x) whose tail is is an element of given K.ähler metric.

We restrict it to V and take its image in H2m+2(V, C), which is zero. Now consider the diagram

—. .L

tail

Hm41'm4tw) .1.

m+I(U)

C). Notice the absence of a canonical morphism from 2-m+L .m41(U)

C). ,m+l Since U is (m + 1)-complete, injective by Proposition 2. In—rn—k deed, the only cohomology groups involved are H m+l+k (U, fl rn—k Cl )

ON COMPLEX SPACES

THE

in H2m+2(U, C) vanm. We know that the image of ishes, since it comes from H2m+2 ( V, C) which is 0. Therefore the image of

for 0 k form

in 11m+1 .m+l(U) vanishes too, and this implies that = for some (m, m)-form w. So finally,

has

the

as required. If we had chosen to use Reiffen's lemma, we would have followed the same

lines but instead of restricting successively from I to V and from V to U, we would work directly on the direct limit of all neighborhoods of Y in

x.

It is clear that our method proves the following, slightly stronger, result: THEOREM 1'. Let X be a complex space, Y a complex subspace of arbitrary elements of X' • '(I). Let dimension m and (1, 1)-forms). Then be the respective tails (= locally their product A is near every compact subset of A corn x ... x (The proof is identical to the above, using the product instead given a (p, q)-form, Now let us examine the dual we want to know if it is the sum of a b-exact and a 0-exact form. In the

same spirit, we define the space := of

"generalized C —.

is

sends

q

(X) to

,

(p. q)-forms." As a hypercohomology group, this ,q) The tail morphism •.. q (X) fl Ker and its image is by definition the (p, q)-forms." By a reasoning of the same kind

space of "good as before, one can prove THEOREM 2. If is a good

(p, q)-form on a complex space

I and Y is any complex subspace of I of dimension <min(p, q), then is the sum of a

a 0-exact form near every compact subset of Y.

Theorem 1 constitutes a crucial step in the solution of a problem of Hironaka. The result is THEOREM 3 [10). Let X be a Kähler space, Y a reduced complex space •

and it: I —' Y a proper open epi morphism of complex spaces having fibers of pure and constant dimension. Suppose that it is flat or Y is normal. Then Y is a Kähler space.

Theorem 2 has no apparent geometric applications but illustrates the kind of results that may be proved by the above-mentioned techniques. The commutatlvlty relation. It is not obvious to compare 'I) x 'I' and 'I' x 1

in the algebra X(X). The truth is that this algebra is indeed anticommutative.

We consider alternating cochains in this paragraph.

J. VAROUCHAS

576

Let U be a fixed covering of a complex space X. There

PRoPosITIoN 3.

are bilinear maps m

*:

such that, if

and (1)2

E

xc1)2

x

m+n—l

n :

we have

E

*CI)2+(—l)

XCI))

(1)

*CI2).

particular, the aLgebra X(X) of generalized doubly closed forms on X is anticommutative, and its diagonal subalgebra em Xm.m(X) is commutative. We will give the complete expression of * c1)2. We first agree that any element involving a superscript outside of its domain of definition (for example for k p) is zero. We conserve the notations used to define We set In

Gk := (_l)m_1

.

:= (

&,I

where

:= (_l)m_I

.

:= (_l)m

.

:= (—1)'"

.

(1)m Now

*

is the element of

defined by the matrix

Ge''.0

...

0

80,q+s—1

\

0p+r—1,q+s—1

We omit the (very long!) verification of the identity X

—

=

X

*

+

*

+

*

Final remarks. The actual setting in which the functors G and 11

be a morphism of sheaves bounded

defined is the following: Let

(i', D) with

from below. Its cone is defined as the

and differential D := (1 the induced morphism

are

—'

from and

to :=

Denote by (Cone am).

THE OLEQUATION ON COMPLEX SPACES

577

whose cokernel is There is a canonical morphism This in turn is the source of a canonical morphism to denoted This morphism is injective (resp. surjective) if =0 (resp. 0) for all k > 0. We agree to omit the left superscript n when it is 0. differential The reason for which we did not give the definition of forms is that the groups Hm(X, K) will finally be the same for all theories. However the reader may consult [3, 4, 8, or 101. The ambiguity lies in the answer to give to the question: Given a polydisk D of a complex subspace

vanish? For the "weak" theory, the answer is to say "when the restriction of to the smooth part of X vanishes" [3], and this only when X is reduced. For the "strong" theory, the answer is "when belongs to the d-invariant self-conjugate ideal [4, 10]. The subspace X does generated by the coherent ideal sheaf D not have to be reduced, but even if it is the two do not coincide. Hypercohomology is defined at many places in the literature (1, 5,6, or 7]. The (correct) functorial point of view is to extend the domain of definition of a functor from objects of an abelian category to complexes of objects of the same category. We will not continue in this direction! The point of view of [5] is to define hypercohomology by tech cochains and this is enough for our purposes.

X of D and a differential form

on D, when does

All of the above-mentioned groups may be indexed by families of supports. The usual operations on sheaf cohomology with supports have their analogue here but will not be discussed. REFERENCES

1. C. Banica and 0. Stanasila,

dans la globale des espaces ccntplexes, Qauthier.Villars, Paris, 1977. 2. D. Barlet, Convexitê au voisinage d'un cycle, Séminaire F. Norguet Ill, I.ecture Notes in Math., voL 807, Springer-Verlag, Berlin and New York, 1980, pp. 102—121. 3. T. Bloom and M. Herrera, De Rham cohomology of an analytic space, invent. Math. 7 (1969), 275—296. 4. A. Fujiki and 0. Schumacher, The moduli space of extremal compact Kdh!er manifolds and generalized Weil-Petersson metrics, Pubi. Res. Inst. Math. Sci. 26 (1990), 101—3 83. 5. P. Oriffiths and .Y. Harris, Principles of algebraic geometry, J. Wiley, New York, 1978. 6. A. Grothendieck, Sur points d'a!gèhre homologique, Tôhoku Math. J. 9 (1957), 119—221. 7. R. Hartshorne,

On the de Rham cohomology of algebraic varieties, Inst. Hautes Etudes

Sci. PubI. Math. 45 (1975), 5—99.

8. B. Maigrange, Sur les fonctions differentiables et les ensembles analytiques, Bull. Soc. Math. France 91 (1963), 113—127. 9. H.-.J. ReilTen, Riemannsche Hebbarkeitsthize für Cohomologie-kiassen mit kompakzem Trdger, Math. Ann. 164 (1966), 272—279. 30. 3. Varciuchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), 13—52.

UNIvEasITE DE NANCY I, FRANCE

Procccdrnp of Symposia ui Pwe Mathematics Volume 52(1991), Pin 2

Fixed Points of Holomorphic Mappings in a Bounded Convex Domain in JEAN.PIERRE VIGUE Abstract. In this paper, we study the set Fix I of fixed points ofaholomorphic mapping f: 1) -. D from a bounded convex domain D in C1' into itself. We prove that Fin j is a connected subinanifold of D and that there exists a holomorphic retraction w: D -. Fiif. If moreover, f is conünuous on we prove, using a result of C. Earle and It. Hamilton, that f has a fixed point in (This gives a very simple proof of Brouwer's theorem in the hotomorphic case).

1. Introdwtion. Let D be a bounded domain in C", and let f: 1) -, D be a holomorphic mapping. The set

Fix!

x}

{x

of fixed points of f has been studied by many people (look, for example, at M. Hervé 151 and E. Vesentini [7,8]). Let us recall first the following theorem proved by F. Vesentmi [8]. Let B be the open unit ball of a complex Banach space E. Suppose that eve?y point x belonging to the boundary JB of B is a complex extreme point of Let f: B —. B be a holomorphic mapping such that THEOREM 1.1.

f(O) =0. Then Fix! = B ii F, where F = {v

EIf'(O) 'V = v}.

The proof heavily depends on the fact that each point of the boundary is a complex extreme point of Now if we only suppose that I) is a bounded convex domain in C" , we will prove the following theorem.

f:

THEOREM 1.2. Let D be a bounded convex domain in C", and let D D a holomorphic mapping. Then the set Fix I is a connected complex sub-

of D. The proof we give does not use complex geodesics (compare with [9] and 1980 Mathematics Subject Classification (1985 Revision). Primary 32H 15, 54H25. This paper is in final form and no version of it will be submitted for publication elsewhere. © 1991

519

Sl.OOiS.2i

Sociely

pcrp.r

JEAN-PIERRE VIGLJt

580

The end of this talk is devoted to existence theorems. Let D be a bounded convex domain in C". A holomorphic mapping f: D —+ D may have no fixed points in D. (For example, the biholomorphic automorphism of the open unit disc

z+a has no fixed points in ii.) By Brouwer's theorem, a continuous mapping f: Using ideas of C. Earle and always has a fixed point in Hamilton [3), we will give a simple and elementary proof of this result when

f is holomorphic in D.

It is also possible to prove a result concerning holomorphic mappings holomorphically depending on a parameter. Part of these results have been generalized for bounded convex domains in reflexive Banach spaces by P. Mazet and i-P. Vigué [6]. (I would like to thank Pierre Mazet for many discussions on the subject of this talk.)

2. A bounded convex domain In C". THEOREM 2.1. Let D bea bounded convex domain in C" andlet f: D —, D be a holomorphic mapping. If Fix f is not empty, there exists a holomorphic retraction w: D —+ Fix f. It implies that Fix f is a complex ana-

lyzic submanifold of D. Moreover, if a

F ={v

Fixf, Fix

is tangent in a at

(a).v =v}.

PROOF. First, we

consider the mapping

defined by

By Montel's theorem, we can find a subsequence converging to a holomorphic mapping (uniformly on compact subsets of D). As D is taut, is a holomorphic mapping from D to D. We have the following lemma. As D is convex,

LEMMA 2.2.

is a holomorphic mapping from D to D.

Let a

E

Fixf. The linear map

F={v€C"If'(a).v=v}.

is a projection onto

Let G bethekernelofço'(a). Then f'(a).Gc

G, and 1 is not an eigenvalue of f'(a)IG.

argument of the proof of the lemma is that by Cauchy's inequalities, the spectral space corresponding to the eigenvalue I of f'(a) is equal to the corresponding eigenspace. Now we consider the sequence of iterates of The central

By Lemma 2.2 and Cauchy's inequalities, we prove that

subsets of D morphic retraction (i.e., = formly on compact

a holoniorphic mapping such that f= iv,and

to

uniis a holo-

converges

Fixf c

FIXED POINTS OF HOLOMORPHIC MAPPINGS

581

By a theorem of H. Cartan [21, in a neighborhood U of a fixed point a of V such that u(a) = 0, and f, we can find a local coordinate chart U: U such that, in the local chart u, y' is a linear projection p. Assume now that we are in the local chart u. Let F = Imp, G = Kerp, and let us use coordinates z (x, y) with x E F and YE G. We have f(x, y) =(x,-f2(x, y)), is not an eigenvalue of (O/Oy)J(O, 0). The same is and by Lemma 2.2, true for every x belonging to a small neighborhood W of 0 in F. By the inverse mapping theorem, the image of id 'G — f2(x, •) contains an open set in G. For x = 0, 0 belongs to this image, and the same is true for every x sufficiently near 0 in G. By analytic continuation, it implies that 4v(D) is 1

equal to Fixf. 3. ExIstence theorems. For general properties of the Carathéodory pseudodistance, we refer to [4]. The following result has been proved by C. Earle and R. Hamilton [3]. D be a Let D be a bounded domain in C" and let f: D holomorphic mapping such that f(D) is relatively compact in D. Then there exists a real number k < I such that f is k-contractive for the Carathéodory integrated distance. it implies that f has a unique fixed point x in D. THEOREM 3.1.

As

usual, x0 is the limit of the sequence of iterates f"(a), for some

aED. let D be a bounded convex domain in C", and let f: D -.' D be A < 1, let a holomorphic mapping. Let a D, and, for every A, C D, and by us consider fA(x) = (1 — A)a + Af(x). It is clear that one has a unique fixed point x1. By careful Theorem 3.1, Now,

proves the followtng lemma. LEMMA 3.2.

(i)

The two following propositions are

f has afixed point in D;

(ii) there exists a sequence

—+ 1

XED.

such that xA

converges to a point

-.

These methods give another proof of Brouwer's theorem for bolomorphic mappings.

Let D be a bounded convex domain in C", and let f: 1Y —s be a continuous mapping, holomorphic in D. Then f has a fixed point in

THEOREM 3.3.

PROOF. Let (1 — A)a + if. We know that JA has a unique fixed point xA. As A —' 1, we can find a converging subsequence x2 —' x0, and

by continuity, x0 is a fixed point of f. In fact, the same method enables us to prove the following result for a family of holoinorphic mappings depending on a complex parameter.

JEAN-PIERRE VIGUE

582

let M be Let D be a bounded convex domain in a connected complex manifold, and let f: M x D —' D be a holomorphic THEOREM 3.4.

M such that f(x0,.) has a fixed mapping. Assume that there exists x0 point in D. Then,for every xEM, f(x,.) hasafixed point in D. z) = PROOF. Let a E D and, forevery A, 0< A < 1 ,Iet us consider —2)a+Af(x, z). For every x E D, fA(x,.) has a unique fixed point it is the limit of the sequence of iterates in D, and as is clear that z q'A(z) is holomorphic. By Montel's theorem, we can find converges to a holomorphic mapping such that a subsequence D, We deduce from Lemma 3.2 that q(x0) 6 D. As D is taut, g(x) and the theorem is proved. M. We have f(x, Vx = (1

1

To conclude this talk, we give the following result proved in a joint paper with M. Abate 111. THEOREM 3.5.

Let D be a bounded convex domain in

and let

to commuting family of continuous mappings from holomorphic in D. Then there exists a coinnion fixed point for the family (f)tEI• a

REFERENCES

I. M. Abate and i-P. Vigué, Common fixed points in hyperbolic Riemann surfaces and convex domains, (to appear).

2. H. Cartan, Sur les rEtractions

"ariElE, C.R. Acad. Sci. Paris Sèr. 1, Math. 303

(1986), 715—716.

3. C. Earle and R. Hamilton, 4 fixed point theorem for holomorphic mappings. Proc. Sympos. Pure Math., no. 16, Amer. Math. Soc., Providence, Ri., 1970, pp. 61-65. 4. 1. Franzoni and E. Vesentini, Holomorphic maps and invariant distances, Mathematical Studies, no. 40, North-Holland, Amsterdam, 1980. 5. M. Hervé, Quelques propriEtEs des applications analyliques d'une boule a m dimensions dans elle-même, J. Math. Pures AppI. (9) 42 (1963), 117—147. 6. P. Mazet ci .1-P. Vigu6, Points fixes d'une applications holomorphe dun domaine bornE dans Iui-même, Acta Math. (to appear). 7. E. Vesentini, Complex geodesics, Compositio Math. 44 (1981), 375—394. 8. —, Complex geodesics and holomorpisic maps, Synipos. Math. 26 (1982), 211—230. 9. J.-P. Viguó, complexes e points fixes d'applica:ions holomorphes, Adv. Math. 52 (1984), 241—247.

10. —,

Points fixes

(applications holomorphes dans un domaine borne convexe de C,

Trans. Amer. Math. Soc. 289 (1985), 345—353. UNtVERSITE DE

Pornus,

FRANCE

Proceedings of Symposia in Pure Mathematics t1991). Par' 2

Holomorphic Functions Taking Their Values

in a q-Space and the Cauchy-Fantappiè Formula L. WAELBROECK

Some functional Analysts study "quotients." A "quotient' is the quotient

of a vector space E with a structure for doing Functional Analysis by a subspacc F with a similar structure, the embedding F C E is Continuous. (The structure of F is stronger than that of E.) A quotient space is the quotient of E modulo F, written ElF, the quotient is "formal." One can define several categories of quotients. F. H. Vasilescu studies quotient Fréchet

spaces (qFré) [5]. We study the category q of quotient bornological spaces [9]. No good category of quotient locally convex complete spaces exists. For those readers who are not Functional Analysts, we shall speak of holomorphic functions on U C C", taking values in a quotient Fréchet space. Some definitions are easier. One can identify a quotient space ElF with the vector space ElF. The category qFré is a subcategory of EV, the category of vector spaces with linear morphisms. In particular, if El F, E' IF' arc quotient Frechet spaces, and u E/F —' E /F is bijective and is a morphism of qFrê, then its inverse is also a morphism of qFré. U, E) is the Fréchet space of functions of class taking values in E

if UcC" isopenand E aFréchetspace; if

is a quotient Fréchet space.

Two definitions of holomorphic functions on U taking their values in a quotient Fréchet space, F. H. Vasilescu [6] calls 4U, EjF) E)I t9(U, F). I call it ElF). And I call ElF) = EIF)l

9°(U, ElF) =

F).

The two definitions are usually not equivalent. I shall show that ElF) if U c C" is a domain of holomorphy as an ElF) 1980 Mathematics Subject Classification (1985 Revision). Primary 46A09. The author works on quotient bornological spaces. He believes that the subject will be part

of Functional Analysis. This paper is in final form and no version of it will be submitted for publication elsewhere. ©

Amcvicak Mathematical Society

0052.0717/91 51.00+5.25 perpav

583

L. WAELBROECK

584

applicatIon of the formula. And I shall begin by reminding the reader of the formula. Part I. The Cauchy-Fantappiè Formula

In 1956, we wanted an n-dimension formula giving f(z), f E 9(U),

z E U C C, to construct holomorphic functional calculus. J. Leray [4] gave one (at a talk in Brussels). L. Fantappié attended the talk and said that he had used it 10 years earlier. I do not find in Fantappit's paper [2), but Leray attributed it to him, and it is called the formula." At this meeting, J. Polking used the "Fantappiè formula." His formula came from Leray's paper. (Fantappiè considered convex subsets of C , Polking studied convex domains in C2.) We used it in our These d'Agrégation (7], then Bourbaki (1] used it to construct a holomorphic function calculus. We use "variables." For most Functional Analysts a variable is a dot, e.g. f(.). We hyphen the variable, and write for example f(2), or 2). If we write f(s, 2), then s is not a variable but 2 a variable.

1. The cohomology space The the with forms of compact support, order.

9(U) is the space of functions of class compact support,

formula uses

ii). with

on

are the spaces àf forms of degree

n,n—1 in

andofdegree0 in

with coefficients in 9(U). by

V)

1. DEFINITION.

where VcU. to be closed in (9)°"(U) for U a Stein

We would like

manifold. The space of cohomology H°"(U) would then be a bornological space.

2. The functions u.(2, and the form th(y, 2, s, df). Let U C C be a domain of holoxnorphy, V U holomorphically convex. Let s E U \ V, E 9(V) such that = 1. If s' is near — to s, then E,(S — 2,)j is uniformly near to I on V. The function 2) = J3(s, — 21)f5(2) is defined when s' is near to s. With a partition of unity, we obtain functions u1(I, 2) E 9(V)) such that

2).

I

Leray used the form

When f€d(U) and z€V heobtained f(z) =

(2irz)

f,w f(s)tb(u;

z,s)ds1 A

THE

FORMULA

U, W has a smooth boundary.

V

3. The

of cohomology w(u; z). We shall not prove the CauchyFantappiè-(Leray) formula, but our (slightly) different one. The cohomology class of wk(u; 2, A = (n + A au,, does not depend

on u,y if

1=

E

u. considered in the paragraph 2 are now called u. We introduce y E such that y(s) = if s E W, let u. = (1 — y(s)). u(s) ifs W, and u1(s, z) =0 if SEW. Then 1

I=

— z1)u1(s, z) + y(s,

with U.

z),

(We write y(s, z)—y(s).)

We use the form of degree n,

(n+k)! k—

Wk(Z,S,dS)

k!

when f E tl(U), (z E V c U).

and use it as kernel for finding

3.2. Using the relations

A..•

A

A

A

A

A

=0; we prove that Wk (u; z, s,

—

Wk_l (u;

z, s, d3) E

A..•A

A

A

L WAELBROECK

In other words,

(n+k)! k' —

y

(n±k —1)!

A•

9(V))).

E

3.3. Assume next that —

z.)u1(z,

s)+y(z, s) = =

s)+y(z, s).

—

1

Then

= where

= u',.u1—u.U1

9(V)),

E

=

,and ft. = y'•u1—u.y

E

9(V)).

We shall prove that

wk(u,y; z,s, d3)—wk(u',y'; z,s,

if 9(V)), and ft.

9(V)). It is sufficient to prove it in the two

E

cases:

(a)u'1=u1—(s2—z2)a,14=u2+(s1—z1)Q,Vi3:u'=u,y'=y, 9(V)))

(Q E

(b)

u =

=y+

u.,

ft.

(z — s),

(ft

E

9'(U, 9(V)))

3.4. In case (a),

+(z1

—(z2 —s2)cz)A

— 8u1 A

A 8u3 A

A

=

3.5.

We have already shown that

In case (b), we assume that k = 1. Of course, ft =

—

—

— fi(z1 —

E

9(V)). A 8142 A••• A

FORMULA

THE

Of course fi(z1

=

587

Next Ou2A•••A Ou,,

(fl(z1

=

A

A t9u2 A••• A = The cohomotogy class of wk(u, y; 2, s, d3) with compact support does

notdependonthechoiceof (U1,..., 2.

noron k. It iscalled

co(U, V; 2). DEFINITIoN. The cohomology class with compact support in V is

w(U, V; 2). If z E VI c V, the inclusion maps co(U, VI)

2)

onto to(U, V, 2).

For those familiar with q-spaces (an object of the category q), w(U, V; 2) H°"(U, H°"(U, el(V)), (or more correctly, w Ex E using the "miracle f'unctor"). 4. The Cauchy-Fantappië formula. Stokes formula shows that (4.1)

2iri

u

w(U, V; z)ds =

14'ZEVCUCC. the integral of w( U, V; 2) A dz1 A ..; A If z E V c U C does notdependon V. Assumethat V= V x V ,where V C U,V CU

U' and U" domains of holomorphy, V' and V" holomorphically convex. Then

w(U, V; z)

(4.2)

V', z') A w(U", V"; z")

... , z,,). Wehave functions u E e9(V')), y" E

u7

y' E such that

'I,

I

=

I

=

—

z.)u(s', z') + y'(s', z'),

It

—

z'5+y"(s", z").

Then

z')

1

+

(si' n'+I

—

z')y'(s' ,

,

z") + y'(s' , z')y"(s" , z")

L. WAEI.BROECK

588

The form (n' + n" + k)! belongs

A••• A

A

A

A

to the cohomology class of w(U, V; z). In function

its degree is n'; u and y' depend only on z1, ...

does

,

not contribute to the form

n... A

A••• A and

(4.3)

w(U' x U" , V' x V" , (z' , z"))

io(U' , V'; z') A w(U" , V" ; z")

fw(U. V; 2)A ds1

(44)

=(2iri)".

When zEV, (4.5)

—

z.)w(U, V; z) = 0,

because —

=

21)wk(u;

2)

A0u1 A••• A

V C U, z E Cauchy-Fantappiè formula

f(z)

V, open holomorphic convex. We have the

f

f(s)w(U, V; z)ds1 A•••A = We choose two polydomains V, W such that z V c W c U. We assume belongs that SuppyE Wand VsE V, y(s)=0. Then VZE W: W). But to the ideal generated by the functions z, — in the algebra V; z) = 0, the integral of o.,(W, V; is equal to (z1 formula above (4.6). and we obtain the (5.6)

Part II. HolQznorphic Functions Taking Their Values in a Quotient Fréchet Space

We remind the reader that our research is about quotient bornological spaces. This category is stable under general direct limits and general inverse limits. The category of quotient Fréchet spaces is stable under countable inverse limits, this is sufficient in most cases in Analysis, and stable under finite direct limits, this is not sufficient. But statements are easier to understand if

E is Fréchet, or ElF is a quotient Fréchet space. One can define two functors of holomorphic functions on U c C", open: ElF) and F) E E ElF)

ThE

FORMULA

= g'(U, F). Both ElF) are Fréchet spaces in a ElF) and natural way. We can prove that (U, EJF) is in general a strict subspace of ElF). EIF) We believe that EIF) if U is a Stein manifold, and shall prove it when U C C" is a domain of holomorphy, giving an application formula. of the

1. Morphisms A

ElF). The morphism

ElF) EIF)

A

ElF)

E) C tf'(U, ElF).

is strict, induced by the inclusion mapping a

2. Themo UcC" bea domain of holomorphy, Vk c U holomorphic convex. Using the Cauchyformula we construct a linear bounded mapping

E). The set

C U. We have functions such that

is open, holomorphically convex

y(s, 2) E

2) e 1=

We write the

2).

2)

—

expression,

with wk(u, y; z, s,

=

A• . .n

E).

bk(f)(2)

shall show that f — (f) E V, F). Let (u,y), (u',y') be such that Vi: U. and y We

1=

+ y(2,

—

=

—

+ y'(2,

s).

We know that Wk

The function b(f) depends on (u, y), but if u', y' satisfy the same relation as u, y, bk(f) — F) because f f(s)(Oço) A ds1 .. A .

=

—

A

A ds1 A ... A ds,,,

comesfrom wk(u',y';z,s, di)). We want to show that

E

F). The property is local, con-

sider W C Vk, openpolydomain. If W cU, onefinds fj Et9(W, E), 12 E W, F) such that ElF). The Caucby-Fantappiè = 11+12 if f E formula maps f1 onto itself, and 12 onto an element of F).

L WAELBROECK

3. The Mittag-Leffler lemma and the morphism B. We use the "Mittag0 be a short exact seLeffler" lemma. '1k E N, let 0 — —. C Ek, the inclusion continuous, and Ek are Fréchet spaces, quence,

is a continuous linear mapping, ak(Ek+l) C and '1k, ak : Ek+l —. At the inverse limit, and is dense in

is onto, inverse limit is left

(One proves only that urn Ek —s lim exact.)

In our situation, for all k E N, we consider a short exact sequence The open sets Vk are holomorphic convex in U, Runge has shown that the F) have dense ranges. The Mittag-Leffler F) —' restrictions lemma shows that V,,, E))I(lim t9'( F)). (This form is used by Hörmander [3, Theorem 9.1.7).) F) hence For all k,we have a morphism ElF) — F). The Mittag-Leffler lim a morphism 9(U, ElF) , F)) , F) (t9( , E)l , E) lemma says that urn F). We have a morphism 9(U,

F).

B

4. The morphisms A and B are inverses each of the other. If E 1(U, E) E)/9(U, F) then A(f1) E belongs to the equivalence class of f because c tl'(U,EjF) and f(U, F) c F). Some more general F) = E) A is injective because F), then E) fl results can be proved. Here it is obvious, let f E

F). It is surjective. Use the Cauchy-Fantappiè formula and the Mittag-Leffler ElF) and f1 an element of the equivalence lemma. Begin with f E ElF). Let (Vk) be a sequence of open subsets class of f, so f1 E holomorphically convex in U. For all fl E N, Cauchyof U, '1k: c U and Fantappièmaps f1 onto The Mittag-Leffler lemma says that

E). f 9(U, ElF) if U C C is a domain

We have proved that 91(U, ElF) of holomorphy and ElF is a quotient Fréchet space.

f

REFERENCES

1. N. Bourbaki, Thioriespectrale, Chapitre 1, Hermann, Paris, 1967. 2. L. Fantappit, Ann. Mat. Pura AppI. 22 (1943). 3. L HOrmander, The analysis of linear partial differential operators. I, Springer-Verlag, Berlin and New York, 1983.

THE CAUCHY.FANTAPPIE FORMULA

591

4. J. Leray, Fonctions de variables complexes. sa reprEsentation comme somme depuissances nEgatives de fonctionnelles lineaires, Acad. Nat di Lincci, classi di Fisici, Mat. e Nat. Seria VII XX (1956), 589—590.

6. F. H. Vasilescu, Spectral theory in quotient FrEchet Spaces. I, Rev. Roumainc Math. Pures App!. 32 (1987), 56 1—579.

7._, Spectral theory in quotient FrEchet spaces. II, .1. Operator Theory 21(1989), 123— 202.

8. L Waelbroeck, Etudes spectra/es des algebres completes, Acad. Roy. Beig. ci Sd. MEm. 9.

Collect. 8° 31(1960), Fasc. 7. ci dernier. , The category of quotient bornological spaces, Aspects of Mathematics (S. A. Barroso, ed.), Elsevier Science Publications, Amsterdam, 1986.

UNWERSITE

DE BRIJXELLES, BELGIUM

Proceedinp o( Symposas in Pure Mathematics Volume 52(1991). Part 2

Schwarz's Lemma and Hermitian Manifolds

with Constant Holomorphic Curvature B. WONG

1. Statement of result. Let D

be

a bounded domain in C". We say that

D covers a compact complex manifold if there exists a discrete subgroup F C Aut(D) = group of biholomorphisms of D, acting freely on D such that

M

D/F is compact. Suppose p is a point on the boundary OD of D; it is called a peak point

if there exists no complex analytic variety of positive dimension lying on OD passing through p. Peak points are very ample in existence for any bounded

domain D in C". One can locate the peak point by looking at the local maximum of the euclidean distance function restricted on OD.

Let p beapointon OD;aboundary neighborhood N of p isan open set in D such that N=UflD, where U isan open set in C" containing

p. A classical theorem asserts that any compact Kähler manifold with constant negative holomorphic curvature is covered holomorphically by the euclidean ball. However, the corresponding assertion for hermitian manifolds with constrant negative holomorphic curvature is unknown. For n = 2, this was settled in [2]. The major difficulty lies in the fact that curvature tensor associated with hermitian connection is rather difficult to use. For instance, even the following special case is an open question. PROBLEM. Let M be a compact hermitian manifold with constant negative holomorphic curvature; is its universal cover biliolomorphic to the euclidean ball? We shall prove the following partial solution to this conjecture using basically the techniques of the Ahlfors-Schwarz lemma [1, 3, 6, 10]. The results due to Stanton [8] and Lempert [4) are also crucial. 1980 Mathematics Subject Classification (1985 Revision). Primary 32R 15, 32H20, 32H25. Research partially supported by an NSF Grant DMS-8 704408. This paper is in final form and no version of it will be submitted for publication elsewhere. Americas Matheinaucal Society 0082.0117/91 51.00 + 5.25 per

593

5. WONG

594

Let D cc C" be a bounded domain covering a compact hermitian manifold M = D/f with constant negative holomorphic curvature. THEOREM.

Suppose there is a peak point p ÔD with one boundary neighborhood N biholomorphic to a convex bounded open set; then D is biholomorphic to the euclidean bail.

Here convexity means that any line segment joining two points of N is completely contained in N. Finally, we would like to mention a related result due to S. Frankel (to appear in Acta Math.) which states that a convex bounded domain covering a compact complex manifold is itself a symmetric domain. 2. Curvature, the Ahlfors-SchwarL lemma, Intrinsic measures, and some

intrinsic characterizations of euclidean balls. Let M be a complex manifold

of dimension n, x E M. DEFINITION. The Kobayashi-Royden differential metric on M is a func-

tion KM: T(M)—R suchthatforall there exists a holomorphic map f: B(R) —. M

KM(x, v) = inf

suchthatf(O)=xanddfo(—(O)) where

B(R) = {w

E

C'lIwI 0 and let (*)

f,:B1

be a holomorphic mapping such that f,(O) = x., df,(tj = v., where t, T0B1 , and < (1 +e)K0(x,, v1). Eventually passing to a subsequence w can assume that f locally uniformly converges to a function 1: B1 —. Since f(O)=p isapeakpoint, f hastobeconstant, f_=p. Thereforeif i is big enough, maps the disc { z C: I z 0, = 0 on B(b), the geodesic ball of radius b around o M. This would prove the theorem. To this end,

we shall use (2) with b as given and with c satisfying b < 2b < c as before. Assume g A log r + B for some positive constants A and B.

POLYNOMIAL FUNCTIONS

Then s(c)

605

A logc + B. Hence (2) implies:

of

*dg C

f

f'

88(b)

8(b)

tdt

{Alogc—Bi(b)}.

t dt/ V(t) = cc; this is because

t dt

I

f

Jb V(i) = 21 V(r)

b2

['t2

V(b)

Jb d

f —1

\

hr2 and V(r)/r2 = 0 by hypothesis. Hence we may let c 00 and apply L'Hopitars rule to conclude that the right side of the previous inequality goes to zero. Hence = 0 in B(b). Q.E.D. We now construct a 2-dimensional complete Kähler manifold M, biholomorphic to C, so that it possesses nonconstant polynomial functions and yet urn V(r)/r2 = 0. The underlying manifold of M is R2 and, relative

to polar coordinates {r, O}, the metric is dr2 + g(r)2 dO2, where g is a C°° function which we proceed to specify. Let exp 2', and ii. for I = 1,2 Pick a small e, e >0, with its "smallness" yet to be determined, and let e, , I = 1, 2 Define [0, co) to be to, cc) the following Continuous function: (a) outside the intervals U1 [/L,, ti

g0(r)=r;(b)ineach[1u1,v,—e1],g0(r)=p1,i=l,2,...;and(c)ineach g0(z1+e1)=si1+e1. Now g0 islinearand let S be the discrete point set U — U + ek}. By a standard smoothing process, we can obtain a C°° nondecreasing functions g such that g = g0 outside an arbitrarily small pre-assigned neighborhood ?I of S. This is the g we are after. Since by choice, g(O) = 0 and g'(O) 1, the C°° character of the metric dr2 + g(r)2 do2 follows from standard arguments.

In the following computations, g will only make its appearance in the integrands of some integrals (see (4)—(6) below); since both e and can be made as small as we wish, it would be perfectly permissible (though at the expense of omitting some nonessential technicalities) to assume instead that g(r) is the following function: (d) g(r) = r outside the intervals zig], i = 1,2 v.], and (e) g(r) = on each Hereafter, we shall work with a metric dr2 + g(r)2 dO2 satisfying (d) and (e). Let p be a point in M with geodesic polar coordinates (r, 0). Define cD: M —. C by c1(p) = Re'°, where R = expfl ds/g(s). c1 is easily seen by a direct computation to be an injective C°° conformal map between M equipped with dr2 + g(r)2 dO2 and C equipped with the flat metric (see [M,

H.WU

606

top of p. 44)). We shall prove: (4) (5)

it follows from (4) that

is a biholomorphic map so that is C (relative to the complex structure of M defined by the isotherfrom Al to so that a polynomial mal coordinates of dr2 + g(r)2 dO2). By (5), R

I of degree k on C (in the usual sense) pulls back via cb to a polynomial of degree 3k on M. Once we have proved that

function

V(L'

Jim

(6)

2

t—•oc

=

M would be a complete Kähler manifold possessing a nonconstant V(r)/r2 = 0. polynomial function such that urn (4) follows immediately from the fact that g(s) s for all s ? 0. Next we prove (6). Because of (d) and (e), then

V(ti1) = V(fl1) + 2irp1(u1 —

+ 2

=7rp1(2

1+1

—1).

(6) follows. Finally, to prove (5), by elementary reasoning dsfg(s) 3 logx it suffices to show that for any x [p1, v1], we have for/=l,2 Indeed, Since ii. =

I\ ds 1" ds P" I I 1f's' —=1 —+1 j———ids \g(s) SI s g(s) =

f

+

LI

2, the fundamental group of a compact manifold with negative bisectional curvature must be an infinite group. This is certainly nontrivIal information. A good intrinsic characterization of manifolds that can be covered by a domain is still unknown. There is a corresponding uniqueness question in this problem. Namely, if we deform the complex structure on the compact complex manifold, how does the complex structure deform on its universal cover? Are they all realizable as domains? For what class of manifolds arc the universal covers isomorphic to each other so that deformation of complex structure corresponds to deformation of discrete group.

If a compact complex manifold is covered by a bounded domain fl, we can cover this manifold by coordinate charts whose coordinate transformations are given by biholomorphic transformations of Conversely, if a manifold admits such a structure, the standard monodromy argument says that there is an immersion of the universal cover of the manifold to Such an immersion is called a developing map. It gives rise to a representation p of the fundamental group ir1 (M) into the group, of automorphisms of Q. with compact quotient. Clearly the image of p(ir1 (M)) acts on It is a difficult question to determine whether the developing map f is an injective or a covering map. If (M)) preserves a metric on then this question can be answered satisfactorily. For in this case, rds2 will be invariant under the deck transformation group that acts on the universal cover of M. Hence tds2 is a complete metric. By passing to the universal cover of fl, if necessary, we may assume that fl is simply-connected. We can now show that p(r1 (M)) acts freely and properly discontinuously on Indeed, if p(g)f(x) = f(x), then for any curve a joining x and g(x), f(a) is a loop at f(x) and can be homotopic to a point in fl. As rds2 is complete and f is an isometry, we can lift the homotopy via f to obtain a contradiction. The same type of argument proves that p(,r1(M)) acts

properly discontinuously.

SHING-TUNG YAU

624

Therefore f induces an immersion from M onto the quotient of the universal cover of by p(ir1 (M)). This is a pretty satisfactory situation. In particular, it is easy to deduce that if is a bounded domain, then M is covered by the universal cover of If M is compact, will in fact be simply connected? The question for uniformization is then reduced to find a "flat" structure which is modeled after some bounded domain. The difficult problem is to find these structures by information on invariants of the algebraic manifolds that are readily compatible. In the Weyl Symposium (Y2], the author has given an algebraic characterization of compact algebraic manifolds that can

be covered by a bounded symmetric domain. There are also recent nice works of C. Simpson [Si] on a similar subject. The basic ideas were based on the analysis of Kahler Einstein metrics constructed by the author, and the Hermitian Yang Mills connections constructed by Donaldson, Uhienbeck, and the author. One may also look at the problem of uniformization from the point of view of Kleinian group. For example, we may look for subgroups of the automorphism group of which act freely and properly discontinuously on some subdomain of cPa. The quotient of such a domain by the subgroup would be an interesting complex manifold. It is a curiosity that no significant research was done on this subject when n> 1. The answer is perhaps that, except in some obvious case constructed from n = 1 , there is no nonstandard action.

REFERENCES

[C]

S. Bando, On three-dimensional compact Kdhler man fildr of non-negative bisectional curvature, J. Differential Geom. 19 (1984), 283—297. M. A. Chinak, Complex manifolds of negative curvature, Functional Anal. AppI. 23

[C-Cl

(1989), 250—251. H.-D. Cao and B. Chow, Compact Käliler manifolds with non-negative curvature operator,

[B]

Invent. Math. 83 (1986), 553—556.

J. Cheeger and D. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature, I. Differential Geom. 6 (1971), 119—128. [C-I] J. Carleson and D. Toledo, Harmonic mappings of Kdhler manifolds to locally symmetric spaces, Preprint. (CCLI Z.-H. Chen, S.-Y. Cheng, and Q.-K. Lu, On the Schwartz lemma for complete Kdhler manifolds, Science Sinica 9 (1979). [F] S. Frankel, Ada Math. 163 (1989). [G) P. A. Griffiths, Complex analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. 94(1971), 21—51. [C-WI R. Green and H. Wu, convex functions and manifolds of positive curvature, Ada (C-G]

Math. 137 (1976), 209—245. [C-MI D. GromoU and W. Meyer, On complete open manifolds of non-negative curvature, Ann. of Math. 90(1969), 75—90. N. Hitchin, On the curvature of rational swfaces, Proc. Sympoo. Pure Math., voL 27, [HI Amer. Math. Soc., Providence, Ri., 1973, pp. 65—80. (HSWI A. Howard, B. Smyth, and H. Wu, Kö.hler ,nanifolds with nonnegative bisectional curvature, Acta Math. 147 (1981), 51—56.

COMPLEX DIFFERENTIAL GEOMETRY

625

[1)

litaka, On algebraic varieties whose universal covering manifolds are complex afline three space, Bull. Amer. Math. Soc. 78 (1972), 737—740.

fJ-YJ

.1.

and S.-T. Yau, Harmonic maps and Kdhler manifolds, Math. Ann. 262 (1983),

145— 166.

(Ml]

N. Mok, An embedding theorem of complete Kã.hler manifolds of positive bisectional curvature onto affine algebraic varieties, Bull. Soc. Math. France 112 (1984), 197—258. [M2] The uniformization theorem for compact Kdhler manifolds of non-negative holomorphic bisectional curvature, J. Differential (3com. 27 (1988), 179—214. [M-ZJ N. Mok and J.-Q. Zhong, Curvature characterization of compact Hermitian symmetric spaces, J. Differential (3com. 23 (1986), 15—67. [Mo) S. Mori, Projeclive manifolds with ample tangent bundles, Ann. of Math. 110 (1979),

_,

593—606.

[R]

Peternell, Calabi- Yau manifolds and a conjecture of Kobayashi, Preprint. H. L Royen, The Ahlfors-Schwarz lemma in several complex variables, Math. Helv. 55

[Sa]

J. Sampson, Applications of harmonic maps to Kdhler geometry, Contemp. Math. 49

[P1

(1980), 547—558.

(1986), 125—134.

W.-X. Shi, Ph.D. Thesis, Harvard University, 1990. C. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to I. Amer. Math. Soc. 1 (1988), 867—9 18. [Su] Y.-T. Siu, Complex analyticity of harmonic maps and strong rigidity of complex Kähler manifolds, Ann. of Math. 112 (1980), 73—111. [S-TI I Y.-T. Siu and S.-T. Yau, Compact Kãhler manifolds of positive bisectional curvature, [Shj (Si]

Invent. Math. 59 (1980), 189—204.

[S-T21 —, [Wa] (WI)

Complete Kdhjer manifolds with non-positive curvature of faster than quadratic

decay, Ann. of Math. 105 (1977), 225—264. H.-C. Wang, Complex pasallisable manifolds, Proc. Amer. Math. Soc. 5(1954), 771—776.

P. M. H. Wilson, Calabi-Yau manifolds with large Picard number, Invent. Math. 98 (1989), 139—155.

[Wi)

B. Wong, Characterization of the ball in C" by its automorphism groups, Invent. Math.

41(1977), 253—257. [W2J —, A class of compact complex manifolds with negative tangent bundles, Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, R.I., 1984, pp. 217—223. [Yl] S.-T. Yau, On Calabi conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sd. U.S.A. 74 (1971), 1798—1799. [Y2J Unsformization of geometric structures, Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, R.I., 1988, pp. 265—274.

_,

[Y-Z1] S.-T. Yau and F. Zheng, On projective manifolds with universal covering space C", Beijing, 1988.

[Y-Z2] —, Negatively

-pinched Riemannian metric on a compact Kdhler manifold, Preprint. [Y-Z3] Y.-U. Ye and Q. Zhang, Preprint, Duke University, 1989. (Z] F. Zheng, Ph.D. Thesis, Harvard University, 1989. HARVARD UNIVERSITY

Several Complex Variables and Complex Geometry (Proceedings of Symposia in Pure Mathematics, Vol.52) (Pt. 2)

Several complex variables and complex geometry,

Algebra and geometry in several complex variables

Analysis and Geometry in Several Complex Variables (Trends in Mathematics)