From Holomorphic Functions to Complex Manifolds (Graduate Texts in Mathematics)

Klaus Fritzsche Hans Grauert From Holomorphic Functions to Complex Manifolds With 27 Illustrations Klaus F n t z s c...

Author: Klaus Fritzsche | Hans Grauert

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Klaus Fritzsche Hans Grauert

From Holomorphic Functions to Complex Manifolds With 27 Illustrations

Klaus F n t z s c h e Be rg i sch e Universitat Wup pertal GauBstraBe 20 D-42 1 1 9 Wuppertal

H a ns Grauert Mathematisches Institut

Geor g-August-Univer sitatGottingen BunsenstraBe 3-5 .

G ermany

D-37073

[email protected]

Germany

Gottingen

Editorial Board S. Axler Mathematics Department

San Francisco

State

University San Francisco, CA

94 132

F.W. G e h r i n g

K.A.

Mathematics Department East Hall

Mathematics Department

University of Michigan Ann Arb o r , MI 48 109

USA

USA

[email protected]

fgehring ® ma th.lsa.

Ribet

University of California, Berkeley Berkeley, CA

94720-3840

USA rib et@ math. berkeley .edu

umich.edu

Mathematics Subject Classification (2000): 32-01, 32Axx, 32D05, 32Bxx, 32Qxx, 32E35 Library of Co ngress Cataloging-in-Publication Data Fritzsche, Klaus. From holomorphic functions to complex manifolds p. em. -(Graduate texts in mathematics ; 213)

I

ISBN 0-387-95395-7 (alk paper) I. Complex manifolds. 2. Holomorphic functions.

Klaus Fritzsche, Hans Grauert.

Includes bibliographical references and indexes,

I. Grauert, Hans, 1930-

III. Series.

II. Title.

QA331.7 .F75 2002 2001057673

5!5'.98---dc21 Printed on acid-free paper.

©

2002 Springer-Verlag New York, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use i n connection with any form of information storage and retrieval, electronic adaptation, computer soft ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Production managed by Michael Koy; manufacturing supervised by Erica Bresler. Photocomposed from the authors' LaTeX files. Printed and bound by Maple-Vail Book Manufacturing Group, York, PA. Printed in the United States of America. 9 8 7 6 5 4 3 2 I ISBN 0-387-95395-7

SPIN 10857970

Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSprin er Science+Business g

Media GmbH

Preface The aim of this book is to give an understandable introduction to the the ory of complex manifolds. With very few exceptions we give complete proofs. Many examples and figures along with quite a few exercises are included. Our intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involved with sheaves, coherence, and higher-dimensional cohomology are avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional co cycles are used. Nevertheless, deep results can be proved, for example the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution of the Levi problem. The first chapter deals with holomorphic functions defined in open sub sets of the space @". Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi mum principle, can be applied directly to obtain corresponding properties of holomorphic functions of several variables. Furthermore, certain properties of differentiable functions of several variables, such as the implicit and inverse function theorems, extend easily to holomorphic functions. In Chapter II the following phenomenon is considered: For n > 2, there are pairs of open subsets H c P c en such that every function holomorphic in H extends to a holomorphic function in P. Special emphasis is put on domains G c en for which there is no such extension to a bigger domain. They are called domains <.f holomorphy and have a number of interesting convexity properties. These are described using plurisubharmonic functions. If G is not a domain of holomorphy, one asks for a maximal set E to which all holomorphic functions in G extend. Such an "envelope of holomorphy" exists in the category of Riemann domains, i.e., unbranched domains over e n . The common zero locus of a system of holomorphic functions is called an analytic set. In Chapter III we use Weierstrass's division theorem for power series to investigate the local and global structure of analytic sets. Two of the main results are the decomposition <.f analytic sets into irreducible components and the extension theorem of Remmert and Stein. This is the only place in the book where singularities play an essential role. Chapter IV establishes the theory of complex manifolds and holomorphic fiber bundles. Numerous examples are given, in particular branched and un branched coverings of e n , quotient manifolds such as tori and Hopf manifolds, projective spaces and Grassmannians, algebraic manifolds, modifications, and toric varieties. We do not present the abstract theory of complex spaces, but do provide an elementary introduction to complex algebraic geometry. For example, we prove the theorem of Chow and we cover the theory of clivi-

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sors and hyperplane sections as well as the process of blowing up points and submanifolds. The present book grew out of the old book of the authors with the ti tle Several Complex Variables, Graduate Texts in Mathematics 38, Springer Heidelberg, 1976. Some of the results in Chapters I, II, III, and V of the old book can be found in the first four chapters of the new one. However, these chapters have been substantially rewritten. Sections on pseudoconvexity and on the structure of analytic sets; the entire theory of bundles, divisors, and meromorphic functions; and a number of examples of complex manifolds have been added. Our exposition of Stein theory in Chapter V is completely new. Using only power series, some geometry, and the solution of Cousin problems, we prove finiteness and vanishing theorems for certain one-dimensional cohomology groups. Neither sheaf theory nor 8 methods are required. As an application Levi ' s problem is solved. In particular, we show that every pseudoconvex domain in e n is a domain of holomorphy. Through Chapter V we develop everything in full detail. In the last two chapters we deviate a bit from this principle. Toward the end, a number of the results are only sketched. We do carefully define differential forms, higher dimensional Dolbeault and de Rham cohomology, and Kahler metrics. Using results of the previous sections we show that every compact complex mani fold with a positive line bundle has a natural projective algebraic structure. A consequence is the algebraicity of Hodge manifolds, from which the classical period relations are derived. We give a short introduction to elliptic opera tors, Serre duality, and Hodge and Kodaira decomposition of the Dolbeault cohomology. In such a way we present much of the material from complex differential geometry. This is thought as a preparation for studying the work of Kobayashi and the papers of Ohsawa on pseudoconvex manifolds. In the last chapter real methods and recent developments in complex an alysis that use the techniques of real analysis are considered. Kahler theory is carried over to strongly pseudoconvex subdomains of complex manifolds. We give an introduction to Sobolev space theory, report on results obtained by J.J. Kohn, Diederich, Fornmss, Catlin, and Fefferman (a-Neumann, subellip tic estimates), and sketch an application of harmonic forms to pseudoconvex domains containing nontrivial compact analytic subsets. The Kobayashi met ric and the Bergman metric are introduced, and theorems on the boundary behavior of biholomorphic maps are added. Prerequisites for reading this book are only a basic knowledge of calculus, analytic geometry, and the theory of functions of one complex variable, as well as a few elements from algebra and general topology. Some knowledge about Riemann surfaces would be useful, but is not really necessary. The book is written as an introduction and should be of interest to the specialist and the nonspecialist alike.

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vii

We are indebted to many colleagues for valuable suggestions, in particular to K. Diederich, who gave us his view of the state of the art in a-Neumann theory. Special thanks go to A. Huckleberry, who read the manuscript with great care and corrected many inaccuracies. He made numerous helpful sug gestions concerning the mathematical content as well as our use of the English language. Finally, we are very grateful to the staff of Springer-Verlag for their help during the preparation of our manuscript. Wuppertal, Gottingen, Germany Summer 2001

Klaus Fritzsche Hans Grauert

Contents P r efa ce I

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Holomorphic Functions 1. Complex Geometry . . . . . . . . . . . . . Real and Complex Structures . . . . . Hermitian Forms and Inner Products Balls and Polydisks . . . . . . . . . . . Connectedness . . . . . . . . . . . . . . Reinhardt Domains . . . . . . . . . . . 2. Power Series Polynomials Convergence . . . . . . . . . . . . Power Series . . . . . . . . . . . . 3 . Complex Differentiable Functions . The Complex Gradient . . . . . Weakly Holomorphic Functions . . Holomorphic Functions . . . . . 4 . The Cauchy Integral . . . . . . . . . . . . The Integral Formula . . . . . . . . . . Holomorphy of the Derivatives . . . . The Identity Theorem . . . . . . . . . . 5 . The Hartogs Figure . . . . . . . . . . . . Expansion in Reinhardt Domains Harto gs Figures . . . . . . . . . . . . 6. The Cauchy-Riemann Equations . . . Real Differentiable Functions . . . . Wirtinger' s Calculus . . . . . . . . . . . The Cauchy-Riemann Equations . . . 7. Holomorphic Maps . . . . . . . . . . . . . . . . . . . . The Jacobian . . Chain Rules . . Tangent Vectors . . . . . . . . . . . . . The Inverse Mapping . . . . . . . . . . 8. Analytic Sets . . . . . . . . . . . . . . . . . Analytic Subsets . . . . . . . . . . . . . Bounded Holomorphic Functions . Regular Points . . . . . . . . . . . . Injective Holomorphic Mappings . ·

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Domain s of H o1omorphy 1. The Continuity Theorem General Hartogs Figures Removable Singularities . The Continuity Principle Hartogs Convexity . . . . Domains of Holomorphy 2. Plurisubharmonic Functions Subharmonic Functions . The Maximum Principle Differentiable Subharmonic Functions Plurisubharmonic Functions The Levi Form . . . . . . . . Exhaustion Functions 3. Pseudoconvexity . . . . . Pseudoconvexity . . . . . . . . . . . . . The Boundary Distance . . . . . . . . Properties of Pseudoconvex Domains 4 . Levi Convex Boundaries . . . . . . . . . Boundary Functions The Levi Condition . Affine Convexity . . . . . . . . . . . . . A Theorem of Levi . 5 . Holomorphic Convexity . . . . . . . . . . . Affine Convexity . . . . . . . . . . . . . Holomorphic Convexity . . . . . . . The Cartan-Thullen Theorem . . . 6. Singular Functions . . . . . . . . . . . . . Normal Exhaustions . . . . . . . . . . . Unbounded Holomorphic Functions Sequences . . . . . . . . . . . . . . 7. Examples and Applications . . . . Domains of Holomorphy . . . Complete Reinhardt Domains Analytic Polyhedra . . . . . . . . . . 8. Riemann Domains over en . . . . . . Riemann Domains . . . . . . . . . . . Union of Riemann Domains . . . . 9 . The Envelope of Holomorphy . . . . . Holomorphy on Riemann Domains Envelopes of Holomorphy . . . . . Pseudoconvexity Boundary Points Analytic Disks . . . . . .

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An alytic Set s 1. The Algebra of Power Series . . . The Banach Algebra Bt . . . . Expansion with Respect to z1 Convergent Series in Banach Algebras Convergent Power Series Distinguished Directions . . . 2 . The Preparation Theorem . . . . Division with Remainder in Bt The Weierstrass Condition . . . . . . . . Weierstrass Polynomials . . . . Weierstrass Preparation Theorem . . . . . . . . . . . . 3. Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . Unique Factorization . . . . . . . . . . . . . . . . . . . . Gauss's Lemma . . . . . . . . . . . . . . . . . . . . Factorization in Hn . . . . . . . . . . . . . . . . . . . . . Hensel's Lemma . . . . . . . . . . . . . . . . . . . . . . . The Noetherian Property . . . . . . . . . . . . . . . . . 4. Branched Coverings . . . . . . . . . . . . . . . . . . . . . . . Germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudopolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euclidean Domains The Algebraic Derivative . . . . . . . . . . . . . Symmetric Polynomials . . . . . . . . . . . . . . . The Discriminant . . . . . . . . . . . . . . . . . . . . . Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . The Unbranched Part . . . . . . . . . . . . . . . . . . . . Decompositions . . . . . . . . . . . . . . . . . . . . . . . Projections . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Irreducible Components . . . . . . . . . . . . . . . . . . . . . Embedded-Analytic Sets . . . . . . . . . . . . . . . . Images of Embedded-Analytic Sets . . . . . . . . . . . Local Decomposition . . . . . . . . . . . . . . . . . . . . Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . The Zariski Topology . . . . . . . . . . . . . . . . . . . . Global Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Regular and Singular Points . . . . . . . . . . . . . . . . Compact Analytic Sets . . . . . . . . . . . . . . . . . Embedding of Analytic Sets Regular Points of an Analytic Set . . . . . . . . . . . . . The Singular Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . Extending Analytic Sets . . . . . . . . . . The Local Dimension . . . . . . . . . . . . . . . . . . . .

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1 05 1 05 1 05 1 06 1 07 1 08 1 09 1 10 1 10 1 13 1 14 1 15 1 16 1 16 1 17 1 19 1 19 120 123 123 124 125 125 126 126 127 130 130 1 32 1 35 1 35 1 37 138 140 141 141 143 143 144 145 147 147 150

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Contents 153 Complex M an ifolds 1 . The Complex Structure . . . . . . . . . . . . . . . . . . . 153 Complex Coordinates . . . . . . . . . . . . . . . . . . 153 156 Holomorphic Functions . . . . . . . . . . . . . . Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . 157 Holomorphic Mappings . . . . . . . . . . . . . . . . . 158 159 Cartesian Products . . . . . . . . . . . . . . Analytic Subsets . . . . . . . . . . . . . . . . . . . . . . . 160 162 Differentiable Functions . . . . . . . . . . . . . . . . . . Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . 164 166 The Complex Structure on the Space of Derivations . . The Induced Mapping . . . . . . . . . . . . . . . . . . . . 167 168 Immersions and Submersions . . . . . . . . . . . . . . . Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 171 2 . Complex Fiber Bundles 171 Lie Groups and Transformation Groups . . . . . . . . . Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . 1 73 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 174 175 Complex Vector Bundles . . . . . . . . . . . . . . 177 Standard Constructions . . . . . . . . . . . . . . . 180 Lifting of Bundles . . . . . . . . . . . . . . . . . . . . . Subbundles and Quotients . . . . . . . . . . . . . . . . 180 182 3. Cohomology Cohomology Groups . . . . . . . . . . . . . . . . . . . . . 182 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . 184 Acyclic Coverings . . . . . . . . . . . . . . . . . . . . . . 1 85 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 1 86 The Singular Cohomology . . . . . . . . . . . . . . . . . 188 192 4. Meromorphic Functions and Divisors . . . . . . . . . . . . The Ring of Germs . . . . . . . . . . . . . . . . . . . . . 192 . . . . . . . . . 193 Analytic Hypersurfaces . . . . . . 196 Meromorphic Functions . . . . . . . . . . . . . . . . . . Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Associated Line Bundles . . . . . . . . . . . . . . . . . . 200 Meromorphic Sections . . . . . . . . . . . . . . . . . . . . 201 5 . Quotients and Submanifolds . . . . . . . . . . . . . . . . . . 203 Topological Quotients . . . . . . . . . . . . . . . . . . . . 203 Analytic Decompositions . . . . . . . . . . . . . . . . . . 204 205 Properly Discontinuously Acting Groups . . . . . . . . Complex Tori . . . . . . . . . . 206 Hopf Manifolds . . . . . . . . . . . . . 207 208 The Complex Projective Space 210 Meromorphic Functions . 21 1 Grassmannian Manifolds . . . ·

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Submanifolds and Normal Bundles Projective Algebraic Manifolds . . Projective Hypersurfaces . . . . . The Euler Sequence . . . . . . . . . . Rational Functions Branched Riemann Domains Branched Analytic Coverings . . . . Branched Domains . . . . . . . . . . Torsion Points . . . . . . . . . . . . Concrete Riemann Surfaces . . . Hyperelliptic Riemann Surfaces Modifications and Toric Closures Proper Modifications . . . . . . . . . Blowing Up . . . . . . . . . . . . . The Tautological Bundle . Quadratic Transformations Monoidal Transformations Meromorphic Maps . . . . . . . . . . . . . . . . . . . . Toric Closures ·

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Stein Theory 1 Stein Manifolds Introduction Fundamental Theorems Cousin-! Distributions . . . . . . . . . . . Cousin-Il Distributions . . . . . . . . . Chern Class and Exponential Sequence Extension from Submanifolds . . . . . Unbranched Domains of Holomorphy The Embedding Theorem . . . . . . . The Serre Problem . . . . . . . . . . . . . . . . . 2 . The Levi Form . Covariant Tangent Vectors . . . . . Hermitian Forms . . . . . . . . . . . . . Coordinate Transformations Plurisubharmonic Functions The Maximum Principle Pseudoconvexity 3 Pseudoconvex Complex Manifolds Examples . . . . . . . . . . . . . . . Analytic Tangents . . . . . . . . . . . . . . 4. Cuboids . . . . . . . . . . . . . . . . . . . . . Distinguished Cuboids . . . . . . . . . . . Vanishing of Cohomology . . . . . . . . Vanishing on the Embedded Manifolds ·

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Cuboids in a Complex Manifold . Enlarging U' . . . . . . . . . . . . . Approximation . . . . . . . . . . . . Special Coverings . . . . . . . . . . . . Cuboid Coverings . . . . . . . . . . The Bubble Method . . . . . . . . . Frechet Spaces . . . . . . . . . . . . Finiteness of Cohomology . . . . . Holomorphic Convexity . . . . . . Negative Line Bundles . . . . . . . . Bundies over Stein Manifolds . . . The Levi Problem . . . . . . . . . . . . Enlarging: The Idea of the Proof Enlarging: The First Step . . . Enlarging: The Whole Process . Solution of the Levi Problem . . The Compact Case . . . . . . . . .

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VI Kahler M anifolds 1. Differential Forms . . . . . . . . . . . . . . . . . . . . . . The Exterior Algebra . . . . . . . . . . . . . . . . . . Forms of Type (p,q) . . . . . . . . . . . . . . . . . . . Bundles of Differential Forms . . . . . . . . . . . 2. Dolbeault Theory . . . . . . . . . . . . . . . . . . . . . . Integration of Differential Forms . . . . . . . . . . . The Inhomogeneous Cauchy Formula . . . . . . . . The 3-Equation in One Variable . . . . . . . . . . . . . A Theorem of Hartogs . . . . . . . . . . . . . . . . . . . Dolbeault' s Lemma . . . . . . . . . . . . . . . . . . . . Dolbeault Groups . . . . . . . . . . . . . . . . . 3 . Kahler Metrics . . . . . . . . . . . . . . . . . . . . . Hermitian metrics . . . . . . . . . . . . . . . . . . . . . The Fundamental Form . . . . . . . . . . . . . . . . . Geodesic Coordinates . . . . . . . . . . . . . . . . . . . . Local Potentials . . . . . . . . . . . . . . . . . . . . . . Pluriharmonic Functions . . . . . . . . . . . . . . . The Fubini Metric . . . . . . . . . . . . . . . . . . . . Deformations . . . . . . . . . . . . . . . . . . . . . . . . 4. The Inner Product . . . . . . . . . . . . . . . . . . . . . . The Volume Element . . . . . . . . . . . . . . . . . . . The Star Operator . . . . . . . . . . . . . . . . . . . . . . The Effect on ( p,q)-Forms . . . . . . . . . . . . . . . The Global Inner Product . . . . . . . . . . . . . . . . Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . .

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A djoint Operators .

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The Kahlerian Case Bracket Relations . The Laplacian . . . . . . . . . . . . . Harmonic Forms . . . . . . . . . . . . Consequences . . . . . . . . . . . . . . Hodge Manifolds . . . . . . . . . . . . . . Negative Line Bundles . . . . . . . . . Special Holomorphic Cross Sections Projective Embeddings . . . . . . . Hodge Metrics . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . Period Relations . . . . . . . . . . . . The Siege l Upper Halfplane . . . . . Semipositive Line Bundles Moishezon Manifolds

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VII B oundary Beh avior 1. Strongly Pseudoconvex Manifolds . . . . . . . . . . . . The Hilbert Space . . . . . . . . . . . . . . . . . . . . . Operators . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Conditions . . . . . . . . . . . . . . . . . . . 2 . Subelliptic Estimates . . . . . . . . . . . . . . . . . . . . . . Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . The Neumann Operator . . . . . . . . . . . . . . . . Real-Analytic Boundaries . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Nebenhullen . . . . . . . . . . . . . . . . . . . . . . . . . . . General Domains . . . . . . . . . . . . . A Domain with Nontrivial Nebenhulle . Bounded Domains . . . . . . . . . . . . . . . tr<2 D omams m \l.. 4. Bo undary Behavior of Biholomorphic Maps . . . . . . . . . . . . . . . . . . . The One-Dimensional Case . . . . . The Theory of Henkin and Vormoor . . . . . . . . . . . Real-Analytic Boundaries . . . . . . . . . . . . . . Fefferman' s Result . . . . . . . . . . . . . . . . . . . . . Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . The Be rgman Metric . . . . . . . . . . . . . . . . . . . .

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References

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Index of Not ation

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Index

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C h ap ter I

Holomorphic Functions 1.

C omplex Geo metry

Real and Complex S tructures . Let V be an n-dimensional com plex vector space. Then V can also be regarded as a 2n-dimensional real vector space, and multiplication by i := A gives a real endomorphism J: V---+ V with J2 = - idv. If {a1, . . . , a n } is a complex basis of V , then { a1, . .. , an, i a1, . . . , ia,} is a real basis of V. On the other hand, given a 2n-dimensional real vector space V, every real endomorphism J: V ---+ V with J2 = - idv induces a complex structure on V by (a+ib) · v:= a·v+b . J(v). We denote this complex vector space also by V, or by (V,J), if we want to emphasize the complex structure.

If a complex structure J is given on V, then -J is also a complex structure.

It is called the - space ( V,-J) is sometimes - conjugate complex structure, and the denoted by V. A vector v E V i� also .1 vector in V . If z is a complex number, then the product z · v, formed itV, gves the same vector as the product z · v in V. Our most important example is the complex n-space e n := {z:= (ZI, · . . ,zn) : Z;

with the standard

E e for i = 1 , . . . , n },

basis

e1 := (1,0 , . . . ,0), .. . ,e n := (0 , . . . ,0, 1) .

We can interpret en as the real 2n-space

together with the complex structure J: IR2n ---+ JR2n , given by

These considerations lead naturally to the idea of "complexification."

1 A row vector is described by a bold symbol, for instance v , whereas the corre

sponding column vector is written as a transposed vector: v

t.

I. Holomorphic Functions

2

Definition. Let E be an n-dimensional real vector space. The com plexification of E is the real vector space Ec == EEB E, together with the complex structure J : Ec -+ E, given by J(v,w) := (-w,v).

Furthermore, conjugation

C

in Ec is defined by

C(v, w) :=(v, -w).

Since CoJ = -JoC, it is clear that C defines a complex isomorphism between Ec and Ec. The complexification of JR" is the complex n-space C" identified with JR2 n in the way shown above. In this case the conjugation C is given by

and will also be denoted by

z H z.

If V = E, is the complexification of a real vector space E, then the subspace Re(V) := { (v,O)

:

v

E E) c V

is called the real part of V. Since it is isomorphic to E in a natural way, we can write V E EB iE. 1f V is an arbitrary complex vector space, then V is the complexification of some real vector space as well, but this real part is not uniquely defined. It is given by the real span of any complex basis of V. "'

Example Let E be an n-dimensional real vector space and E* := HomiR (E, JR) the real dual space of linear forms on E. Then the complexification (E*)c can be identified with the space HomiR(E,q of complex-valued linear forms on E. In the case E

=

JR n , a linear form

,\

E E* is always given by

t \ . . A .VHV a ,

with some fixed vector a E JRn . An element of the complexification ( E*)c is then given by V H v · z t with z = a + ib E (JRn )c = en . Now let T be an n-dimensional complex vector space and F(T) :=HomiR(T, C) the space of complex-valued real linear forms on T. It contains the subspaces T' := Homc(T,C) of complex linear forms andT' :=Homc(T,q of complex antilinear forms 2. 2

real linear map >. : T-+ 1C is called complex anti linear if >. ( c . v ) = c . >.( v ) for c E IC. Therefore, T 1 can be viewed as the set of complex antilinear forms on T.

A

l.

Complex Geometry

3

Let {a 1 , .. . , an} be a complex basis of T, and b; : = ia;, fori = 1, ... , n. Let { o:1, . .. , O:n, fJ1 , ..., f3n} be the basis of T* = HomiR (T, IR) that is dual to {a 1, ..., an , b 1 , . .. , bn} Then we obtain elements

-

A; Claim. PROOF: Then

:= a1

+ i {3; E F (T),

i

= 1, ... , n .

Theforms A; are complex-linear. Consider an element z = z1 a 1 + . +znan E T with z; = x; -tiy; E C.

n n Ak ( :2::: x;a; + :2::: y;b;) i=1

n

i=1 n

i=1 i=1 Xk + i:Uk = Zk.

Now the claim follows.

.

•

It is obvious that the A; �re.Jinearly independe!!t. Therefore, ( A 1, . . , , In} is a basis of T ' , and {A1,. ··;An} is a basis ofT'. Since it is also obvious that T' nT' = {0}, we see that every element A E F(T) has a unique representation n

n

A= I:c;A; + I:d;A;, with c;,d; i=1

i=1

E C.

Briefly,

A = A1 + A11, with A1 E T ' and A11 E T ' , Here A is real; i.e., A E HomiR(T, IR) if and only if A11 = N, Hermitian Forms an d Inner Products Definition. Let T be an n-dimensional complex vector space. A Her mitian fo rm on T is a function H : T x T -> C with the following properties: 1. v >-+ H(v, w) is C-linear for every wE T . 2. H(w,v) = H (v, w) forv, w E T.

It follows at once that w >-+ H (v, w) is C-antilinear for every v E T, and H(v,v) is real for every v E T. If H(v,v) > 0 for every v 1- 0, H is called an inner product or scalar product.

4

I.

Holomorphic Functions

There is a natural decomposition

H (v, w ) = S(v, w J+ iA(v, w), with real-valued functions S and

S(w,v) + iA( w v) ,

=

A.

H(w, v)

Since =

H (v, w) = S(v, w )- iA(v, w),

it follows that S is symmetric and A antisymmetric. Example If k is a field, the set of all matrices with p rows and q columns whose elements

lie in k will be denoted by Mp,q( k) and the set of square matrices of order n by Mn (k) . Here we are interested only in the cases k = lR and k = C . A Hermitian form on en is given by

H: (z,w) H zHwt, where HE Mn (C) is a Hermitian matrix, i.e., H t = H. The associated symmetric and antisymmetric real bilinear forms S and A are given by

-

and

A(z,w)=Im(zHwt) =

� (zHwt-wHzt).

If H is an inner product, then S is called the associated Euclidean z n n e r product. The identity matrix En yields the standard He rm itian scalar product

(zlw/ = z · wt

=

n

L:>vWv· v =l

Its symmetric part Cziw)2 := Re ((z l w )) is the standa rd Euclidean scalar z = x + iy and w = u + iv, with x,y , u, v E lRn, product. In fact, if we write n then 2

1

-t +w .z -t) (z .w n 1 L 2 (vvWv + WvZv ) v =l n

L (XvUv + YvVv)·

v =l

1. Complex Geometry

5

If the standard Euclidean scalar product on IR. n is denoted by (· ..1 ..·) n , we obtain the equation

B alls an d Polydisks Definition .

The Euclidean norm of a vector z E e n is given by

the Euclidean distance between two vectors z, w by dist(z, w ) :=liz- wll· An equivalent norm is the sup - n o rm or modu lus of a vector: lzvl· lzl := v=max l, ... ,n

This norm is not derived from an inner product, but it defines the same topo logy on en as the Euclidean norm. This topology coincides with the usual topology on IR2 n . We assume that the reader is familiar with it and mention only that it has the Hausdorff property. Definition . Br(zo) := { z E e n : dist(z,z0) < r} is called the ( op e n ) ball of radius r with center z0. A ball in en is also a ball in IR2 n , and its topological boundary 8Br(zo) = { z E en

:

dist(z,z0) = r}

is a (2n- I)-dimensional sphere. Definition . en . Then

Let r

=

(r1, . .. ,rn ) E !R n , all rv

>

O,zo

p n ( Zo' r) : = { z E en : I Zv - z�O) I < Tv for v

=

=

(zi0l, . .. ,z�0l ) E

1' . . . , n }

is called the (op e n ) polydisk (or polycylinde r) with polymdius rand center zo. If r E IR+ and r := (T, . . . ,r ) , we write P�(z0) instead of P n (z0, r). Then P�(zo) = { z E en : lz- zo l < TJ. If Ddenotes the open unit disk in e, then pn := Pr(o)

called the unit polydisk around 0.

=

Dx . n

·

.

x Dis

times

6

I. Holomorphic

Functions

We are not interested in the topological boundary of a polydisk. The fo llowing part <.f the boundary is much more important: Definition. the set

The distinguished boundary <.f the polydisk pn(zo, r) is

Tn(zo,r) = { z

E en : \zv

- z�0l\ = rv for v = 1 , ..., n} .

The distinguished boundary of a polydisk is the Cartesian product of n circles. It is well known that such a set is diffeomorphic to an n-dimensional torus. In the case n = 1 a polydisk reduces to a simple disk and its distinguished boundary is equal to its topological boundary.

Connectednes s . Both the Euclidean balls and the polydisks form a base <.f the topology <.f en . By a region we mean an ordinary open set in en . A region G is connected if each two points of G can be joined by a continuous path in G. A connected region is called a domain. If a real hyperplane in IR.n meets a domain, then it cuts the domain into

two or more disjoint open pieces. For complex hyperplanes in the complex number space (which have real codimension 2) this is not the case: 1 . 1 Propositi o n . Let G c en be a domain and E

:=

{Z =

(zi, · .. ,z,)

Th e n G ' := G- E is again a domain.

E en

: ZI

= 0) .

PROOF: Of course, Eis a closed set, without interior points, and G' = G- E is open. Write points <.f en in the form z = (zi, z*), with z* E e n - I _ Given two points v = (vi , v*) and w = (wi,w*) in G ' , it must be shown that v and w can be joined in G ' by a continuous path. We do this in two steps. Step 1 : Let G = pn(z0, c ) be a small polydisk. Then G ' is the product <.f a punctured disk and a polydisk in n - 1 variables. Define z := (wi,v* ) . Clearly, z E G ', and we can join vi and WI within the punctured disk, and v* and w* within the polydisk. Therefore, v and w can be joined within G ' . Step 2: Now let G be an arbitrary domain. There is a path t.p :I -+ G joining v and w. Since t.p(I) is compact, it can be covered by finitely many polydisks UI , . . . , Uz such that U>. C G for A = 1 , ... ,1.

It is easy to show that there is a o > 0 such that for all t ', t" E I with it' - t"l < 6, t.p ( t' ) and t.p(t") lie in the same polydisk Uk . Then let a = t o < ti < . . . < tN = b be a partition of I with \tj - tj - I \ < o for j = 1 , . . . ,N. Let Zj : = t.p(tj ) and .X (j) E ( 1 , ..., l} be chosen such that U>.(j ) contains Zj and Zj - I (it can happen that .X (ji) = .X (h) for ji 1- j2 ). By construction Zj- I lies in U>. (j ) n U>.(j - I J , and thus U>.(j ) n U>. (j - I) - E is always a nonempty open set.

1.

Complex

7

Geometry

We join v = zo E U>. (I J and some poi nt Z I E U>. (I J n U>.(2 J E by a path 'P I interior to U>. (I) E By (l )this is possible. Next we joi n Z I and a poi nt z2 E U>. ( 2 J n U>. ( 3) E by a path 1p2 interior to U>. ( 2 J E, and so on. Finally, 'P N joins ZN - I a nd w = ZN within U>. (N) - E. The compositio n of 'P I , ip N co nnects v and w in G ' . • -

-

.

-

-

. . . ,

Rein hardt Domains Definition.

The poi nt set

Y : = {r = (ri , .

n) E lR n

. .,r

: rv

> 0 for v = 1 , . .. , n}

will be called absolute space, the map T :en (lzii, . .. , lzn I ) the natural projection.

---+

"f/

with T(Z I , , . . , Z ) n

:=

The map T is co nti nuous and surjective. For a ny r E Y, the preimage T- I (r) is the torus Tn(o, r). For z E en, we set Pz := pn(o, T(z )) and Tz := Tn(O, T(z)) = T- I (T(z )) (see Figure 1 . 1) . Definition. A domai n G C en is called a Reinhardt domain if for every z E G the torus T z is also co ntai ned in G.

Figure 1.1. A

polydisk

in

absolute space

Rei nhardt domains G are characterized by their images in absolute space: T- I T( G) = G. Therefore, they can be visualized as domains in Y. For exam ple, both balls and polydisks arou nd the origi n are Rei nhardt domains.

8

I. Holomorphic Functions

Example

Let zo E en, with l z�0l l > l for v = l , .. . ,n . The n T(e;e .zo) = T(zo), but l e;e . z0 - z0 I = le;e - 11 . lzo I > l e;e - 11, and for suitable () this expression may be greater tha n E. So pn (zo, c ) is not a Rei nhardt domain.

Definition. Let G c en be a Rei nhardt domain. 1 . G is called proper if 0 E G. 2. G is called complete if V z E G n (C*t : Pz C G (see Figure 1 . 2).

Later o n we shall see that for any proper Rei nhardt domain G there is a smallest comp lete Rei nhardt domain G containing G. �

(a)

Figu r e 1.2.

(b)

(a) Complete and (b) noncomplete Rei nhardt domain

Exercises 1 . Show that there is a n ope n set B C e2 that is not co nnected but whose image T(B) is a domai n in absolute space. 2. Which of the following domains is Rei nhardt, proper Rei nhardt, complete Rei nhardt? (a) G1 := { z E e 2: 1 > lz1l > lz2l}, (b) G2 : = {z E e2 : lz1 l < l a nd lz2 l < 1 - l zii}, (c) G3 is a domain in e2 with the property z E G ===;. eit · z E G for t E JR. 3 . Let G c en be an arbitrary set. Show that G is a Rei nhardt domain 1 -¢:=? 3 Q C "f/ ope n and connected such that G = T- (Q). 4. A domai n G c en is called conwex, if for each pair of poi nts z,w E G the line segment from z to w is also co ntai ned in G. Show that a n arbitrary domain G is co nvex if and o nly if for every z E 8G there is a n affine linear fu nctio n >. :en --+ lR with .X (z) = 0 a nd >.ic < 0.

2.

2.

Power Series

9

Power S erie s

Polynomials . In order to simplify notatio n, we i ntroduce multi-i ndices. For v = (v1, ... ,vn) E zn and z E en define n

a nd The notatio n v > 0 (respectively v > 0) means that vi > 0 for each (respectively v > 0 and vi > 0 for at least one i).

�

A fu nction of the form z r-+ p (z) =

L

lv l<m

av z v , with

a,

E C for lvl < m,

is called a polynomial (of degree less than or equal to m ) . If there is a v with lvl = m a nd a, 1- 0, the n p (z) is said to have degree m . For the zero polynomial no degree is defined. An expression of the form a v zv with a, 1- 0 is called a monomial of degree m == lvl. A polynomial p (z) is called homogen eous of degree m if it consists only of monomials of degree m . 2. 1 Propositi on. and only if

PROOF :

A

polynomial p (z) 1- 0 of degree m i s homogeneous if

p (Az) = Am · p (z),

fo r all A E C.

Let p (z) = avz v be a monomial of degree m . The n

The same is true for finite sums of monomials. O n the other hand, let p (z) = 'I:I i
r-+

p (Az) =

N

Pi (z) . A' L i= O

and

A r-+ Am · p (z)

are equal. This is possible only if the coefficients are equal, i.e., Pm (z) =p (z) a nd Pi (z) = 0 for i 1-m . Sop = Pm is homogeneous. •

Convergence. If for every v E N3 a complex number c v is given, o ne can consider the series 'I:v >o c, and discuss the m atter of co nvergence. The trouble is that there is no cano nical order o n N3.

I.

10

Holomorphic Functions

Definition. The series :Z:::: v >o cv is called convergent if there is a bi jective map t.p : N --+ N3 such that "'£': 1\cO c, is convergent ! cv ! {L vEI

:

I

if

and only

C N� finite

if

}

is a bounded set. The proof is trivial. 2 . 3 Pro p osit io n . If the series :Z::: v�o Cv converges to the complex number c, then for each E > 0 there exists a finite set Io C N� such that: 1.

L: ! cv ! < E,fo r anyfin ite

2.

L c,

vEI

set K C N3 with K nio = 0 .

- c < E, fo r any fin ite set I with Io C I C N3

PROOF : We choose a bijective map t.p : N --+ N�. Then I:� 1 c 0 there exists an io E N o such that I:� ;0 jc
Then for any finite set I with Io C I c N�,

L Cv - C vEI

=

( L Cv - C ) + L vEio

Cv < vEio

vEI -Io

vEI-Io •

Example

Let q1, ..., qn be real numbers with 0 < q; < 1 for i = 1 , .. . , n, and q : = v q�n is a positive real number. (q1 , ... ,qn)· Then for any v E N3, q = qr1 .

•

.

If I c N� is a finite set, then there is a number N such that I c {0, 1 , . . . ,N}n, and therefore

2.

Power Series

11

Since the partial sums are bounded, the series is convergent. It is absolutely convergent in any order, and the limit is n L: qv = II

i=

v>O

l

1 . 1 -q 2

We call this series the generalized geometric series. Now let M c en be an arbitrary subset, and Uv : v E N()} a family of complex-valued functions on M. We denote by ll!viiM the supremum of l fvl on M. De fin ition . Th e series :Z::::v>o fv is called normally convergent on M if the series of positive real numbers :Z::::v>ollfviiM is convergent.

2.4 Prop ositi o n . Let the series :Z:::v>O fv be normally convergent o n M. Then it is convergent for any z E M, and for any bijective map t.p : N--+ N0 the series 2::� 1 f : o l fv (z) I is convergent for any z E M. But then 2::v>o fv(z) is also convergent, and there is a complex number f(z) such that I:� 1 f'P(i)(z) converges to f(z), for every bijective map t.p :N--+ N0. If an

E

> 0 is given, there is an io such that I:� io II !'P(i) liM < E. Then m

L f'P(i) (z) i=k

'!!!

<

L II!'P(i) liM < ·i=k

E,

for m > k > i0 and z E M.

Therefore. "'

L f
00

L

i=k+l

This proves the uniform convergence.

f'P(i) (z)

< E,

for k > i0. •

Power Series. Let { a, : v E N0} be a family of complex numbers, and Zo E en a point. Then the expression

is called a (formal) power series about z0. It is a series of polynomials. 1f this series converges normally on a set M to a complex function f, then as a uniform limit of continuous functions f is continuous on M.

12

I. Holomorphic Functions

2.5

Abel 's lemma. Let P' cc P C e n be polydisks around the origin.3 If the power series :Z::: v >o a v zv converges at some point d the distinguished bo undary of P, then it converges normally on P'. Let w E Bo P be a point where :Z::::v>O a v wv is convergent. Then there is a constant c such that lav wv I < c for all v E N0. PROOF:

We choose real numbers q; with 0 < q; < 1 such that lz;l < q;l w;l for any z = (z1, . . . , Zn ) E P' and i = 1 ... , n. It follows that ,

iav z v l <

q"

. c, for q = (q1, . . . qn ) ,

,

z

E P', and

v E N().

Then lla v zv iiP' < q" . c as well, and from the convergence of the generalized geometric series it follows that :Z::: v >O a v z v is normally convergent on P'. • Definition. We say that a power series :Z:::: v >o a v (z - zo) v converges compactly in a domain G if it converges normallyon every compact subset K c G. 2.6 Corollary. Let P c e n he a polydisk around the origin and w be a point of the distinguished bo undary of P. If the power series :Z::: v 7o a v zv converges at w, then it converges compactly on P. Let K C P be a compact set. Then there is a q with 0 < q < 1 such that K c q · P cc P. Therefore, the series is normally convergent on • K.

PROOF:

Let S (z) = :Z::: v >O av z v be a formal power series about the origin, and B :={Z E e n

:

S (z) convergent} .

2.7 Propositi on. The interior BO ts a complete Reinhardt domain, and S (z) converges compactly in B0• Let w be a point of B0• There is a polydisk p n (w ,c) C B0 and a point v E pn (w, c) n (C*r' such that w E Pv(O) . Then Tw C B , and if wE (C*) n , then also Pw(O) C B".

PROOF:

"

To see that B0 is a complete Reinhardt domain, it remains to show that it is connected. But this is very simple. Every point of B0 can be connected to a point in B0 n (C* ) n , and then within a suitable polydisk to the origin. 3

The notation U CC V means that U lies relatively compact in compact set which is contained in V.

V;

i.e., U is a

2. Power Series

13

From these considerations it follows that B0 is the union of relatively compact polydisks around the origin. Therefore, S ( z ) converges compactly o n B . • "

The set B0 is called the domain of convergence of S(z ). 2.8 Propositi on. Let G be the domain of convergence of the power series S ( z ) = :Z::: v >O av z v . Then

L av

v >O v1>o

·

VI

VJ· z1

V

J· · · z.

J

1

· · · z

V

n

n

also converges compactly on G.

PROOF: Let w be any point of (C*) n n G, and lavwv l < c for every v E N�. If 0 < q < 1 and z = q .w, then

Now,

is convergent. Therefore, SzJ ( z ) is convergent, and it follows that SzJ is normally convergent on Pz(O). Since every compact set K C G can be covered by finitely many polydisks of this kind, Sz. is compactly convergent on P. • J

Definition. Let B c en be an open set. A function f : B --+ C is called holomorphic if for every z0 E B there is a neighborhood U = U (z 0 ) c B and a power series S (z ) := :Z::::v>o a v (z - z0 t that converges on U to ! ( z). -

The set of holomorphic functions on B is denoted by 0( B).

It is immediately clear that every holomorphic function is continuous. Exercises 1 . Let j, g be two nonzero polynomials. Prove that

deg(f ·g)

=

deg(f) + deg(g) .

2. Let f = h · · .!k be a homogeneous nonzero polynomial. Show that fi is homogeneous, for i = 1 .. . , k. ,

4

I. Holomorphic Functions

3. Fi nd the domain of co nvergence for the following power series:

4.

Determine the limit and the domain of convergence of the series v z'' w . (2zt + F(z, w) = tL>O v >O

L )

L(

5. A polyradius r (r1, . . ., r,) E "f/ is called a radius cf convergence for the power series f(z) l:: v >O a v zv if f(z) is co nvergent i n P pn (O, r), but not co nvergent in any po lydisk P' pn (O, r') with P cc P' . =

=

=

=

Prove the following generalizatio n of the root test: r is a radius of convergence for f (z) if a nd o nly if lmnl 1'\/lavlr v

3.

=

1.

C om plex Differ entiable Functions

The C omp lex Gr adient Definit ion. Let B c en be an ope n set, Zo E B a poi nt. A fu nctio n f : B --+ e is called complex differentiable at zo if there exists a map A : B --+ en such that the following hold: 1 . � is co nti nuous at zo . 2. f(z) = f(zo) + (z - zo) . � (z) t for z E B .

Complex differe ntiability is a local property: For f to be complex differen tiable at z0 it is sufficie ntthat there is a small neighborhood U U(zo) C B such that the restrictio n flu is complex differentiable at zo . =

3 . 1 Proposit ion. Iff is complex differentiable at zo, then the val ue f unction A at zo is uniq uely determined.

cf

the

Assume that there are two maps �� and � 2 satisfying the condi tio ns of the defi nition. The n PROOF:

In particular, there is a n E > 0 such that the equ atio n holds for z zo +te, 1 , ... , n. 1f �.\ ( �i-'>, ... , ��>->) , the n with t E e, It! < E, and i =

=

t· (��'> (z) - ��2>(z ))

=

=

0 for It! ,

< E,

z = Zo +te, a nd i

=

1 , . ..,n .

Because the � �>-> are co ntinuous at z0 , it follows that ��(zo) = � 2 (zo ) .

•

3. Complex Differentiable Functions

D efi n i t i on . Let f : B exists a representatio n

---+

e be complex differentiable

at

15

z0. If there

f(z) = f(z o) + (z -zo) . �(z) t ,

with A continuous at z0, the n the uniquely determi ned numbers M (zo) Bzv

= fzv (zo) : = ev . � (zo) t

are called the partial derivatives o ff at z0. The vector

is called the complex gradient of f at z0. Rem ark s 1 . If f is complex differentiable at z0 , the n f is continuous there as well. 2 . A fu nctio n f is called complex differentiable in a n ope n set B if it is complex differentiable at each poi nt of B. The n the partial derivatives of f define fu nctio ns fzv o n B. 1f each of these partial derivatives is again complex differentiableat zo , the n f is called twice complex differentiable at z0, a nd o ne obtai ns second derivatives

[)2j OZvOZJL

(zo)

= fzvz, (zo) ·

By inductio n, partial derivatives of arbitrary order may be defined. 3. Sums, products, and quotients (with no nvanishi ng denominators) of com plex differe ntiable fu nctio ns are again complex differentiable.

Weakly Holomorphic Functions . Let B c en be an ope n set, z0 E B a poi nt, a nd f a complex-valued fu nction o n B. For w -=!= 0 let 'P w : e ---+ en be defined by 'Pw(()

:= z o + (w.

The n f o t.pw ( () is defined for sufficie ntly small (. Iff is complex differentiable at zo , the n we have a representatio n f(z) = j(zo ) + (z -zo) . � (z) t, with A co ntinuous at z0. It follows that

and f o 'P w is complex differe ntiable at ( = 0, with

16

I. Holomorphic Functions

This is the complex directional derivative of f at z0 in the directio n w . We de note it by Dwf (zo) · In particular, fzv (zo) = Dev f (zo) for v = 1 .. . , n. ,

A n arbitrary fu nctio n f is called partially diflerentiable at z0 if all partial derivatives Dev f ( zo ) exist for v = 1 . . . , n. ,

A fu nctio n f is called weakly holomorphic on B if it is continuous and partially differe ntiable o n B. The n for z = (z1 , . . . , zn ) E B and v = l , . . . , n the fu nctio ns ( r-+ f(zl, . . . , Zv- 1 , C, zv+ l 1 , zn ) •

•

•

are holomorphic functions of one variable. I ff is complex differentiable o n B, the n f is also weakly holomorphic o n B. Later o n we shall see that weakly holomorphic functions are always complex differe ntiable, in co ntrast to the behavior of real differentiable fu nctio ns.

H olomor phic Funct ion s 3 . 2 Prop o sit io n. Let P C en be a polydisk around the origin, and S(z) = :Z::: v >o av z v a power series tha t converges compactly on P to a function f . Th en f is complex differentiable a t 0 , with fz1 (0 )

= a1, 0 , . . , 0 , · . . , fzn (0) = ao, . . . ,O, l ·

We choose a small polydisk Pc: cc P arou nd the origin such that S(z) is normally co nvergent o n P,. But the n the series obtai ned by any rearrangement of the terms is also normally co nvergent, and it co nverges to the same limit. We write PROOF:

f (z)

L:> v zv

v>O a o , o . . . . , o + z1

+ Z2 ·

v1 > 0 vz , .. . ,vn > O

+

· · ·

+ Zn ·

vl = . . . = Vn-1=0 Vn > O

Since the series ll 1 ( z) , . . . ,ll n (z) converge normally o n Pc: to conti nuous fu nc • tio ns, f is complex differentiable at 0 , with fzv (O) = flv ( O ) . 3 . 3 Cor ollary. If B c en is an open set, and f : B function, then f is complex diflerentiable on B.

-+

e

a holomo rphic

4.

The Cauchy Integral

17

Let z0 E B be a n arbitrary poi nt. There is a power series S (w) co nverging compactly near 0 to a holomorphic fu nctio n g such that

PROOF:

f ( zo + w )

=

g( w)

=

g(O) + WI . �I (w) + ·..+ Wn .�n (w) ,

with conti nuous fu nctio ns �I , .. . , A,. It follows that f is complex differen• tiable at zo . Exercises 1 . Show that there is a fu nctio n f : e n --+ e that is complex differe ntiable at every poi nt z = ( zi , . . . , zn ) with Zn = but is nowhere holomorphic. 2. Prove the following chain rule: If G c en is a domain, J : G --+ e a complex differentiable fu nctio n, and t.p = ( 'P I , . . . ,t.p n ) : A --+ G a map with holomorphic compo nents t.p ; , the n f o t.p : A --+ e is a holomorphic fu nctio n, with (! o t.p ) ( () = V f ( t.p ( () ) . t.p' ( () t .

0,

'

4.

The C auch y I ntegr a l

The Integr al Formula. Let r = (ri , .. . , rn ) be a n element of iR+. , P = p n ( o, r ), T = Tn ( o, r ) , a nd f a co ntinuous fu nctio n o n T. The n kt (z , ()

:=

f( () ( ( z) (I, . . ,I) _

defines a co nti nuous functio n kf : P x T --+ C. Defi nit ion.

(2:i) J kt (z , () d( J n

T

((n - Z n )

is called the Cauchy integral o ff over T. Obviously, Cf is a co nti nuous fu nctio n on P. 4 . 1 Theorem (C a u chy integral formula). Let P, T be as above, and U = U(P) be an open neighborhood cf the closure of P. Iff is weakly holomorphic on U, then Ctlr (z) = f(z ) fo r an y z E P.

Drn (0), we may assume that U = UI x · . . x Un , with ope n neighborhoods U; = U; (Dr, ( 0)) for i = 1 , .. . PROOF:

If P

=

0

(O ) x . . . x

,

, n.

18

I.

Holomorphic Functions

Since f is weakly holomorphic, we can fix z ' = ( z1 , . . . , Zn - d E U1 x · . · x Un -1 and apply the Cauchy i ntegral formula in o ne variable to (n r-+ f(z', (n ) · For Zn E Drn ( 0) it follows that f(z 1 , Zn )

=

1 21r i

J \(n\= rn

f (z' , (n ) d(n (n - Zn

Similarly, for the penultimate variable Zn - 1 and z" = (zl , . . . , Zn - 2 ) E ul . . . x Un - 2 we obtain f(z11 , Zn - l , Zn )

�

=

2 i

and after n steps, f(z)

j

=

f(z" , (n- 1 , Zn ) dl"n - 1 (n - 1 - Zn - 1 '>

\(n- l l =r n-1

f(z " , (n - 1 , (n ) dl" ":,n - 1 , ":,n dl" ) ((n - 1 - Zn - I ) ( (n - Zn

J

J

X

CJ I T(z) , for z E P.

•

4.2 Theorem (power series exp ansion). Let P = pn (o, r) c en be a polydisk and T its distinguished boundary. Iff : T -+ C is a continuous function, then there is a power series l:: v 7 o av z v that converges to Ct (z) in all of P. Th e coefficients a, of this series are given by

Q,V1 · · · Vn

PROOF:

=

( -1 ) n j 27r l

"

J((l , · · · , (n ) dl"l · · · d�" + v1 + V '> n · l l '> n . . · (n I" T '> 1

Setti ng 1 := ( 1 , .. . , 1 ) E N�, for z E P a nd (

1

( ( -z) l

1 ( (I - zl) · · · ((n - Zn )

2:

v, = O

•

T it follows that

1 Zl · · · · 1 (n (I (I

( �� ) VI l: ( �: ) Vn •

E

(

) · · · ( 1 - Z(n ) n

•

If r = (r 1 , . . . , rn ) , the n for fixed z E P and arbitrary ( E T we have

Zj (j

-

=

Since T is compact and l f(() l

<M

z - l jl qj ·.rj

--

f

.1 , . . . , n. or J f 1 , <

continuous o n T, there is a co nstant M with o n T. The n L: v 7 o (J(C)/C v+ l ) z v is domi nated o n T by the

4.

The Cauchy Integral

19

convergent series ( M/r1 n =v >O q, , where q = ( ql ' . . . , qn )' and therefore it is normally convergent as a series of functions on T with limit f ( ( ) j ( ( -z) 1 . We can interchange summation and integration:

( 1 )n r Ct (z) = 1r

"' v, ! (() = z av � l Jr ( ( z) d( v >O

2 i

with av .

·

_

- ( 27r1 i )n lrr ! (() d(.

The series converges for each

z E

c+ l

P.

•

4.3

Osgood's theorem. Let B c en be an open set. Th e following state ments about a function f : B ---+ e are equivalent:

f is holomorphic. 2. f is complex differentiable. 3. f is weakly holomorphic. 1.

We already know that a holomorphic function f is complex differ entiable, and it is trivial that then f is weakly holomorphic. PROOF:

On the other hand, let f : B ---+ e be weakly holomorphic, and z0 E B an arbitrary point. There is a small polydisk P around z0 that is relatively compact in B . If T is its distinguished boundary, then J I P = Cf/ T, and the Cauchy integral is the limit of a power series. So f is holomorphic. •

In addition, if f is weakly holomorphic on B , z 0 E B a point, and P CC B a polydisk around z o , then there is a power series S(z) = :Z::: v > o av (z - z0) v that converges to f on all of P.

-

Holomorphy of the Derivatives

4.4 Weierstrass ' s convergence theorem. Let G c en be a domain, and (ik ) a sequence of holomorphicfunctions on G that converges unzformly to a fu nction f . Th en f is holomorphic. The limit function is continuous. Let zo E G be a point, P cc G a polydisk around z0, and T its distinguished boundary. Then PROOF:

Since T is compact, we can interchange integral and limit. Thus, for any fixed z E P, lim Cfk/T (z) C lim fk / T (z) Cf/T (z) .

k--+CXJ

=

k -+ =

=

20

I . Holomorphic Functions

Since f is continuous on T, the Cauchy integral Cf iT has a power senes • expansion in P . Therefore, f is holomorphic at z0 .

4 . 5 Prop o sition . Let S(z) = 2:: v >o av z v be a power series and G its do m a in of convergence. Then the limitfunction f of S(z) is holomorphic on G, and the formal derivative Szi (z)

=

L av · Vj Z�1

v >O v1> o

v· -1 V ZJ J . . znn •

converges to fz1 In particular, all partial derzvatzves of f are likewise holo morphzc. •

Since S(z) converges compactly on G, f is locally the uniform limit of a sequence of polynomials. Then it follows from Weierstrass' theorem that f is holomorphic. But also S'z (z) converges compactly on G, and its limit function g must be holomorphic on G. PROOF:

.1

Now let z0 be an arbitrary point of G. Since G is a complete Reinhardt domain, there is a polydisk P around the origin with z0 E P C C G. We define f* (z)

=

r] 9j (Zl , · . . , Zj-I , (, Zj+l , · . . , zn ) d( + f( zl , · . . , 0, . . . , z n)· Jo

For the path of integration we take the connecting segment between 0 and Zj in the zFpl ane. Then f* is defined on P .

Let S(z) = I:� 0 p; (z ) be the expansion into a series of homogeneous polyno mials. Then Bz1 ( z) = I:� 0 (p;)zj (z) , and this series converges uniformly on the compact path of integration we used above. Therefore, we can interchange summation and integration, and consequently,

00

L p; (z)

=

f(z) ,

i =O

for z E P. Hence fz1 (zo) = J;j (zo)

=

9j (zo) .

•

4 . 6 Cor ollary. Let G c en be a dom a in and f : B --+ C a holomorphic function. Th en f is infinitely complex differentiable in G. Let v = ( v1 , . . . , vn) be a multi-index. Then we use the following abbrevia tions:

4 . The Cauchy Integral

21

1 . VI. '. = VI I. . . · Vn .I .

2 . If f is sufficiently often complex differentiable at z 0 , then 81 v l j v D f(zo) : = 8 v1 8 vn ( z o). · · Zn z1 ·

4. 7 Iden tity theorem (for power series). Let f(z) = :Z::: v >O avz v and g (z) = l: v >O bv z v be two convergent power series in a neighborhood U = U(O) c en . Ifthere is a neigborhood V(O) c U with f l v = 9 l v, then a, = b, fo r all v . PROOF: We know that f and g are holomorphic. Then D v f (0) = D v g ( O ) for all v , and successive differentiation gives Dv f ( O )= v! . av and Dv g ( O ) = v! . b,. •

4 . 8 Cor ollary. Let G c e n be a domain, Zo E G a point, and f : B --+ e a holomorphic fu nction. If f ( z) = :Z::: v >o av (z - zo) v is the (uniquely deter m ined) power series expansion near z0 -E G, then a, =

� . Dv f(zo ) , fo r each

v.

v

E N�

4 . 9 Cor ollary (C auchy ' s in equ alities). Let G c e n be a doma in, f : G --+ e holomorphic, z0 E G a point, and P = p n (z 0 , r ) C C G a polydisk with distinguished boundary T. Th en

Let f (z) = :Z::: v >O a v (z - z0 ) v be the power series expansion of f at z 0 . Then Dvf(zo) = v!av and 1 n !( ( ) r d av Jr ( ( zo) v+l ( , = 27r i and therefore PROOF:

( )

I Dv f(zo )l

-

< v'

(27r ) n v'·

J [0 ,27r] n

' f(z1< o) + r 1 c i t t ' . . . ' zn + rn ci t n )l rV

dt 1 · · · dtn

< r_v · sup ! fl . T •

22

I.

Holomorphic Functions

The Identity Theor em . Let G c e n be always a domain. The con nectedness of G will be decisive in the following. 4. 1 0 Identity theorem (for h olomorphic fu nctions).

Let !I, h be two holomorphic functions on G. If there subset U C G with !1 l u = h l u , then !I = h · PROOF:

is

a nonemp ty open

We consider f := !I - h and the set N := {z E G : D" f ( z ) = 0 for all v}.

Then N 1- 0, since U c N. Let zo E G be an arbitrary point, and f(z)

=

"""' _!_ Dv f(zo ) (z - zot � v! v >O

the power series expansion of f in a neighborhood V = V(z0) C G. 1f zo belongs to N , then flv 0, and also V C N . This shows that N is open. Because all derivatives D" f are continuous, N is closed. Since G is a domain, • we get N = G and !I = h -

Remark. In contrast to the theory of one complex variable, it is not suf ficient that !I and h coincide on a set M that has a cluster point in G. Consider for example, G = e2 and M = { ( z1 , z2) : z2 = 0). The holomor phic functions !I ( z1 , z2 ) := z2 ( z1 - z2 ) and h ( z1 , z2 ) := z2 ( z1 + 2 2 ) are equal on M, but fi (O, 1) = - 1 and /2 (0, 1) = 1 . 4. 1 1 Theorem (maximum p r in ciple).

Let f : G ---+ e be a holomorphic function. If there is a point zo E G such that IJ I has a local m aximum at zo , then f is constant. We consider the map t.p w : e ---+ en with 'Pw(( ) = zo + (w, for an arbitrary w 1- 0. Then f o 'Pw is a holomorphic function of one complex variable, defined near ( = 0. Now, since I f o 'P w l has a local maximum at the origin, this function must be constant in a neighborhood of the origin. But the direction w was chosen arbitrarily, so f also has to be constant in a neighborhood of 0 E en . The identity theorem implies that f is constant on • G. PROOF:

Exer cises 1 . Prove Liouville's theorem : Every bounded holomorphic function on en is constant. 2. Prove that if f E O(en ) and l f(z) l < C · llz v ll for some C > 0 and some v ENS , then f is a polynomial of degree at most l v l .

5.

The Hartogs Figure

23

3 . Let G c en be a domain and f E O( G) not constant. Prove that then !( U) C
5.

The

Hartogs Figure

Exp an sion in Reinhar dt Domain s. Let r� , r� be real numbers with 0 < r� < r� for 1 < v < n. We define p

Q

:

-

{z E en : l zv l < r� for all v } , {z E en : r� < lzv l < r� for all v }

Clearly, P and Q are Reinhardt domains. Let f be a holomorphic function in Q. Then for all r E T(Q), the Cauchy integral cf iTr is a holomorphic function in P, and therefore a fortiori in P (see Figure 1 .3 ) .

------� � p

Figu r e 1 . 3 .

We have

•

•

r

' r1

Expansion in

5 . 1 Prop o sitio n . independent of r . PROOF :

I I I

the

I I I

polydisk

Th e fun ction fr : P

---+

e given by fr(z)

:=

Cfi Tr (z) is

24

I. Holomorphic Functions

fr ( z )

=

( 2�i

In each variable (v the integrand !( C,)j ( (v -Zv ) is holomorphic on the annulus { (v : r� < ((v I < r� }. From the Cauchy integral formula for one variable it follows that if r� < rv < r� < r�. This yields the proposition.

•

5.2

Propo sition . Let G c c_ n be a proper Reinhardt domain, f holomor phic on G . Th en for every z E G n ( C*)" the Cauchy integral Cf iT. coincides with f in a neighborhood of the o rigin. PROOF: G n ( C*)n is a Reinhardt domain. Therefore, G0 := T ( G n (C* ) n ) is a domain in the absolute space.

Let B : = { r E Go : CfiTr coincides with f in the vicinity of 0 } . Then B 10, because there is a small r E Go such that Pr (O) c G.

B is open: 1f ro E B , we can find sets P, Q as we did at the beginning of this section such that r0 E Q C Ga. Then for r E Q, fr = Cfi Tr is a holomorphic function on P, and independent of r. But fro coincides with f near the origin. Therefore, Q C B .

Also, Go B is open. The proof goes as above. Since Go is connected, that a implies that B = G0 . -

5.3

Cor ollary. Let G be a proper Reinhardt dom a in, f holomorphic in G. Th en there is a power series S(z) which converges in G to f.

PROOF: Let z0 E G be arbitrarily chosen. Then there is a point w E G n (C* ) n with Zo E Pw· The holomorphic function g := cfiTw has a power series expansion g(z) = � v >O av z v in Pw· Since g coincides with f in a small neighborhood of the origin, the coefficients a , are those of the Taylor series of f about 0. Since z0 was arbitrary, it follows that the series converges in • all of G. By the identity theorem its limit is equal to f . Definitio n .

If G is a proper Reinhardt domain, then

6:= U

zEGn(IC*)"

is called the complete hull of G.

P,

25

5. The Hartogs Figure

Remark s 1 . Every complete Reinhardt domain is proper, but the opposite is in general false. For n = 1 , Reinhardt domains are open disks around 0, and there is no difference between proper and complete domains. 2. The complete hull G of a proper Reinhardt domain G is again a domain containing G. And it is Reinhardt: For z E G there is some z1 with z E P, C 6. But then also Tz C Pz1 C G. The same argument shows that G is complete. 3. Let G1 be another complete Reinhardt domain with G C G 1 . For z E G n (C* )n, z also lies in G 1 , and by the completeness of G1 it follows that P, C G1 . So G C G�, and we see that G is the smallest complete Reinhardt domain containing G. �

�

�

�

An immediate consequence is the following:

5.4 Theor em. Let G be a proper Reinhardt dom a in and f be holomorphic in G. Then there is exactly one holomorphicfunction f i n G w ith Jlc = f .

Har togs Figures. In the case n = 1 the situation above cannot appear. For n > 2 we can choose sets G and G in en such that G 1- 6. This reflects an essential difference between the theories of one and several complex variables. Now let n > 2, P" the unit polydisk, q1 , . . . , q real numbers with 0 < qv < 1 n for v = 1 . , n , and ,

H

=

_

H

( q) : = { z

E

pn

:

I z1 I > ql or I z I < ql' for p, I'

=

2 , . . , n} . .

Then ( P n , H) is called a Euclidean Hartogs figure (see Figure 1 .4). H is a proper Reinhardt domain and P" its complete hull.

(a)

( b)

i z3 [ . . .

92 r-------1

q]

93

H

.

. . .

I I

q2 Y_ /

: lz2l

/

/

1i . .

I I I I I I

/

l z1 l

Figure 1 . 4 . (a) 2-dimensional, and (b) 3-dimensional Hartogs figure

'

- .._

I

26

I. Holomorphic Functions

5 . 5 Har t ogs' t h eor em. Let (P", H) be a Euclidean Hartogs figure. Th en any holomorphicfunction f on H has a holomorphic extension f on P". �

The theorem follows immediately from our considerations above. Exer cises 1. For 0 < q < l let G1 , G2 C e2 be defined by

G1 G2

- { ( z , w ) : q < l z l < l and lw l < 1 } , .- { (z , w) : l z l < l and lwl < q} .

:

( a) Prove that every holomorphic function f on G1 has a unique repre sentation J (z , w )

00

=

'2::

n =�CXJ

an (w)z n , with a,

E O( D ) .

(b) Prove that every holomorphic function g on G2 has a unique repre sentation 00

g (z, w ) =

'2:: bn (w) z n

n=O

,

with bn E O(Dq ( O)) .

(c) Use ( a) and (b) to prove that every holomorphic function f on G1 UG 2 has a unique holomorphic extension to the unit polydisk. 2. Let G c en be an arbitrary Reinhardt domain, f E O(G) . Show that there exists a uniquely determined "Laurent series" I:: vEzn av zv converg ing compactly in G to f .

6.

The C auch y-Riemann E qu ation s

Real Differ entiable Fun ction s. Recall the following from real an alysis:

Let B c en be an open set and z a point of B. A function f : B -+ lR is called o differentiable ( in the real sense) if there is a real linear form L : en -+ lR and a real-valued function r with: 1 . f (z) = f(zo ) + L(z - zo ) + r (z - zo). . r(w) = O. 2· l�o llwll

The real linear form Df ( z 0 ) It can be given in the form

:= L is

called the (total) derivative of f at zo

+ iv >--+ u . \1 f ( zo) t + v . V'y f (z o ) t , with 'Vxf(zo) = (fx1 (zo ) , . . · , fxn (zo)) and 'Vy/(zo) = ( /y1 (zo) , . · . , !Yn (zo) ) . We call (Y'xf(zo), Y'y/(z0 )) the real gradient of f at zo . L

:

u

x

If f = g + ih : B -+ C is a complex-valued function, then f is called differen tiable ( i n the real sense), if g and h are differentiable. The (real) derivative of f at zo is defined to be the complex-valued real linear form Df(zo) : = Dg (zo) + i Dh (z0 ) .

6 . 1 Pr op ositio n . A func tion f : B -+ C is (real) differentiable a t zo only if there are maps A ' , 6." : B -+ en such that:

=

A' and A" are continuous at z0 . 2. f (z) f(zo) + c z - zo ) . D..' (z) t + (z - zo ) . D.. " (z) t for The values 6.' (z 0 ) and D.." (zo) are uniquely determined. 1.

z

if

and

E B.

PROOF: ( 1 ) Let f be differentiable at z0. Then there is a complex linear form A' and a complex antilinear form A" such that Df(zo)

= ' + A" . A

The decomposition is uniquely determined, and there are vectors D..' (zo) and 6." ( z0 ) such that A' (w) = w . D..' (z0) t

and

A" (w) = w · D.." (z0 ) t .

Now we define 6.' (z)

.-

D.." (z)

. _

r(z - zo) . 6. , (zo) + (z - zo) , 2 2ll z - zo ll r (z - zo) . 6." ( z o ) + (z - z o) · 2 ll z - z o ll2 _

_

It is easy to see that

(a:A' and 6." are continuous at z0 , ( bY(z) = f(zo) +

(Z

- z o ) . D.. ' (z) t + (z - z 0 ) . D.. " (z) t

( 2 ) Now let the decomposition be given, and define L(w) r(w) Since

· - w · D..' (z o) t + w · D.." (zo) t , := w . (D.. ' (z) - D..' (z o)) t + w · (D.. " (z) - D..'' (zo)) t .

� ��il < II D..' (z) - D..' (zo ) ll + II D.." (z) - D.." (zo ) ll ,

it follows that f is differentiable at

z0

with derivative L.

•

I. Holomorphic Functions

28

Wir tin ger ' s Calculus Definition . Let f : B representation

--+

e be real differentiable at zo . If we have a

f(z) = f(zo) + (z - zo) . �' (z) t + (z - zo) . �" (z) t , with �' and � " continuous at z0, then the uniquely determined numbers

and

of

Ozv

(zo) = fzv (zo) := ev - �"(zo) t

are called the Wi rtinger derivatives of f at zo. The complex linear (respectively antilinear) forms (8f)z o : en (8 f)zo : en --+ e are defined by

n

and

(8f)z 0 (w) := 2: fzv (zo) w..,

v= l

(8f)z0 (w)

:=

--+

e and

n

2: hd zo)w.., ,

V=l

and the differential of f at zo by (df)zo := (8f)z0 + (8f)zo · Obviously, Df(zo) = ( df), .

If we introduce the holomorphic (respectively antiholomorphic) gradient '\lf : = (fz 1 , · · · , fzn )

and

Vf : = (Jzl , . . . , fzJ ,

then (8f)z0 (w) = W . '\lf(zo ) t and (8f)z0 (w) = W · 'Ilf(zo) t. 6.2 Pr op ositio n . Let f be a (c omplex-valued) func tion that is real differ· entiable at zo . Th en fzv (zo )

_

fzv (z o ) PRooF:

and

Let be

L : = Df (zo)-

1

; uxv (zo) - i fyv (zo ) ) , 1

; uxv (zo) + i fyv (zo) ) .

Then

6 . The C auchy-Riemann Equations

29

Putting things together we obtain

•

Rema rk. Use these formulas with care ! The derivatives fxv and fY v in general are complex-valued. So the equations do not give the decomposition of fzv and fzv into real and imaginary parts, respectively!

The C auchy-Riemann Equation s 6 . 3 Theor em. Let f : B ---+ C be a continuously real d�fferen tiablefunction. Th en f is holomorphic if and only if f:iv(z) - 0 on B , fo r v = 1 , .. . , n. PROOF: (a) If f is holomorphic, then f is complex differentiable at every point z0 E B . Comparing the two decompositions f(z) = f(zo) and

+

(z

_

z o ) . � (z) t

f(z) = f (zo) + (z - zo) - � ' (z) t + (z - zo) - �" (z) t

we see that �'(zo) = � (z0) and �"(zo ) = 0. The latter equation means that hd zo) 0 for v = 1 , . . . , n. (b) If fzv (z) - 0, then f is holomorphic in each variable and is consequently • holomorphic.

Remark. (8f)z 0 = 0

Now the following is clear: If f is holomorphic near z 0 , then and

Df(zo) (w) = (df)z0 (w) = (8f)z0 (w) =

n

L fzv (zo)wv .

v =l

The equation (af)z = 0 is the shortest version of the Cauchy - Riem ann dif feren tial equations. In greater detail, these are the equations hd z)

0, for v = 1 , .. . , n .

Finally, if f = g + ih, then we can write the Cauchy-Riemann equations in their classical form:

30

I . Holomorphic Functions

Exer cises 1 . Derive the Cauchy-Riemann equations in their classical form. 2 . Let f : G -+ e be real differentiable. Prove the formulas (f ) = ( f )Zv

Zv

and

fz.:z .. = fz.. z.. for v, = 1 , . . . , n . •

r

r

v

f"" "

3 . Let G c e n be a domain and h , . . . , Jk : G -+ e holomorphic functions. Show that if 2::::�= 1 fJ fj is constant, then all fJ are constant. ah 2 a2 1 h l 2 . . . Hmt: If h IS holomorphlC, then a = - ,:::. ZjUZi UZj

Holom or p hic M ap s

7.

The J a cob ia n . Let B c e n be an open set. A map f = ( fl , . . . , fm ) : B -t e m is called holomorphic (respectively real differentiable) if all components f; are holomorphic (respectively real differentiable).

7 . 1 Proposition . The map f : B -+ em is holomorphic if and only if fo r any z 0 E B there exists a m ap A : B -+ Mm , n (C) with the following properties: 1 . A is continuous at zo . 2. f(z) = f (zo) + (z - zo) . 6. (z) t , fo r z E B . The value 6.(z 0 ) is uniquely defined. PROOF:

The map f is holomorphic if there are decompositions

with A, continuous at z 0, for

p, = 1

,

.. . , m.

Then A is given by 6. (z) t = (6. I (z) t , . . 6. m (z) t ) . We leave the further details to the reader. • ,

,

Definition . If f : B -+ em is holomorphic, then Jr (zo) := 6. (zo) is called the complex Jacobian (ma trix) of f at zo . The associated linear map f' (zo ) : <en -+ em is called the (complex) derivative off at zo. It is given by f' (zo) (w) = W . Jr (zo) t .

7. Holomorphic Maps

Explicitly,

we

31

have

Jr (z ) = This matrix is also defined for differentiable maps. Definition . If f = g + ih : B ---+ e m is a differentiablemap, then the real Jacobian m a trix JIR ,r (zo) E M2 m , 2n (IR) is the real matrix associated to the real linear map (Dg(z o ), Dh(z0 )) : e n = JR2 n ---+ JR 2m .

The real Jacobian of f = g + ih is given by

JIR. f = ( h m ) x , (hm ) y 1 · ' ' (hm ) y, ( hm ) xl The EX-linear map Df (z) : en ---+ en is defined by Df ( z) := Dg(z) + i Dh (z) .

Setting (8f )z : = ((afr )z , . . . , (8fm )z) and (8f )z := ((afr )z , . . . , (8fm ) z ) , we obtain

Df (z) = ( a f) , + (8f)z ·

7 . 2 Theor em . A dzfferentiable m ap f = g + ih : B ---+ em is holomorphic zj and only iJDf(z ) is C- linear for every z E B . If f is holomorphic and n = m , then det (JIR ,r(z)) = ldet Jr (z) i 2 . The map f is holomorphic if and only if (8f ) z = 0 for every z . Then Df (z ) = (8f )z , which is complex linear. In this case we have the Cauchy Riemann equations PROOF:

and therefore

If n =

m,

A + i C.

then JIR, f =

(�

B D

) w1th. B = -C and

A

= D, and Jr =

32

I . Holomorphic Functions

By elementary transformations,

( A +C i C ( A + iC det C

d

-

et

l det ( A + ;cw . •

It follows that holomorphic map s are orientation preserving !

Chain Rules. Let B c e n be an open set, f : B ---+ e m a differentiable map, and g a complex-valued differentiable function that is defined on the image <.f f . Then g o f : B ---+ e is differentiable, and the following holds:

7 . 3 P r op o sition (comp lex chain r u le). m

m

(g 0 f) zv

L )9w/L

0

f) · ( JJ" ) Zv +

L (9w/L

0

f) ·

(fi"Lv '

0

f) .

(fI")Zv

m

m

L (9w/L

1"= 1

0

f) · ( f/" )zv +

L (9w/L

M= l

One can use the well-known proof for the chain rule in real analysis, consid ering Zv and Zv as independent variables.

7 . 4 Cor ollary. I ff and g are holomorphic, then (g o f)zv ( z ) (g o f)zv (z)

_

0

m

(i. e. , g o f is holomorphic),

L 9wJ f(z)) . (fl" )zv (z ) .

J"= l

The second equation can be abbreviated as V' (g o f) ( z )

=

V'g (f( z) ) . Jr (z)

Tangent Vector s. In this paragraph we use the term differentiable for infin itely differentiable. Defin ition. A tangent vector at a point z E en is a pair t (z, w) , where the direction w <.f t is an arbitrary vector of en . If the base point z is fixed, we simply write w instead of t or (z , w ) . =

The set Tz <.f all tangent vectors at at z .

z

is called the tangent space (of C ) "

7. Holomorphic Maps

33

The notation "tangent vector" is motivated by the following:

Let B C en be an open set and I C lR an interval containing 0 as an interior point. If a (a1 , . . . , an ) : I --+ B is a differentiable path, then a' (0) is the direction <.f the tangent to the curve a at the point a (O) . Therefore, =

a (O)

:=

(a(O), a'(O) )

is called the tangent vector of a at z a(O) . Each tangent vector (z,w) E Tz can be written in the form a(O ), e.g. , a (t) : = z + t w . =

The tangent space Tz carries in a natural way the structure <.f a complex vector space: (z, w1) + (z, w2) A · (z,w)

(z , WI + w2), :- (z,A · w ) , for >. E e .

Every tangent vector t (z, w) operates linearly on the algebra C(B) <.f differentiable functions on B by =

t[j]

= Df(z) (

w .

)

This is the directional derivative, also denoted by Dw f(z). If t a(O ) , for some differentiable path a, then t[f] = ( f o a ) ' (0) , due to the chain rule. =

The operator t : C(B) --+ lR satisfies the product rule:

t[f · g] = t[f] · g ( z ) + f (z) · t[g] .

In general, a linear operator satisfying the product rule is called a derivation. In Chapter IV we will show that the tangent space is isomorphic to the vector space <.f derivation s.

The Inver se M app in g. Let B 1 , B2 c en be open sets, and f : B1 --+ B2 a holomorphic map. Definiti o n . The map f is called biholomorphic (or an invertible holo morphic m ap ) i ff is bijective and r - 1 holomorphic.

7.5 Inverse mapping theorem. Consider a point zo E B 1 and its image Wo = f( zo) . Th en the following are equivalent:

There are open neighborhoods U = U(zo) C B 1 and V = V(w0) C B2 such that f : U --+ V is biholomorphic. 2. det Jr (zo) 1- 0 . 1.

I. Holomorphic Functions

34

PROOF: 1f flu : U ---+ V is biholomorphic, then (flu )- 1 o f = i d u and 1 det ( En ) det ( J(fl u ) -1 (wo) . Jr (z0 )) = det (J(fl u)-1 (wo)) . det ( Jr (zo )), and therefore det ( Jr (zo)) f= 0. 1f det (Jr (zo )) 0, then also det (JIR,r (zo ) l det Jr (zo)l 2 0. It follows from =

=

f= f= real analysis that there are open neighborhoods U = U(zo) C B1 and V =

=

V (w0 ) c B2 such that flu : U ---+ V is bijective and g : = (fl u ) - 1 : V ---+ U

a continuously differentiable map (in the real sense). Then f o g idv is a holomorphic map, and if � write f = (h ' . . . , !n ) and g (9 1 , . . . , 9n ) , then =

=

n

o

= Uv o g) w'" = 2) Uv ) X=l

o g) . (9 >-.)w'" , for v, M = 1 , . . . , n.

z-'

In the language of matrices this means that

0 = Jr

\79 1 ·

Since Jr is invertible, it follows that \79>-. = 0 for each g is holomorphic.

.X.

Therefore, the map

•

7 . 6 Implicit fu nction theorem. Let B c e n X em be a n open set, f = ( h , . . . . fm ) : B ---+ e m a holomorphic mapp ing, and (zo , wo) E B a point with f(zo, wo) = 0 and

f det ( o az: (zo , wo)

p, = 1, . . . , m v = n + 1, . . . , n + m

) f= O.

Th e n there is a n open neighborhood U = U' x U" C B and a holomorphic m ap g : U' ---+ U" such that

{ (z, w) E U' x U"

:

f(z, w) = 0 } = { (z, g(z))

: z

E U' } .

PROOF: We write Jr (z o, wo ) = ( J' I J" ) , with J' E Mm ,n (C) and J" E Mn (C), and define F : B ---+ e n x e m by F (z, w ) : = (z,f(z, w ) ) . Then det Jp (zo , wo) = det

( �� J,, ) f= 0.

Therefore, there are open neighborhoods U = U(zo , wo) C B and V V(z0 , 0) C e n+ m such that F l u : U ---+ V is biholomorphic. Obviously, y - 1 (u, v) (u, h(u, v )) . We may assume that u U' X U" c e n X em and V = U' x W, with some open neighborhood W W(O) C e m . Defining g : U' ---+ U" by g (z) := h(z , O) , it follows that =

=

=

=

7. Holomorphic M aps

f(z, w)

=0

� � �

35

F(z, w) = (z, 0) (z, W ) = F - 1 (z, 0 ) w = h(z, 0) = g( z ) .

This completes the proof.

•

Rem ark. We can exchange the coordinates in the theorem. If rk Jr (zo, w0 ) = m, then there are coordinates zi, , . . . ,zi n such that r - 1 (0) is the graph <.f a map g = g(zi, . . . , zi n ) near (zo , wo) . Exercises G P" C e 2 be the unit polydisk and f (!1 , fz) : G --+ G a holomorphic map with f(O) = 0. ( a ) Show that if f( z ) = z+l::: n > 2 Pn (z ) with pairs Pn ( z ) = (P�n ) ( z ) , p�n ) (z) ) z. Hint: Use <.f homogeneous polynomials <.f degree n , then f( z ) Cauchy's inequalities and consider the iterated maps fk = f o . . . o f (k times). (b) Show that if f is biholomorphic, then fr , fz are linear. Let G1 , G2 c e n be two domains. A continuous map f : G 1 --+ G2 is called proper if for every compact subset K c G2 the preimage f- 1 (K ) is a compact subset <.f G1 . (a) Show that every biholomorphic map is proper. Give an example of a proper holomorphic map that is not biholomorphic. (b) Let G1 and G 2 be bounded. Show that a continuous map f : G1 --+ G2 is proper if and only if for every sequence (z k ) in G 1 tending to 8G1 , the sequence (f(z k )) tends to 8G2 • (c) Let G' , G" C e be bounded domains and f : G' x G" --+ G2 a proper holomorphic map onto a bounded domain G2 C e2 • Show that z >-+ fw (z, w) cannot vanish identically on G'. Let z0 E 8G' be an arbitrary point and ( Zk ) a sequence in G' tending to z0. Show that the sequence of holomorphic maps 'P k : G" --+ G2 with t.p k (w) := f(zk , w) has a subsequence converging compactly on G" to a holomorphic map t.po : G" --+ e2 with t.po (G") C 8G2 . Show that there must exist at least one point z0 E 8G' such that the corresponding map t.po is not constant. Use the results of the last exercise to prove that there is no proper map ping from the unit polydisk to the unit ball in e2 • Let G1 c en and G2 c em be domains and f : G 1 --+ G2 a biholomorphic map . Show that m = n. Let G c en be a domain and D : G"(G) --+ � a derivation, i.e. , an �-linear map satisfying the product rule at zo E G . Show that D [f] depends only on f l u , U an arbitrary small neighborhood <.f z0 .

1. Let

2.

3. 4. 5.

=

=

36

I. Holomorphic Functions

6. Let G c en be a domain, f = (!1 , Jm ) : G ---+ e m a holomorphic mapping, and M : { (z, w ) E G x em : w f (z) }. Prove the following: If g : G X em ---+ e is a holomorphic function with 9 IM = 0, then for every point (z 0, wo ) E M there is a neighborhood U and an m-tuple of holomorphic functions in U such that ( a1 , , cy •

•

.

=

.

•

•

m

g (z , w )

8.

=

=

L:>IL (z, w )

)1= 1

·

(w!L - f!L (z) ) for (z , w ) E U.

An alytic Set s

An alytic S ub sety - Let B c en be an arbitrary region. If U c B is an open subset, and 1 ' . . . , fq are holomorphic functions on U, then their common zero set is denoted by N( JI , .

.

.

, q)

f = {z E U : fi (z) = . . . = Jq (z) = 0 } .

Defin it ion . A subset A C B is called analytic if for every point zo E B there exists an open neighborhood U = U (zo) C B and holomorphic functions h , . . . , fq on U such that U n A = N (h , . , fq ) · .

.

If z0 is a point of B - A then we can choose an open neighborhood U = U ( zo) and holomorphic functions h ' , fq on U such that ,

.

:=

zo E U'

.

.

U - N ( h , . . . , Jq ) c U c

B.

Since the zero set N (!1 , , fq ) is closed in U, it follows that B - A is open and A closed in B . Therefore, an analytic set in B could have been defined as a closed subset A C B such that for any z0 E A there exists a neighborhood , fq E O(U) with A n U = N ( h U and functions II , . . , Jq ) · .

.

.

.

.

.

.

.

Example In general, analytic sets cannot be given by global equations. We consider the domain G :=G1 U G 2 with : l z1 l < .

1

2

and l z2 l <

: l z1 ! < l and

· -

1

2

1},

< lz2 ! <

1 }.

For the analytic set we take A = { ( z1 , z2) E G2 : z1 = z2} (see Figure =

The sets G1 , G2 give an open covering of G with A n G1 = { (z1 , z2 ) : z1 - z2 0 } So A is an analytic subset of G. =

.

o

1.5).

and A n G2 =

8. Analytic Sets

37

A

Figur e 1 . 5 .

A

not

globally defmed analytic set

Iff is a holomorphic function in G that vanishes on A, then f can be analyt ically continued to the unit polydisk P 2 , since ( P2 , G) is a Euclidean Hartogs figure (up to the order of the coordinates). Let f be the continuation. Since g (z) := f(z, z) vanishes for � < l z l < 1, it also vanishes for 0 < l z l < � This means that f vanishes on A = {(z 1 , z2 ) E G : z1 = z2 } . Any zero set of finitely many holomorphic functions in G that vanish on A must contain A. So A itself cannot be given by global holomorphic functions. In the next chapter we define special domains in en each of which possesses a holomor phic function that cannot be analytically extended to a larger domain. Those domains are called domains of holomorphy. On such domains the global rep resentation of analytic sets is possible. The proof of this fact is not contained in this book, because it requires sheaf theory. One has to show that the sheaf of germs of holomorphic functions that vanish on A is "coherent" (cf. [GrRe84] , Section 4.2). Then every stalk of this sheaf is generated by global sections (Cartan' s theorem A, cf. Chapter V in this book, and [GrRe79] , Sec tion IV .5). From that it can be proved that A is the zero set of finitely many global holomorphic functions. �

�

�

Definition . A subset M of a domain G is called nowhere dense in G if the closure of M in G has no interior points. Since an analytic set A C G is always closed in G, it is nowhere dense if in every neighborhood of every point z E G there are points outside of A .

8 . 1 Pr op osit ion . Assum e that A is an analytic set in a domain G c en . If A has an interio r point, then A G If A is nowhere dense in G, then G - A is connected. =

.

PROOF: To start with we assume that G B is a ball and that there are holomorphic functions h , . . . , fq on B with A = N ( !1 , . . . , Jq ) . =

38

. Fu ctions I. Holomorph1c n

i nter ior point of A , we consider an arbitrary complex line L lf zo E B is�; the all vanish on L n B and identity theorem the functions f i through zo · B. th er efore in

If

A is nowhere dense in B and L an arbitrary complex line, then either L n B c A or A has only isolated points on L n B . So any two points of L nB outside <.f A can be connected in L n (B - A) .

Now let G be an arbitrary domain. If zo E G is an interior point <.f A , and w0 E G an arbitrary point, then we can join these points by a continuous path a : [0, 1] --+ G. The compact image <.f this path can be covered by finitely many balls B c G such that B n A is the zero set <.f holomorphic functions on B. Successively it follows that every ball is contained in A. So A = G . If A is nowhere dense in G, then we consider zo , wo E G A and use the same continuous path. It is clear from above that any point z in the first ball B that is not an element of A can be joined in B - A to z0 . Applying this • successively we obtain a curve between z0 and wo in B - A . -

If n = l , then a nowhere dense analytic set consists only of isolated points.

B oun ded H olomorphic Fun ction s. Assume that G c en is a do main and A C G a proper analytic subset. 8 . 2 Riemann extension theorem . If f is a holomorphic fun ction in G - A that is bounded in a neighborhood of every point of A, then f can be holomorphically extended to G. PROOF: Since A # G, A is nowhere dense in G. Let zo E A be an arbitrary point. Then there is a complex line L through zo that in a neighborhood <.f z0 intersects A only in zo. After a linear change <.f coordinates we may assume that zo = 0 and that L = Ce 1 is the zr-axis. We can find a polydisk 1 P = {z = (z 1 , z') E ([ x c n - : l z1 l < r 1 , lz ' l < r} CC G such that A n { z : l z1 1 = r 1 , l z'l < r-} is empty. For any c' E c n - 1 with lc'l < r, the set D = {z : l z 1 1 < r1 and z' = c '} is a 1-dimensional disc such that D n A contains only isolated points, since otherwise D c A (see Figure 1 .6) . By the classical Riemann extension theorem in one variable f can be extended to a function f (z 1 , z') that is holomorphic in z 1 . By the classical Cauchy integral formula we have f( (, , z') d(,, for l z 1 1 < r 1 and l z' l < r. r � = , z') J(z1 27rl }IC,I=n (, - Z1 �

The integrand on the right side is holomorphic on P . Consequently, the left side is differentiable (in the real _.§ense ), and since integration and differenti ation by z i can be exchanged, f is holomorphic on P . If we carry this out

8. Analytic Sets

. A

r c'

39

'

'' ' '

t----,,.·-----1 . <

'' Figur e

' '' ''

'

''

D

r1

:;;..

l z1 l

1 .6 . Riemann extension theorem

at every point z0 E A , by the identity theorem we obtain the desired global extension of f to G. •

Regular P oin t s. Let G c e n be a domain, and z E G a point. If j1 , . . , , Jq are holomorphic functions in a neighborhood <.f z , then we define rkz ( !I , . . . , Jq)

:=

rk Ju, , ...

,

fq )

(z) .

Defin ition . An analytic set A C G is called regular of codimension q at z E A if there is a neighborhood U = U(z) C G and holomorphic functions h, . . . , Jq on U such that : 1 . A n U = N(f1 , . . , , Jq) · 2. rkz (JI , . . . , fq ) = q. The number n - q is called the dimension of A at z.

The set A is called singular at z if it is not regular at that point. The set <.f regular points <.f A is denoted by Reg( A) or A, the set of singular points by Sing(A) .

. It is clear that A is open in A, and therefore Sing( A) C A closed.

8 . 3 Theorem (local parametr ization of regular points). Let A c G be analytic, z0 E A a p o int. A is regular of coilimenszon q at z0 if and only zj there are open neighborhoods U = U(z o) C G and W = W (O) C e n and a biholomorphic m ap F : U ---+ W such that F (z o) = 0 and

F (U n A) = {w = (wl , . . . ' W ) E w n

:

Wn - q+ l = . . . = Wn =

0 }.

PROOF: Let A be regular at z0. There is an open neighborhood U = U (zo) such that A n U = N( JI , . , . , fq) and rkz0 (!I , . . , , fq ) = q. By renumbering the coordinates we can achieve that

40

I. Holomorphic Functions J(fi , ... ,fq)

(z 0 ) = (J' ] J" ) '

with f E Mq, n -q (C) , J" E Mq (C), and det J" # 0. Then define F : U --+ e n

by

Consequently, the Jacobian has the form ]p ( zo )

=

(

En - q J'

and therefore det IF 1- 0. Shrinking U if necessary, we have our biholomorphic map F : U --+ W, with F ( zo ) = 0 and w = F ( z) for some z E u n A

�

W n- q+ l = . . . = Wn = 0. •

The other direction of the proof is trivial.

Up to this point it is not clear whether or not there exist regular points. In Chapter III we will show that the set of singular points of an analytic set A is a nowhere dense analytic subset of A . At the moment we want only to demonstrate that the zero set of a single holomorphic function contains at least one regular point (and then, of course, a nonempty open set of regular points). 8 . 4 Propositi on. Let G c e n be a dom a in, and f a nonconstant holomor phicfunction o n G. Th en the analytic set N ( f) contains a regular point. PROOF:

The case n

=

1 is trivial. Therefore, we assume n > 1 . V' f (z)

0 on A . Since f is not constant, it is impossible that there is a point z such that D v f ( z) = 0 for every multi-index v . Therefore, we can find a point z0 E A, an integer no , a multi-index vo , and some A E { 1 , . . , n } such that 1 . ] vo ] = no and (D v0 f)z,.. (zo) f- 0 , 2. D v f ( z ) = 0 for every z E A and every Y with ] v i < no. The set M := {z E G : D vo f (z) = 0} is analytic in G and regular of codimension 1 at z0 . We may assume that zo = 0 and M = { z = ( z1 , z' ) E G : z1 = 0}, making G sufficiently small.

If every point of A := N(f) is singular, then

.

We have A C M , and we want to show equality near z0 . It is clear that the function ( H f ( (, 0 ') has exactly one zero at ( = 0, and it follows easily from Rouche's theorem that for z' sufficiently close to 0' the functions ( H f( (, z') also have exactly one zero. This means that there is a neighborhood V = V(O) c U such that V n A = V n M . In particular, z0 is a regular point • of A .

41

8. Analytic Sets

Definiti on . A k-dimensional complex submanifold of a domain G C en is an analytic set A C G such that A is regular of codimension n k at every point. -

If A C G is a k-dimensional complex submanifold, then for every point z E A

there is an open neighborhood U = U (z) C G, an open set W C ek, and a holomorphic map t.p : W ---+ U such that: 1 . rk J'P (w) = k for w E W . 2 . t.p(W) = U n A . 3 . t.p : W ---+ U n A i s a topological map.4 The proof follows immediately from the local parametrization theorem. The map t.p is called a local parametrization.

Inj ective Holomorphic Mapping s. Let G c en be a domain, and f = ( h , . . . Jn ) : G ---+ en a holomorphic map. 8. 5

Iff is

Theorem.

injective, then det lr (z) 1- 0 everywhere.

We use induction on n. The case n = 1 is well known. We consider the case n > 1 and define h == det lr . PROOF:

Assume that N (h) 1- 0. Then there exists an open subset U C G such that M == U n N (h) is a nonempty ( n 1 )-dimensional complex submanifold of U. -

We claim that Jr l M 0. To prove this, we assume that there is a point zo E M with Jr (z0) 1- 0. Without loss of generality, we may assume that 8J a n ( zo ) f- 0.

Zn Let F : G ---+ en be defined by F( z' ' Zn ) == (z' ' fn ( z' ' Zn ) ) . Then det ]p (zo) 10, and there are connected ooen nei�hborhoods U ofz0 and V of wo : F (zo ) such that F : U ---+ V is b1ho1omorpliic. There is a holomorphic map f : V --+ =

en- l such that

and we define g

= (9 1 , · . . gn 1 ) : by g (w ') := f(w', w;;'>) . 4 A map I{! : X

,

-

w

== { w' E en - 1

:

(w', w,�>) E V } ---+ en - 1

Y between topological spaces is called

topological or a homeo m orph ism if it is continuous and bijective and the inverse mapping ip- 1 : Y ---+ X ---+

is also continuous.

42

I.

Holomorphic Functions

Since g is injective, we can apply the induction hypothesis and conclude that det Jg (wb) 1- 0. Now

Jr (zo) . Jp-1 (wo) =

( Jr (wo) ) = ( Jg (wb) O' en

Therefore, h ( zo ) = det h (zo) · det Jg (wb) 1- 0 as well. This is a contradiction. We have demonstrated that Jr ( z) = 0 for every z E M . Since f is holomor phic, also Df(z) = 0 on M , and using a local parametrization of M we obtain that f i M is locally constant. But this is impossible, since f is injective. The • set N(h) must be empty. 8.6 Corollary. 1f G c e n is a domain, and f : G --+ (51 " a n injective holomo rphic mapping, then also f(G) is a doma in, and f : G --+ f(G) is biholomorphic. PROOF: Let w0 := f(zo) be a point of G' := f (G) . Then det Jr (zo ) 1- 0, and there are open neighborhoods U = U(z0 ) c G and V = V(wo) c (51 " such that f : U --+ V is biholomorphic. It follows that w0 is an interior point • of G' and that r- 1 is holomorphic at Wo . Exercises 1 . Prove the following properties: (a) Finite intersections and unions of analytic sets are analytic. (b) I ff : G1 --+ G2 is a holomorphic map between domains and A C G2 1 an analytic set, then r- ( A) c G1 is analytic as well. (c) If A 1 C G1 and A2 c G2 are analytic sets, then A1 x A2 is an analytic subset of G 1 x G2 . 2. Let U c (51 " be an open neighborhood of the origin and A c U be an analytic subset containing the origin. For 1 < k < n - 1 and I = { i 1 , . . . , i k } c ( 1 , . . . , n} let pI : (51'' --+ c_ k be defined by Prove: 1f A is regular of codimension n - k at the origin, then there exists an I and open neighborhoods V = V(O) c U, W = W(O) c c_ k such that PI : A n V --+ W is bijective. 3. Show that A == { (w, z1 , z2) E C3 : w2 = z1 z2} is an analytic set that is regular of codimension 1 outside the origin and singular at 0. 4. Let A 1 , A2 be two analytic sets in a neighborhood of the origin in (51 " such that 0 E A : = A1 n A2 . Suppose that U n A1 1- U n A2 for every neighborhood U of 0 . Show that A is singular at 0.

C h ap ter 11

Domains of Holomor phy 1.

The C ontinuity The orem

General Hartogs Figures. The subject of this chapter is the contin uation of holomorphic functions. We consider domains in en , for n > 2. A typical example is the Euclidean Hartogsfigure (P", H), where pn = p n ( O , 1 ) is the unit polydisk, and H = { z E P"

:

1 z1 1

> q1 or

l zvl < qv for v = 2, . . . , n} .

Here q1 , . . . , qn are real numbers with 0 < qv < 1 for v = 1 , . . . , n. Every holomorphic function f on H has a holomorphic extension jon P".

general Hartogs figure. We use the symbolic picture that appears as Figure IL l Z2 , . . .

, Zn

'-------'-- - Figur e Il. l . General Hartogs figure 1 . 1 C ontinuity

theorem. Let G c (51 " be doma in, (P, H) a general Har togs figure with ii C G, f a holomo rphicfTf_n ction on G. lf GnP is connected, then f can be continued uniquely to G u P.

44

II. Domains of Holomorphy

PROOF:

Let g : pn ---+ en be an injective holomorphic mapping such that P == g(Pn) and H == g ( H ) . The function h = = f o g is holomorphic in H . Therefore, there exists--exactly one holomorphic function h on P" with h i H = h. Si!2 ce g : P" ---+ P is biholomorphic, the function fo : = h o g - 1 is defined on P, and it is a holomorphic extension of fi H - We define -

-

�

f(z) :=

{

f(z) fo(z)

for z E G, for z E 2;.

Since G n P is connected and f = fo on H, it follows from the identity theorem that f is a well-defined holomorphic function on G U P. This is the desired extension of f . • -

�

-

Example

Let n > 2 and P' c c P be polydiscs around the origin in en . Then every holomorphic function f on P - P' can be extended uniquely to a holomorphic function on P. For a proof we may assume that P = pn is the unit polydisk, and P' = pn(O, r), with r = (r1 , . . . , r ) and 0 < rv < 1 for v = l, . . . , n . It is clear n that G := P P' is a domain. Given a point zo = ( zio > , . . . , z�o) ) E G with I z;;'> I > rn , we choose real num bers q1 , . . . , qn as follows: For v = 1 , .. . , n - 1 , let qv be arbitrary numbers, with r, < qv < 1. To obtain a suitable q , we define an automorphism T of n the unit disk D by ( - z;;'> T (() : = z;:'>( - 1 This automorphism maps z�0> onto 0 and a small disk D C { ( E e : rn < ICI < 1} around z;:'> onto a disk K c D with 0 E K. Notice that 0 need not be the center of K. We choose qn > 0 such that Dqn (0) C K •

If we define H : = {z E P" : l z1 1 > q1 or l zv l < qv for v = 2 , . . . , n } , then (P", H) is a Euclidean Hartogs figure. The mapping g : pn ---+ pn defined by g(z1 , . . . , zn ) : = ( z1 , . . . , Zn - 1 , T- 1 ( zn ))

is biholomorphic, and (P, ii) = (P", g(H)) is a general Hartogs figure, with

-

Since P n G = G is connected, the continuity theorem may be applied. The preceding example is a special case of the so-called Kugelsatz which we shall prove in Chapter VI.

1.

45

The Continuity Theorem

(0 ) Zn g

K Figur e 1 1 . 2. A

Hartogs figure for concentric polydiscs

Removable Singularities . Let G c ccn be a domain. If A c G is an analytic set and f a holomorphic function on G - A that is locally bounded along A, then by Riemann's extension theorem f has a holomorphic extension to G. If n > 2 and A is a complex linear subspace of codimension greater than or equal to 2, then every function holomorphic on G - A has such an extension.

1 . 2 Theorem. Let P" = pn (o, 1 ) be th e unit polydisk in ccn , n > 2, k > 2, and E := {Z = (z l , . . . , zn ) E ccn : Zn - k+l = = Z n = 0}. Th e n every holomorphic function f on pn - E can be holomorphically ex tended to pn . .

.

.

PRooF:

Set P' := {z' := (z1 , . . . , zn - k) : l z' l < 1 } , and for 0 < r < 1 define ?;' := {z" = ( Zn - k + l , . . . , zn ) : l z" l < r } . Let P" : = P{' and fix an E with 0 < E � 1 . Then P" n E c P' x P�', and for w E P' the function fw (z") == f (w, z") is holomorphic on P" - Pf:l From the example above we know that fw has a holomorphic extension fw to P" . Now define f : pn ---+ CC by f(w , z " ) := fw (z"). On P" - E, f is equal to f and is therefore holomorphic. �

_......._

_......._

_......._

_......._

For w E P' take a small open neighborhood U = U(w ) c c P ' .Then U x 8P�' is compact. By the maximum principle we conclude that

I j(z', z" ) I

=

l iz, (z " ) I < l l fz" ll aP;' < llfiiK <

oo ,

K

for (z' , z " ) E U X P�' - E. �

From Riemann's extension theorem it follows that f is holomorphic on pn .

•

46

II. Domains of Holomorphy

1.3 Corollary. For n > 2, every isolated singularity of a holomorphic func tion of z1 , . . . , Zn is removable.

Riemann's extension theorem is false if we drop the condition "f bounded along the analytic set." For example, let G c e n be a domain, g : G ---+ C a holomorphic function , and let f : G - N (g) ---+ C be defined by f (z) 1 /g(z). Then f is holomorphic on G - N (g) but cannot be extended to any point of :=

N (g ) .

Things look quite different if there is a little hole in the hypersurface:

1 . 4 Proposition. Let n > 2, G0 C c_n - 1 a domain, g : Go ---+ C a contin uous function, and f := {z = (z' , Zn ) E Go X c : Zn = g (z' ) } the graph of g in G := G0 x C . In addition, let z0 be a point of f and U = U ( z 0 ) C G a small neighborhood.

If f is a holomorphic function on ( G- f) UU, then f has a unique holomorphic extension to G.

PROOF: The uniqueness of the extension follows from the identity theorem. For the proof of existence ( which is only a local problem ) we may assume that Go = {z' E c_n - l : l z' l < 1 } and that there is a q with 0 < q < 1 such that l g (z') I < q for z' E G0. It also may be assumed that U is connected. Then it is clear that G' : = ( G - f) U U c pn pn ( 0 , 1) is connected. Since g : z' r--+ (z' , g (z')) is continuous, U' g - 1 (U ) is an open neighborhood of z� with (U' X D) n f c u and therefore U' X D c G' . For v = 1 , . . . ' n - 1 let Tv be the automorphism of D defined by =

:=

Then h : pn ---+ pn with h(z1 , . . . , zn ) := (T1 (z2) , . . . , Tn - 1 ( zn ) , z! ) is holo morphic, h(O) (z�, O) , and h({z E pn : ! z1 l > q}) C {w E pn : lwn l > q } . =

q '

r,, '

/

I

zo

lz' l

Figure 11.3. Extending a holomorphic function across a hypersurface

1.

We define q1

:=

The Continuity Theorem

47

q, and for v = 2, . . . , n choose qv such that h (D

X

Dq2 (0)

X · · · X

Dq, (O)) C U'

X

D.

2, . . . , n} Then ( Pn , H) with H {z E pn : l z1 l > ql or l zv l < qv for V is a Euclidean Hartogs figure, and (P, H) ( Pn , h(H)) is a general Hartogs figure, with H C G' (see Figure II.3). Since P n G' G' is connected, the proposition follows from the continuity theorem. • :=

=

-

=

-

=

Sometimes we wish to use a family of analytic disks instead of a Hartogs figure. The Continuity Principle.

Definition. A family of analytic disks is given by a continuous map t.p : D X [0, 1] ---+ en such that 'Pt (() : = t.p((, t ) is holomorphic in D, for every t E [0, 1] . The set St : = 'Pt (D) is called an analytic disk, and bSt := t.pt (8D) its boundary.

Observe that in general bSt is not the topological boundary of St . Definition. A domain G c en is said to satisfy the continuity prin ciple if for any family { St , t E [0, 1] } of analytic disks in en with U o < t
Example

Let pn be the unit polydisk and { St, t E [0, 1] } a family of analytic disks in en with U o< t < l bSt c p n and So c pn . Because So and the union of all boundaries bSt-are compact sets , there is an c > 0 such that

U

O
bSt

c Pn (o, 1 - c )

and

S0

c Pn (o, 1 - c ) .

We assume that U o
It is clear that t o > 0, St0 rt pn , and St C pn for 0 < t < to. Then St 0 contains a point Zo = ( ziO) ' . . . ' z�O) ) E apn . If the family of analytic disks is given by the map t.p : D X [0, 1] ---+ en ' and WI" denotes the jLth coordinate function, then fM ,t ( ( ) := w'" o t.p((, t) is continuous on D and holomorphic in 0 D . Choosing JL such that l z�o ) l = 1 , there is a (u E D with fJ",to ((u) = z� l and lfM ,t o ((o ) l = 1 . But by the maximum principle we have l fM , t ( (o) l < sup l fM , t l < 1 8D

c,

for t < to.

Since t r--+ f!" , t ( (o) is continuous, a contradiction is reached, and therefore pn satisfies the continuity principle.

48

II. Domains

of Holomorphy

H artogs Convexity .

A domain G c en is called Hartogs convex if the following holds: If (P, H ) is a general Hartogs figure with H C G, then P C G. Definition.

An immediate consequence of the definition is the following: The biholomorphic image of a Hartogs convex domain is again Hartogs convex. 1.5 Theorem. Let G c en be a domain that satisfies the continuity prin ciple. Then G is Hartogs convex.

PROOF: Let (P, H) be a general Hartogs figure with H C G. We assume

that it is the biholomorphic image (g(P n ) , g(H)) of a Euclidean Hartogs figure ( Pn , H) with

In order to define analytic disks we choose some r with q1 introduce the affine analytic disks Dw

:=

{z

=

(z1 , z") E pn

=

P'

x

P" : 1 z1 1 < r and z"

<

=

r

<

1 and

w}.

Since Dw c pn for every w E P"' we can define t.p..;., : D X [0, 1] --+ en by setting 'Pw ( (, t) : = g (r(,, tw) . Then a family { St ( w) : 0 < t < 1} of analytic disks in P is given by St (w)

:=

t.pw (D

X

{ t} )

=

g(Dtw ) .

It follows that bSt (w) C G for every w E P" and every t E [0, 1] , and in addition, S0 (w ) = g (Do ) C G. The situation is illustrated in Figure II.4.

z" w

I'

St(w)

bSt (w)

'

/

Figure 11.4. Analytic disks in a Hartogs figure

49

1 . The Continuity Theorem

Since G satisfies the continuity principle, we obtain that g ( Dw) = SI (w) is contained in G. This is valid for every w E P". Therefore, P C G, and G is • Hartogs convex. 1.6 Corollary.

The unit polydisk pn is Hartogs convex.

Domains of Holomorphy

Definition. Let G c e n be a domain, f holomorphic in G, and Zo E 8G a point. The function f is called completely singular at z0 if for every connected neighborhood U = U(zo) C e n and every connected component c of u n G there is no holomorphic function g on u for which g l c = fi e ·

Example

Let G : = e - { x E lR : x < 0} and let f be a branch of the logarithm on G. Then f is completely singular at z = 0 but not at any point x E lR with

X < 0.

A domain G c e n is called a weak domain of holomorphy if for every point z E 8G there is a function f E 0( G) that is completely singular at z. Definition.

The domain G is called a domain of holomorphy if there is a function f E 0( G) that is completely singular at every point z E 8G. Examples

1 . Since e n has no boundary point, it trivially satisfies the requirements of a domain of holomorphy. 2. It is easy to see that every domain G C e is a weak domain of holomor phy: If Zo is a point in aG, then f(z) 1/(z - zo) is holomorphic in G and completely singular at z0. :=

For G D we can show even more! The function f(z) := -z:=;: 0 zv! is holomorphic in the unit disk and becomes completely singular at any boundary point. Therefore, D is a domain of holomorphy. At the end of this chapter we will see that every domain in e is a domain of holomorphy. 3 . If f : D ---+ e is a holomorphic function that becomes completely singular at every boun�ry point, then the same is true for f pn = D x · · · x D ---+ e, defined by f( Z1 , . . . , Zn ) : = f (zl ) + · · + f(zn ) . In fact, if zo is a bound ary point of pn , then there exists an i such that the ith component z<0> is a boundary point of D. If f could be extended holomorphically across z0, =

�

·

�

:

2

50

II.

of

Domains

Holomorphy

�

�

then fi ((, ) : = f(zi0> , . . . ,(,, . . . , z�o > ) would also have a holomorphic extension. But then f could not be completely singular at zJ0> . Therefore, the unit polydisk is a domain of holomorphy. 4. If (P", H) is a Euclidean Hartogs figure, then H is not a domain of holo morphy. 1.7

Let G c e n be a domain. Iffo r every point Zo E ac there is an open neighborhood U = U(zo) c en and a holomorphicfunction f : G u U -+ e with f(zo) = 0 and f(z) # O for z E G, then G is a weak domain of holomorphy. Proposition.

PROOF:

We show that 1/f is completely singular at zo. For this assume that there is a connected open neighborhood V = V(z0 ) , a connected component C c V n G, and a holomorphic function F on V with Fie = (1/f) 1 9 · The set V' := V - N(f) is still connected and contains C. By the identity theorem the functions F and 1/ f must coincide in V'. Then F is clearly not • holomorphic at z0• This is a contradiction.

1 .8 Corollary.

Every convex doma in in en is a weak domain of holomor

phy. PROOF:

If zo E aG, then because of the convexity there is a real linear form A on e n with .X(z) < .X(zo) for z E G . We can write >. in the form n

n

�=>l:vZv + L avzv, with v=l So >. = Re h (z) , where h(z) : = 2 .'z=�= l avZv .X (z) =

u=l

a

:= (al ,

. . . , a n ) -=/= 0 .

is holomorphic on e n .

Since the function f(z) : = h(z) - h(zo) is holomorphic on e n , f(zo) = 0, and • f(z) -=/= 0 on G, the proposition may be applied. We will show that every weak domain of holomorphy is Hartogs convex. As a tool we need the following simple geometric lemma, which will be useful in other situations as well.

1 .9 Lemma (on boundary compo nents) . Let G c e n be a domain, U c en an open set with U n G -=/= 0 and (en - U) n G -=/= 0. Th en G n ac n au # 0 for any connected component C of U n G .

We choose points z1 E C c U n G and z2 E (en - U ) n G. There is a continuous path 1 : [0, 1 ] -+ G with r( O) = z 1 and 1( 1) = z2 . Let to := sup {t E [0, 1 ] : r(t) E C } and zo : = r(to ) . Clearly, z o E ac n G, but zo tf_ C. Since C is a connected component of U n G, zo cannot lie in U n G and therefore even not in U. Since r(t) E U for t < to , it follows that PRooF:

zo E

au.

•

1 . The Continuity Theorem

1 . 1 0 Theorem. Hartogs convex.

Let G c

en

51

be a weak dom ain of holomorphy. Then G is

PROOF:

Assume that G is not Hartogs convex. Then there is a general Hartogs figure ( P,H ) with H C G but P n G # P. We choose an arbitrary z0 in H and set C : = CP nG (zo ) . 1 Since H lies in P n G and is connected, it follows that H C C. Furthermore, C � P .

Since P n G -=/= 0 and ((Cn - G) n P -=/= 0, by the lemma there is a point Z l E ac n ac n p (see Figure 1 1.5).

F igu r e 11.5. G is not Hartogs convex

Let f be an arbitrary holomorphic function in G. Then fie is also holomor phic, and by the continuity theorem it has a holomorphic extension F on P. Since P is an open connected neighborhood of z 1 , we obtain that f is not completely singular at z 1 . This completes the proof by contradiction. • It follows, for example, that every convex domain is Hartogs convex. As a consequence, we see that every ball is Hartogs convex. 1.11

Theorem.

Every domain

d

holomorphy is Hartogs convex.

The proof is trivial. For the converse of this theorem one has to construct on any Hartogs convex domain a global holomorphic function that becomes completely singular at every boundary point, something that is rather difficult. It wa<; done in 1910 by E.E. Levi in very special cases. The general case is called Levi 's problem.

In 1942 K. Oka gave a proof for n = 2. At the beginning of the 1950s Oka, Bremermann, and Norguet solved Levi's problem for arbitrary n. It wa<; gen1

We denote by CM (z) the connected component of M containing z .

52

II. Domains

of

Holomorphy

eralized for complex manifolds (H. Grauert, 1958) and complex spaces (R. Narasimhan, 1962). Finally, in 1965 L. Hormander published a proof that used Hilbert space methods and partial differential equations. Exer cises

1 . Prove the following statements: (a) Finite intersections of Hartogs convex domains are Hartogs convex. (b) If G1 c G2 c G3 C · · · is an ascending chain of Hartogs convex domains, then the union of all Gi is also Hartogs convex. 2 . Let G c e n be a domain, 0 < r < R, and a E G a point. Let U = U(a) c G be an open neighborhood and define Q == {W E em : r < JwJ < R ) . Prove that every holomorphic function on (G x Q) U (U x P" (0, R ) )has a unique holomorphic extension to G x pm ( 0, R ) . 3 . Let 0 < r < R be given. Use Hartogs figures to prove that every holo morphic function on BR (O) - B r (O) has a unique holomorphic extension to the whole ball BR(O) . 4. For E > 0, consider the domain

Ge == f ( z ,w) E

P2 (0, 1 ) : JzJ

<

Jwj2 + c}.

Prove that Ge is Hartogs convex if and only if E = 0. 5 . Let G C en be a domain and f : G --+ D R (O ) C e a function, r = { (z, w) E G x D R (O) : w = f(z)} its graph. Sow that if there is a holomorphic function F in G x D R(O ) that is completely singular at every point of r, then f is continuous. (With more effort one can show that f is holomorphic.) 6 . Show that the "Hartogs triangle" { ( z,w) E e 2 : JwJ < J z J < 1} is a weak domain of holomorphy.

2.

Plur isubharr nonic Fu n ction s

Subharmonic Functions . Recall some facts from complex analysis of one variable. A twice differentiable real-valued function h on a domain G C e is called harmonic if hzz (z) = 0 on G. The real part of a holomorphic function is always harmonic, and on an open disk every harmonic function is the real part of some holomorphic function. If D = D r ( a) c e is an open disk and f3 : lR --+ lR a continuous periodic function with period 21r, then there is a continuous function h : D --+ lR that is harmonic on D such that h(rei t ) = f3(t) for every t (Dirichlet' s principle).

An upper semicontinuous function t.p : G --+ lR U {-oo} is said to satisfy the weak mean value property if the following holds:

For every a E G there is an r > 0 with Dr (a) CC G and

2. Plurisubharmonic Functions

for 0 < (} <

53

r.

Remarks 1 . If t.p : G --+ lR U { - m } is an upper semicontinuous function, then the sets Uv == { z E G : t.p ( z) < v} are open, and therefore t.p is bounded from above on every compact subset K c G. It follows that the integral in the definition always exists. 2. Harmonic functions satisfy the weak mean value property (even the stronger mean value property with "=" instead of "<" ) . 3 . If f : G --+ C is a nowhere identically vanishing holomorphic function, then logJJJ satisfies the weak mean value property. In fact, the function t.p == logJJJ is harmonic on G - N(f) , because it can be written locally as Re(lo g f) , with a suitable branch of the logarithm. And at any point z0 E N(f) we have t.p(zo) = -oo, so the inequality of the weak mean value property is satisfied. 2 . 1 P r op o sition . Let t.p : G --+ lR satisfy the weak mean value property. If t.p has a global m aximum in G, then t.p is constant. PROOF: Let a E G be any point with c := t.p(a) > t.p(z) for z E G. We choose an r > 0 such that

"' t.p( (}e i t) dt for 0 < 2 �1 a+

D r (a ) C C G and t.p (a ) < 2

Assume that there is a b E D r ( a) with t.p(b) < t.p(a) . We write and get

12-rr t.p(a

1 t.p ( a) < 2 7f 0

+

12-rr t.p(a)

1 (}e i t ) dt < 2 7f 0

(} b

< r.

= a + (}e i to

dt = t.p(a) .

This is a contradiction, so t.p must be constant on D,. (a). Now we define the set M == {z E G : t.p(z) = c ). Obviously, M is closed in G and not empty, and we just showed that M is open. So M = G, and t.p is constant. • Defin ition . Let G c C be a domain. A function s : G --+ lR U { -oo} is called subharmonic if the following hold: 1 . s is upper semicontinuous on G. 2. If D cc G is a disk, h : D --+ lR continuous, hJD harmonic, and h > s on 8D, then h > s on D. 2 . 2 Pr opo sition . Let sv : G --+ lR U {-oo} be a monotonically decreasing sequence of subharmon icfunctions. Then s := limv-+oo sv is subharmonic.

54

II. Domains

of

Holomorphy

PROOF: The limit s = limv--+oo Sv = inf{sv} is upper semicontinuous. Let D c c G be a disk, h : D --+ lR continuous and harmonic on D, with s < h on aD. For fixed E we consider the compact sets Kv == {Z E aD : Sv (z) > h(z) + c}.

Then Kv+l c Kv and n � l Kv = 0. Therefore, there is a Vo E N with Kv = 0 for v > Vo . This means that for v > Vo , Sv < h + E on aD, and therefore the same is true on D. Since the sv are decreasing, s < h + E on D . • This holds for every E > 0, and consequently s < h on D. 2 . 3 P ropo sition . Let (sa)a EA be a fam ily of subharmonic functions o n G. If s : = sup sa ist upper semicontin uous and fin ite everywhere, then s is subharmon ic. PROOF: If s < h on aD, where D cc G and h : D --+ lR is continuous and harmonic on D, then sa < h on aD for every a E A. Since the sa are subharmonic, it follows that s, < h on D for every a E A . But then s < h • on D as well. Examples 1 . Clearly, every harmonic function is subharmonic. 2 . Let s : G --+ lR be a continuous subharmonic function such that -s is also subharmonic. Then s is harmonic. To show this, we look at an arbitrary point a E G and choose an r > 0 such that D == Dr (a) cc G. Then there is a continuous function h : D --+ lR with hlcw = s l av that is harmonic on D (Dirichlet' s principle). It follows that s < h on D. But because -h is also harmonic, we have -s < -h on D as well. Together this gives s = h on D. 3 . Let f : G --+ C be a holomorphic function. Then s : = log lfl is subhar monic. In fact, if f(z) = 0 on G, then we have s(z) = oo and there is nothing to prove. Otherwise, s is harmonic on G - N(f) , and we have only to look at an isolated zero a o f f . We choose D = Dr(a) cc G and a function h that is continuous on D and harmonic on D, with s < h on aD. We know that s , and therefore also s - h, has the weak mean value property on D , and it is certainly not constant. So it must take its maximum on the boundary aD. This means that s < h on D. 4. Let G c C be an arbitrary domain. The boundary distance t5c : G --+ IR+ U { +oo} is defined by -

,

t5c (z) := sup{r E R : Dr (z) C G) . Claim: s := - log t5c is subharmonic on G. PROOF: If G = C , then s(z) = -oo and there is nothing to prove. If G # C, then s is real-valued and continuous. For w E ac we define

2. Plurisubharmonic Functions 01

55

: G ---+ lR by setting sw (z) := - log /z - w / . Then s(z) = sup { s ( z)

•

w

w E aG} . By Proposition 2.3 the claim follows.

•

The M aximum Principle 2 .4

Theorem. Let s : G ---+ lR U { -oo} be a subharmonic fu n ction o n a dom a in G C C. If s takes its m axim u m o n G, then it must be constant.

PROOF: Assume that c := s(a ) > s ( z) for every z E G. As in the case of functions that have the weak mean value property it suffices to show that s is constant in a neighborhood of a . If this is not the case, there is a small disk D = Dr(a ) CC G and b E aD with s(a) > s (b) . Since s is upper semicontin uous, there is a continuous function h on aD with s < h < c and h(b) < c. Solving Dirichlet' s problem we can construct a harmonic continuation of h on D. Now

h ( a)

r = __!__ 21r J

o

h(a + reit ) dt < c = s(a).

This is a contradiction

•

For later use we give the following criterion for a function to be sub harmonic: 2.5

Theorem. Let G c C be a doma in and s : G ---+ lR U { -co} an upper semicontinuous fun ction. Suppose that fo r every disk D C C G and every fun ction f E O(D) with s < Re(f) o n aD it follows that s < Re(f) o n D . Th en s is subharmonic. PROOF: Let D = Dr ( a ) c c G, h : D ---+ lR continuous and harmonic on D , and s < h on aD. For simplicity we assume a = 0. For v E N, a harmonic function h, on D, := D ( v /( v - l ))r(O)

::J

D is given by

Then (h, ) converges on D uniformly, increasing monotonically to h. Further more, for every v there is a holomorphic function fv on D, with Re(fv) = h,. Let E > 0 be given. Then there is a vo such that / h - hv I < E on D for v > v0. Therefore, s < h v + c- = Re(fv + c) on aD for v > Vo . By definition it follows that s < h, + E on D. Since (h,) is increasing, it follows that s < h + c and therefore s < h on D. •

Differentiable S ubharmonic Functions 2.6 Lemma. Let s : G ---+ lR be a �2 fun ction such that Szz > 0 o n G. Then s is subharmonic.

II. Domains

56

of

Holomorphy

PROOF: Let D Dr (a) c c G and let a continuous function h : D -+ lR be given such that h is harmonic on D and s < h on 8D. We define r.p := s - h. =

Assume that r.p takes its maximum at some interior point zo of D . Then we look at the Taylor expansion of r.p at zo in a small neighborhood about zo:

r.p(zo + z)

= r.p (zo ) + 2 Re Q(z) + 'Pz-z(zo)zz + R(z) ,

where Q(z) : = 'Pz (z0)z + � r.pzz (zo)z2 is holomorphic and R(z) / l z l 2 -+ 0 for z -+ 0. The function 'ij;(z ) := 2 Re Q(z) is harmonic, with '1/J(O) = 0. Since it cannot assume a maximum or a minimum, it must have zeros arbitrarily close to but not equal to 0. On the other hand, r.p(zo + z) - r.p ( zo) < 0 and 'Pz-z (zo)zz > 0 outside z = 0. This is a contradiction. Thus r.p must assume • its maximum on the boundary of D , and s < h on D. 2. 7

Theorem. Let s : G -+ lR be and only if Szz > 0 on G .

a

�2 function. Then s is subhamwnic if

PROOF: (a) Let Sz-z(z) > 0 for every z E G. Then we define s" on G by setting s" := s + ( 1 /v)zz. Obviously, (sv)zz = Szz + ( 1 /v) > 0. Then s" is subharmonic by the above lemma. Since (s, ) converges, monotonically decreasing, to s , it follows that s is subharmonic. (b) Let s be subharmonic on G. We assume that Sz-z (a ) < 0 for some a E G. Then there is a connected open neighborhood U = U(a) c G such that Sz-z < 0 on U. By the lemma it follows that -s is subharmonic on U. Then • s must be harmonic on U. So Sz-z(a) = 0, contrary to assumption.

Pluri subharmonic Functions. We return to the study of domains in arbitrary dimensions. Let G c e n be a domain and ( a,w) a tangent vector at a E G. We use the holomorphic mappmg cta,w : e -+ e n defined by cta, w( ( ) := a + ( w. Definition. Let G c e n be a domain. An upper semicontinuous func tion p : G -+ lR U { oo} is called plurisubhannonic on G if for every tangent vector ( a,w ) in G the function -

Pa,w ( ( ) := P o cta, w( ( ) = p(a + (w ) is subharmonic on the connected component G (a , w ) of the set a a, !... (G) e containing 0. R e m arks 1 . Plurisubharmonicity is

a local property.

c

57

2. Plurisubharmonic Functions

2. If f E O(G), then log l f l is plurisubharmonic. 3 . If PI , P2 are plurisubharmonic, then so is PI + P2 . 4. 1f p is plurisubharmonic and c > 0, then c . p is plurisubharmonic. 5 . If (Pv) is a monotonically decreasing sequence of plurisubharmonic func tions, then p : = limv--too p , is also plurisubharmonic. 6. Let (Pa )a EA be a family of plurisubharmonic functions. If p := sup (pa ) is upper semiconlinuous and finile, Lhen il is also plurisubham10nic. 7. 1f a plurisubharmonic function p takes its maximum at a point of the domain G, then p is constant on G.

The Levi Form Definition. Let U C e n be an open set, f The quadratic form2 Lev( !) : Ta --+ lR with

2 ( U ; JR), and a

E 9&'

E

U.

is called the Levi form of f at a. Obviously, Lev(!) is linear in f . Examples

1. In the case n = 1 we have Lev(s) (a, w) = S z-z ( a) ww . So s is subharmonic if and only if Lev(s) (a, w ) > 0 for every a E G and w E C. 2. Let f( z) : = llz ll 2 = 2::: 7 I Zi Zi . Then Lev(f) (a, w ) = llwl/ 2 for every a. 3 . If f E 9&'2 ( U; lR) and g : lR --+ lR is twice continuously differentiable, then

Lev ( g o !) (a, w) = g11 ( f (a) ) · l (8f)a( w) l 2 + g' ( f (a) ) Lev(f)(a, w) . ·

4. If F : U --+ V c em is a holomorphic map and g

E

9&'2 ( V; JR), then

L ev (g o F) (a, w) = Lev(g ) (F(a) , F'(a) ( w))

5 . For

f E 9&'2 ( U ; JR) the Taylor expansion at a E U gives f ( z ) = f (a) + 2 Re (QJ (z -

where Q J ( w ) = I:�= l fz (a) w v quadratic polynomial, and v

2

If H :

T

x

T --+ C

a))t L ev ( f ) (a, z - a) + R(z - a),

+ � I: v fzv z (a) w ,J.L

. R(z - a) hm = Z--Ta II Z - a 11 2 o .

"

"

w J.L

is a holomorphic

is a Hermitian form on a complex vextor space, then the associated quadratic fo rm Q : V --+ lE. is given by Q(v) :=H(v, v).

58

11.

Domains of Holomorphy

2.8

Theorem. Lev(f) (a, w) >

A function ! E 1&'2 (U; JR) is plurisubhamwnic if and only if Ofor every a E U and every w E T,.

PROOF: Let (a,w) be a tangent vector in G and Q := cta, w · Then f oa(O) f(a) and (fo a)(((O) = Lev ( f o a )( O , l) = Lev ( f) ( a w ) .

=

,

Now, f is plurisubharmonic if and only if f o a is subharmonic near 0 for any a = aa, w · Equivalently, (t o a)(((O) > 0 for any such a. But this is true if • and only if Lev(f) (a, w ) > 0 for any tangent vector ( a,w) in G. 2.9 a

Corollary. Let G1 c en and G2 c em be domains, F : G1 -+ G2 holommphic map, and g E 1&'2 (G1 ; JR) plurisubharmonic. Then g o F zs

plurisubharmonic on G1 .

PROOF:

This is trivial, because of the formula in Example 4 above.

•

Exhau stion Function s . For every domain G c e the function log 8c is subharmonic. In higher dimensions it is in general not true that this func tion is plurisubharmonic for every domain G. -

Definition. Let G C en be a domain. A nonconstant continuous func tion f : G -+ lR is called an exhaustionfunction for G if for c < sup c ( f ) all sublevel sets G ( f) := {Z E G : f (z ) < C) c

are relatively compact in G.

Example

For G = en , the function f (z) := llzll 2 is an exhaustion function. For G # en , we define the boundary distance 8c by 8c(z) : = dist (z, en - G).

Then -8c is a bounded, and log 8c an unbounded, exhaustion function. We only have to show that 8c is continuous: -

For every point z E G there is a point r(z) E en 8c (z)

-

G such that

= dist(z, r(z)) < dist(z , w) for every w E en

Then for two arbitrary points u, v E G we have

-

G

.

2. Plurisubharmonic Functions

8c (u) = ll u - r (u) ll < and in the same way 8c (v ) <

59

llu - r (v) ll < llu - vii + Dc (v) , llu - vii + 8c ( u).

Therefore, l8c ( u) - 8c (v) l < llu - v i i · A function f E 1&'2 ( G; JR) is called strictly plurisubharDefinition. monic if Lev(f) (a, w ) > 0 for a E G, w E Ta, and w #- 0. For a proof of the following result we refer to 4. 14.

[Ra86] , Chapter II, Proposition

2. 1 0 S m o o th i n g lemma . Let G c en be a domain, f : G --+ lR a continu ous plurisubharmonic exhaustionfunction, K c G compact, and E > 0. Then there exists a '1&'00 exhaustionfunction g : G --+ lR such that: 1. g

> f on G. 2. g is strictly plurisubharmonic. 3. lg(z) - f(z) l < E on K.

Exercises 1. Let G C e be a domain. Prove the following statements: (a) If f : G --+ e is a holomorphic function, then If I" is subharmonic for a > 0. (b) If u is subharmonic on G, then uP is subharmonic for p E N.

2. 3. 4. 5.

(c) Let u ¢ - oo be subharmonic on G. Then { z E G : u (z) = - oo } does not contain any open subset. Let G c e be a domain, s ¢ -oo a subharmonic function on G, P := { z E G : s (z) = -oo }. Show that if u is a continuous function on G and subharmonic on G - A , then u is subharmonic on G. Let U C en be open, f : U --+ e k a holomorphic map, and A E Mk (JR) a positive seniidefinite matrix. Show that cp(z) := f(z) . A · f(z) t is pluri subharmonic. Let G = { ( z ,w ) E e2 : l w l < l z l < 1} be the Hartogs triangle. Prove that there does not exist any bounded plurisubharmonic exhaustion function on G. Are the following functions plurisubharmonic (respectively strictly pluri subharmonic)?

llzll 2 ), for z E en, llzll 2 ), for llzll < 1 , llz ll 2 e- Re (zn ) , for Z E en . 6. Consider a domain G c en and a function f E 1&'2 (G) . Prove that f is strictly plurisubharmonic if and only if for every open set U CC G there is an c > 0 such that f(z ) - c:((zll 2 is plurisubharmonic on U. Pl (z) P2 (z) P3 (z )

log( 1 + _ - log( 1 -

60

3.

1 1.

Domains of Holomorphy

P seu d ocon vexity

P seudoconvexity A domain G c en is called pseudoconvex Defin ition . strictly plurisubharmonic �= exhaustion function on G.

if there is a

Remarks

1. By the smoothing lemma the following is clear: 1f - log 8c is plurisub harmonic, then G is pseudoconvex. 2. Pseudoconvexity is invariant under biholomorphic transformations.

3 . 1 Theorem. If G c en is a pseudoconvex domain, then G satisfies the continuity principle. Let p : G --+ lR be a strictly plurisubharmonic exhaustion function. Suppose that there exists a family { St : 0 < t < 1} of analytic disks given by a continuous mapping ip : D X [0, 1] --+ en such that So c G and bSt c G for every t E [0, 1] , but not all St are contained in G. PROOF:

The functions p o 'Pt : D --+ G are subharmonic for every t with St c G. It follows by the maximum principle that p / St < maxbs, p for all those t.

We define to := inf{t E [0, 1] : St C/- G}. Then to > 0, St0 C G, and St0 meets aG in at least one point z0. We can find an increasing sequence (tv) converging to to and a sequence of points z v E Stv converging to zo. So p ( z v) --+ c0 := sup c (p ) , but there is a c < co such that P/bs, < c for every • t E [0, 1] . This is a contradiction. 3 . 2 Corollar y.

If G is pseudoconvex, then G is Hartogs convex.

The Boundary Distance

3 . 3 Theorem. If G c en is a Hartogs convex domain, then - log 8c plurisubharmonic on G. PROOF:

For z E G and

u

E en with

8c , u ( z ) :=sup{t > 0 :

is

/lui/ = 1 we define z

+ Tu E G for /T/ < t } .

Then 8c (z ) = inf { 8c , u ( z ) : // u// = 1}, and it is sufficient to show that - log 8c , u is plurisubharmonic for fixed u . (a) Unfortunately, 8c,u does not need to be continuous, but it continuous:

is

lower semi

3. Pseudoconvexity

61

Let zo E G be an arbitrary point and c < 8c, u (z0 ) . Then the compact set K := {Z = zo + Tu : ITI < c} is contained in G, and there is a 6 > 0 such that {z : dist(K, z) < 8 } C G. For z E Bo (zo) and ITI < c we have ll (z + T U)

-

(zo + Tu) ll

= l i z - zo ll

< S, and therefore 8c,u (z) > c.

(b) The function - log 8c ,u is upper semicontinuous, and we have to show that s(() : = - log 8c, u (zo + (b) is subharmonic for fixed u ,z0, b . First consider the case that u and b are linearly dependent: b = X u , >. -=/=- 0. Let Go be the connected component of 0 in { ( E C

:

z0 + (b E G}. Then

sup{t > 0 : zo + (b + Tu E G for ITI < t } sup{t > 0 : ( + T/A E Go for I T I < t } 1>-1 . sup{r > 0 : ( + J E Go for IJI < T J 1>-1 · 8c o (() ,

8c , u (zo + (b)

and this function is in fact subharmonic. (c) Now assume that u and b are linearly independent. Since these vectors are fixed, we can restrict ourselves to the following special situation:

Zo = O, h = e1 ,

n = 2,

and

u = e2 .

Then s(() = - log sup{t > 0 : ((, T) E G for ITI < t}. We use holomorphic functions to show that s is subharmonic. Let R > r > 0 be real numbers such that ( (, 0) E G for I( I < R, and let f : DR (O) -+ C be a holomorphic function such that s < h :=Re f on 8Dr (O) . We have to show that s < h on Dr (O ) . We have

the following equivalences:

s (() < h(()

<===?

<===?

sup{t > O : ((, T) E G for i T I < t} > e - h(() ( ( , c e- f(( ) ) E G for c E D. ·

(d) Define a holomorphic map F by F (z1 , z2 )

:= (rz1 , z2 e- f(r z, ) ) .

Then F is well defined on a neighborhood of the unit polydisk P 2 = P 2 (0, 1 ) . It must be shown that F (P 2 ) C G. We already know the following: 1 . F (z1 , z2) E G for l z1 l = 1 and lz2 l < 1 , because s (t) < h (t) on 8Dr (O ) . 2 . F (z1 , 0 ) E G for l z1 l < 1 , because ( (, 0) E G for 1(1 < T.

62

1 1.

Domains of Holomorphy

These facts will be used to construct an appropriate Hartogs figure. First, note that

JF ( z1 , z2 ) =

(:

� )

e-f rzl )

,

SO

det JF(ZI , Z2 ) � 0.

By the inverse function theorem it follows that F is biholomorphic. For 0 < 6 < 1 we define ho : C 2 --+ C 2 by h0 ( z1 , z2 ) to the compact set

c

:=

:=

(z1 , 8 z2 ) and apply h0

{ ( zl , z2 ) E C 2 : (/zl l < 1 , Z2 = 0) or ( /zi / = 1 , /z2 /

I

1)} c P 2 .

Consequently,

Co := ho (C) = { (zl , Z2) E C2 : (/zl / < 1 , Z2 = 0) or ( /zl / = 1, / z2 / < 8)} . Then F (C0) C G, as we saw above, and therefore Co C F - 1 ( G) . For 0 < c < min(8, 1 - S) we define a neighborhood U6 of C0 by U6

{ ( zi , Z2 ) E C2 : ( /zi / < 1 + c, / z2 / < c) or ( 1 - c < /zi / < 1 + c, /z2 / < D + c ) } . If we choose small enough, then U6 c F- 1 (G ) . 1 Finally, we define He; :=hi (U6 n P 2 ) n P 2 (see Figure 1 1 .6). Then c

H e;

{ (z1 , z2 ) E P 2 : (z1 , 8z2 ) E U6 n P2 } = ( z1 , z2 ) E C2 : (/ zi / < 1 , / z2 / < ) or ( 1 - · E< / z1 / < I , / z2 / < 1 )

{

�

c

b .

.

}.

. .. . . . ..

Figure 1 1 .6. Construction of the Hartogs figure Since (P2 , H 6) is a Euclidean Hartogs figure, (F o h0 ( P2 ) , F o ho (H c; )) is a general Hartogs figure with Fo h0 (H6) c F (U n P 2 ) c G. Since G is Hartogs c;

3. Pseudoconvexity

63

pseudoconvex, it follows that F o h6(P 2} c G. This is valid for every 8 < 1 . But P2 = Uo
Thefollowing p roperties of a domain G c en are equivalent:

G satisfies the continuity p rinciple. 2. G is Hartogs pseudoconvex. 3. - log 8c is plurisubharmonic on G . 1.

4 - G is pseudoconvex. PROOF:

( 1 ) ===? ( 2) ===? (3) ===? (4) ===?

(2) is Theorem 1 . 5, (3) is Theorem 3 .3, (4) follows from the smoothing lemma, ( 1) was proved in Theorem 3 . 1 .

•

Propertie s of Pseudoc onvex D omains 3.5 Theorem.

pseudoconvex.

lf Gl' G2 c en are pseudoconvex domains, then Gl n G2

The statement is trivial if one uses Hartogs pseudoconvexity.

PROOF:

IS

•

3.6 Theorem. Let G1 c G2 c . . . c en be an ascending chain if pseudo convex domains. Then G : = U ::" 1 Gv is again pseudoconvex. This follows immediately from the continuity principle.

PROOF:

•

3 . 7 Theorem. A domain G c en is pseudoconvex if and only if there is an open covering ( U,),EI of G such that U, n G is pseudoconvexfor every [ E /. PROOF: ===?

is trivial. The other direction will be proved in two steps. At first, we assume that G is bounded. "

"

For any point Zo E ac there is an open set u, such that Zo E u, and G n u, is pseudoconvex. If we choose a neighborhood W = W(zo) c U, so small that dist(z,8UJ > dist(z,zo) for every z E W n G, then 8c(z) =DcnuJz) on W n G. This shows that there is an open neighborhood U = U(8G) such that - log 8c is plurisubharmonic on U n G (we use the fact that 8G is compact). Now. G U CC G. We define -

64

II.

Domains

of

Holomorphy

c : = sup{- log b"c (z)

and

zEG-

U} ,

2 p(z) : = max( - lo g 8c (z) , / / z // + c + 1) .

Then p is a plurisubharmonic exhaustion function, and by the smoothing lemma, G is pseudoconvex. If G is unbounded, we write it as an ascending union of the domains G.., : = B..,(O) n G . Each G, is bounded and satisfies the hypothesis, so 1s • pseudoconvex. Then G is also a pseudoconvex domain.

E xer ci ses 1. Suppose that G 1 c en and G 2 c e m are domains. ( a) Show that if G 1 and G 2 are pseudoconvex, then G 1 x G 2 is a pseu doconvex domain in en+ m (b) Show that if there is a proper holomorphic map f : G 1 -+ G 2 and G 2 is pseudoconvex, then G 1 is also pseudoconvex. 2. Let G c en be a domain and g : G -+ lR a lower semicontinuous positive function. Prove that �

G := {(z' , w) E G

x

e

:

/w/ < g(z ')}

is pseudoconvex if and only if - log g is plurisubharmonic. 3 . A domain G c en is pseudoconvex if and only if for every compact set K c G the set

{

Kpl := z E G

:

p(z) < su pp for all plurisubharmonic functions p on G K

}

is relatively compact in G.

4.

Levi Con vex Bou n d ar ies

B oundary Functions Definition . Let G C en be a domain. The boundary of G is called smooth at Zo E ac if there is an open neighborhood u = U(zo) c en and a function g E '6'= ( U; lR) such that: 1. U n G = {z E U : g(z) < 0). 2. (dg)z -=/:- 0 for z E U. The function g is called a local defining fun ction (or boundary fun ction).

Remark. Without loss of generality we may assume that {!yn -=/:- 0. Then by the implicit function theorem there are neighborhoods

4.

(O) l ' x(O) ) U1 of (zO1 ' x
E

Levi Convex Boundaries

en - l

X 11])

.J.&.,

and a 't'= function "( : U1 --+ U" such that { ( z1 , Xn , Yn ) i yn ) = 0} = { (z 1 , Xn , "f ( Z1, Xn )) : (z 1, Xn ) E U1 } .

E

65

U" of yn
E : ( z1, x ) E U1 and Y Making the neighborhood U := { (z 1 Xn + i{) n n U"} small enough and correcting the sign if necessary, one can achieve that ,

u n G = { ( z1 , Xn

+ i yn ) E u :

Yn

< "f ( Z1, Xn ) } .

In particular, U n 8G = { z E U : g(z) = 0) is a (2n - I)-dimensional differentiable submanifold of U. 4.1

Let ac be smooth a t Zo, and let {! 1 ' {!2 be two local defining functions o n U = U(z0 ) . Then there is a 't'00 function h o n U such that: Lemma.

h > 0 o n U. 2. Q1 = h · [!2 o n U. 1.

3.

(d th)z = h(z) . (dg 2 )z fo r z

E

U n 8G.

PROOF: Define h = = {! 1 I {!2 on u ac. After a change of coordinates, we have zo = 0 and [!2 = Yn · Then g ( t) : = g l (z 1 , Xn + it) is a smooth function that vanishes at t = 0. Therefore, -

g ( yn ) - g ( O )

{Yn

Jo

g1 (s) ds

=

Yn ·

{1 Jo

g1 ( tyn ) dt

Q2 ( Z1 , Xn + i y n ) · h(z 1, zn ) ,

where

I

h( z , Xn + •I Yn )

=

10

1

8{!1 - ( z I , Xn + 1. tyn ) dt 8Y n

is smooth. For z E 8G we have (dgi )z = h (z) . (dg 2 ) z . Therefore, h(z) f- 0, and even greater than 0, since h(z) > 0 by continuity. • 4 . 2 Theor em. Let G c c e n be a bounded domain with smooth bounda ry. Then 8G is a differentiable submanifold, and there exists a global defining function.

PROOF: We can find open sets Vi CC Ui C e n , i = 1 , .. . , N , such that: 1 . { V1 , . . . , VN } is an open covering of ac. 2. For each i there exists a local defining function gi for G on Ui .

11.

66

Domains of Holomorphy

3. For each i there is

a

smooth function > 0 in general.

'Pi

:

Ui

-+ lR with

'Pil v,

- 1,

'Pi kn - u, 0, and 'Pi Define
0 on 8G) and '1/Ji 'Pi/'P· Then I:i 'l/Ji l on 8G. The system of the functions '1/Ji is called a partation of unity on 8G. The function g := 2::: �- I '1/Ji(!i is now a global defining function for G. We :=

leave it to the reader to check the details.

•

The Levi C ondition. For the remainder of this section let G c c en be a bounded domain with smooth boundary, and g : U = U(8G) -+ lR a global defining function. Then at any Zo E ac the real tangent space of the boundary is a

(2n -

I)-dimensional real subspace of Tz0 . The space

is called the complex (or holomorphic) tangent space of the boundary at zo. It is a (2n - 2)-dimensional real subspace of 7�0 , with a natural complex structure, so an ( n - 1 )-dimensional complex subspace3 . Definition. The domain G is said to satisfy the Levi condition (respec tively the strict Levi condition) at zo E 8G if Lev(g) is positive semidef init-e (respectively positive definite) on Hz 0(8G) . The domain G is called Levi convex (respectively strictly Levi convex) if G satisfies the Levi con dition (respectively the strict Levi condition) at every point z E 8G. Remark. The Levi conditions do not depend on the choice of the boundary function, and they are invariant under biholomorphic transformations. If

{! 1 = h . {!2,

with h > 0, then for

Z

E 80,

Lev(gi )(z , w ) = h (z) . Lev(g2) (z , w ) + 2 Re{ (8h)z ( w ) . (8g2)z (w ) } . So on Hz (8G) the Levi forms of g1 and g2 differ only by a positive constant. Aftme C onvexity. Recall some facts from real analysis:

A set M c JRn is convex if for every two points x , y E M , the closed line segment from x to y is contained in M . In that case, for each point xo E JRn - M there is a real hyperplane H c JRn with x0 E H and M n H = o . This property was already used in Section 1. 3

Hz (8G)

is often denoted by Tl •0 (8G).

4.

If a

E �n , U

=

Levi Convex Boundaries

67

U (a) is an open neighborhood and r.p U --+ � is at least �2 , :

then the quadratic form

is known as the Hessian of r.p at 4.3

a.

Proposition. Let G CC �n be a domain with smooth boundary, and {! a global defining function with ( dg) x -=!=- 0 fo r X E ac. Then G is convex if and only if Hess(g) is positive semidefinite on every tangent space Tx (aC) . Let G be convex, and Xo E ac an arbitrary point. Then T : = Tx0 (aC) is a real hyperplane with T n C = 0. For w E T and a(t) :=xo + tw we have (g o a) " ( O ) = Hess(g) (xo , w) .

PROOF:

Since g(xo) = 0 and g o a(t) > 0, it follows that g o a has a minimum at t = 0. Then (g o a)" (O) > 0, and Hess(g) is positive semidefinite on T. Now let the criterion be fulfilled, assume that 0 E G, and define g6 by

For small E and large N the set G, := {x : g6(x) < 0} is a domain. We have G, C C6 C G for E' < c:, and Uc;> O C6 = G. Therefore, it is sufficient to show that G, is convex. ,

The Hessian of {!c; is positive definite on Tx (ac) for every X E ac. Thus this also holds in a neighborhood U of aC. If E is small enough, then aC6 C U. We consider S := { (x, y) E C6

x

G, : txt ( 1 - t ) y E G, for O < t < 1 ) .

Then S is an open subset cf the connected set G, x G,. Suppose that S is not a closed subset. Then there exist points x0 , y0 E G, and a to E (0, 1) with toxo+ ( 1 - to)Yo E ace; . So the function t H {!c; 0 a(t) , with a (t) :=txo + ( 1 - t)yo , has amaximum at to. Then (g6 oa) " (t0) < 0 and Hess(g6) (a(t0 ) , x0 - y0) < 0. This is a contradiction. •

A domain G = {g < 0) is called strictly convex at xo E ac if Hess(g) is positive definite at x0. This property is independent of g and invariant under affine transformations. Now we return to Levi convexity. 4.4

Lemma.

Let U C en be open and r.p E � 2 (U; �) . Then

Lev( r.p) (z, w)

=

1

4

(Hess( r.p ) (z , w ) + Hess( r.p ) (z, iw ))

68

II.

Domains

oE

Holomorphy

This is a simple calculation!

PROOF:

•

4.5

Theor em. Let G c c en be a dom ain with smooth boundary. Then the following statements are equivalent: 1.

G is strictly T�evi convex. 2. The re is an open neighborhood U = U( EJG) and a strictly plurisubhar monic function g E 'l!f= (u; JR) such that U n G = {z E U : g(z) < 0) and (dg)z -=/:- 0 fo r z E U. 3. For eve ry z E EJG there is an open neighborhood W = W(z) C en , an open set V c en, and a biholomorphic map F : W --+ V such that F(WnG) is convex and even strictly convex at every point of F(W n EJG) . PROOF:

( 2) : We choose a global defining function g for G, and an open neighborhood U = U ( EJG) such that g is defined on U with (dg)z -=1- 0 for z E U. Let A > 0 be a real constant, and {!A : = eA e - 1. Then {!A is also a (1 )

===?

global defining function, and

Lev(gA)(z , w) The set K : = EJG

X

= Ae

A e (z ) [Lev(g) (z, w) + A l (8g)z (w) l 2 ]

•

S2n- l is compact, and

K0 : = {(z, w) E K : Lev(g) (z, w) < 0 } is a closed subset. Since Lev(g) is positive definite on Hz ( EJG) , we have ( ag)z (w) -=1- 0 for (z,w) E K0 . Therefore,

min Lev(g) (z , w) > - oo ,

M

K

min l (8g)z (w) l 2 > 0 .

C

Ko

We choose A so large that A ·

C +M

> 0. Then

Lev(gA )(z , w) = A · [Lev (g) (z, w) + A l (8g)z(w )l 2 ] > A · (M + AC) > 0 for (z, w)

E

K0 , and

Lev (gA)(z , w) > A 2 · 1 (8g)z(w)l 2 > 0 for (z, w) E K - Ko. So Lev(gA)(z, w) > 0 for every z E EJG and every w E en - {0}. By conti nuity, {!A is strictly plurisubharmonic in a neighborhood of EJG. (2 )

===?

( 3 ) : We consider

transformations:

a

point zo E EJG and make some simple coordinate

4.

Levi Convex Boundaries

69

By the translation z H w = z zo we replace zo by the origin, and a permutation of coordinates ensures that (!w1 (0) -=/=- 0. -

The linear transformation gives u1 = w . V' g(O) t, and therefore g( u )

-

2 Re (u . V'(g o w) (O) t ) + term sofdegree > 2 2 Re (u . Jw (O) t . V' g(O) t ) + terms of degree > 2

2 Re (w . V' g(O) t ) + terms of degree > 2 - 2 Re( u1) + terms of degree > 2. -

Finally, we write g( u) = 2Re(u1 + Q(u)) + Lev(g) (O, u) + . . . , where Q is a quadratic holomorphic polynomial, and make the biholomorphic transforma tion It

follows that

+ Lev(g) (O, v) + terms of order

g(v) = 2Re(v1)

> 3.

By the uniqueness of the Taylor expansion g(v)

=

Do(O) (v)

+ �Hess(g) (O, v) + terms of order

> 3,

and therefore Hess(g)(O, v) = 2 . Lev(g) (O, v) > 0 for v -=/=- 0 (in the new coordinates). Everything works in a neighborhood that may be chosen to be convex. (3)

===?

( 1 ) : This follows from Lemma 4.4: Hess(g) > 0 on Tz (8G)

===?

Lev(g) > 0 on Hz (8G) .

The latter property is invariant under biholomorphic transformations.

•

A Theorem of Levi. Let G cc en be a domain with smooth boundary. If G is strictly Levi convex, then it is easy to see that G is pseudoconvex.

We wish to demonstrate that even the weaker Levi convexity is equivalent to pseudoconvexity. For that purpose we extend the boundary distance to a function on en. dc(z)

:=

8c(z) 0 -8cn _c (z)

for z E G, for z E 8G, for z !f. G.

70

II.

Domains of Holomorphy

4.6 Lemma.

-de is

a

smooth defining function fo r G

PROOF: We use real coordinates x = ( x 1 , . . . , X N ) with N = 2n. It is clear that G = { x : -de( x) < 0} . Let Xo E ac be an arbitrary point and (! : U (xo) -+ JR. a local defining function. We may assume that Qx N (xo) -=/=- 0. Then by the implicit function theorem there is a product neighborhood U' x U" of xo in U and a smooth function h : U' -+ JR. such that { (x' xN ) E U' x U" : g(x' , xN ) = 0 } = { (x jl,( x')) : x' E U' } ,

'

It follows that O = "Vx' Q(x', h(x')) + gxN (x' , h (x')) . Vh(x ') . At the point (x' , h(x')) E 8G the gradient "Vg(x', h(x')) is normal to 8G and directed outward from G. Every point y in a small neighborhood of the boundary has a unique representation y = x + t . V g(x) , where t = -de( Y ) and X is the point where the perpendicular from y tO 80 meetS the boundary. Therefore, we define the smooth map F : U' x JR. -+ JR.N by y

F (x ' , t )

=

=

(x', h(x' )) + t . V g(x' , h (x')) .

Then there are smooth functions =

JJR,F (x', t) and therefore

A

and b such that

( VEhN(x') - I + t A(x') + t b(x') ·

.

E det ( N - I "V h (x')

)

h (x')) V h (x ') t ' (x gx , N det JJR,F(x' , 0) Qx N (x' , h(x')) "V h (x') t EN - 1 - (!x N (x' , h (x )) . de t O' 1 + II "V h (x' ) l l 2 - (!xN (x', h(x'))( l + 11Vh( x') l l 2 ) -=/=- 0. 1

·

(

)

It follows that there exists an c > 0 such that F maps U' x ( - s, s) diffeo morphically onto a neighborhood W = W (x0 ) , and U' x {0} onto ac n W . Moreover, since de (x + t V g(x) ) = -t for I t I < c and c small enough, it follows that de = ( t)J y - 1 is a smooth function near ac. If p ' is defined by p' (x', t ) := (x' ,0), then the projection ·

-

p = p'

o

y- 1 : X + t \l g(x) f-+ X, for X E 80, ·

is a smooth map, and de is given by de( Y ) = CJ . IIY - P( Y ) II , where CJ = 1 for y E G and CJ = - 1 elsewhere. For y # 8G we have

4.

Levi Convex Boundaries

71

N

- Pk ( y )) (8k v - (Pk )yv (Y)) ( · Yk : � II IIY (Y) p ( y ) j p (Y)) N J Y) IIY -:( y )ll · [Yv - Pv ( - (y _

Yv

-

and therefore

'

a . [y - p ( y) - Dp ( y ) ( y - p (y )) ] . IIY - P ( Y ) II Since g(p( y )) 0, it follows that Dp( y )("V g( p ( y ))) = 0. But y - p ( y ) is a multiple of \1 g( p ( y )). Together this gives . "Vg( p ( y)) y - p ( y) "V de(Y ) = a · II Y - (Y) II = ± II g( ( ))ll . P "V p y If y tends to 8G, we obtain that "Vde( Y ) of. 0. • "Vde(Y)

=

E.E. Levi showed that every domain of holomorphy with smooth boundary is Levi convex, and locally the boundary of a strictly Levi convex domain G is the "natural boundary" for some holomorphic function in G. Here we prove the following result, which is sometimes called "Levi's theorem". 4.7

Theorem. A domain G with smooth boundary is pseudoconvex if and only if it is Levi convex.

PROOF :

( 1 )Let G be pseudoconvex. The function -de is a smooth boundary function for G, and - log de = log 8e is plurisubharmonic on G, because of the pseudoconvexity. We calculate -

1

1

Lev( -de )(z , w) + I ( 8(de )) z (w) 1 2 · · de (z)2 de(z ) This is nonnegative in G. If z E G, w E T,, and ( 8 (de)) z (w) = 0, it follows that Lev(-de) (z , w) > 0. This remains true for z -+ 8G, so - de satisfies Lev( - log de )(z, w)

=

the Levi condition.

(2) Let G be Levi convex, and suppose that G is not pseudoconvex. Then in any neighborhood U of the boundary there exists a point z0 where the Levi form of - log 8e has a negative eigenvalue. This means that there is a vector wo such that

'P(((O) = Lev(log 8e) (z0, wo) > 0, for cp(() := log 8e (zo t (wo) . Consider the Taylor expansion

cp(()

) cp(O) + 2 Re(cp((O)( + 2 cp(((0 )( 2 ) + 'P(((O) I ( I 2 + - cp(O) t Re(A( + B(2 ) + >-1( 1 2 + . . . ,

72

II.

Domains of Holomorphy

with complex constants A , B and

a

real constant >. > 0.

We choose a point Po E 8G with 8e (zo) = IIPo - zo l l , and an arbitrary c > 0. Then an analytic disk 'ljJ : Dc:(O ) --+ en can be defined by

'lj;(() := Zo + ( wo + exp (A( + B(2) (Po - zo). We have 'lj; (O) = p0 , and we wish to show that 'ljJ (() E G, for 0 < ICI < c and E sufficiently small.

Since cp(( ) > cp(O) + R e(A( + B( 2 ) + (>. / 2 ) 1(1 2 near ( 8e (zo + (wo)

=

0, it follows th at

exp(cp( ( )) > exp ( cp (O)) I exp (A( + B( 2 ) I exp > 8e(zo) I exp (A( + B( 2 ) I - ll exp (A( + B( 2 ) (Po - zo) II , ·

·

·

C 1(1 2)

for ( small and -=f. 0 . This means that we can choose the E in such a way that 't/J(() E G, for 0 < ICI < c. The analytic disc is tangent to ac from the interior of G. Now f( () = de ( 't/J ( ( )) is a smooth function with a local minimum at ( Therefore ( 8de)p0 ('t/J'(O)) = (8! )0 (1) = 0 , and

f( ( )

=

Re (!((( 0 ) ( 2 ) + !((1(1 2 +terms of order >

Since Re ( f 0 for every t, it follows that Lev( de )(po , 't/J' (0 ))

=

0.

3.

f(((O) > 0 . This is a contradiction to the Levi condition at p0, because -de is a defining function for G.

=

•

Exer cises 1 . Prove Lemma 4.4. 2. Assume that G c c C2 has a smooth boundary that is Levi convex outside a point a that is not isolated in 8G. Show that G is pseudoconvex. 3. Assume that G C C2 is an arbitrary domain and that S c G is a smooth real surface with the following property: In every point of S the tangent to S is not a complex line. Prove that for every compact set K c G there are arbitrarily small pseudoconvex neighborhoods of S n K. 4. Assume that G c c C2 is a domain with smooth boundary. Then G is strictly Levi convex at a point Zo E ac if and only if the following condition is satisfied: There is a neighborhood U = U (zo), a holomorphic function cp : D --+ U with cp (O) = zo and cp'(O) -=f. 0, and a local defining function g on U such that (g o cp)(( ) > 0 on D - {0} and (g o cp) (((O ) > 0 .

5.

Holomorphic Convexity

73

5. Let G c c en be a domain with smooth boundary. If G satisfies the strict Levi condition at zo E 8G, then prove that the following hold: (a) There is no analytic disk r.p : D --+ en with

r.p (O) = z0

and

lim (-+ o

bc( r.p (()) llr.p ( ( ) - r.p (O) II 2

=

0

(b) There are a neighborhood U = U(z0) and a holomorphic function f in U with G n { z E U : f (z) = O} = {zo } . 6. A bounded domain G c en is called strongly pseudoconvex if there are a neighborhood u = u (ac) and a strictly plurisubharmonic function g E '6'2 (U) such that G n U = { z E U : g(z) < 0} . Notice that a strongly pseudoconvex domain does not necessarily have a smooth boundary ! Prove the following results about a strongly pseudoconvex bounded do main G: (a) G is pseudoconvex. (b) If G has a smooth boundary, then G is strictly Levi convex. (c) For every z E 8G there is a neighborhood U = U (z) such that U nG is a weak domain of holomorphy. 7. Let G c en be a pseudoconvex domain. Then prove that there is a family of domains G, C G such that the following hold: (a) G, CC Gv+l for every v. ( b) U:;" 1 Gv = G. (c) For every v there is a strictly plurisubharmonic function fv E '6'00 (Gv+I ) such that G, is a connected component of the set

{z

5.

E Cv+ I

:

fv (z) < 0} .

Holomorp h ic C on vexity

Affine C onvexityWe will investigate relationships between pseudocon vexity and affine convexity. Let us begin with some observations about convex domains in lRN . Let 2 be the set of affine linear functions f : JRN --+ lR with

f(x) = a 1x1 + . . . + aN X N t b,

a 1 , . . . , aN , b E lR.

If Jvf is a convex set and x0 a point not contained in M, then there exists a function f E 2 with f(x0) = 0 and J I M < 0. For any c E JR, the set {x E JRN : f (x) < c} is a convex half-space.

Definition .

Let

H( M)

M

c JRN be an arbitrary subset. Then the set

:= {x E JRN : f (x) < sup f, for all f E 2

is called the affine convex hull of M.

M

}

74

11.

Domains of Holomorphy

5. 1 Pr oposition. Let M, M1 , M2 c lRN be arbitrary subsets. Then 1.

M c H (M) . 2. H ( M) is closed and convex. 3.

H ( H ( M ))

4. If M1

c

=

H ( M) .

M2 , then H(M1 )

c

H(M2). 5. If M is closed and convex, then H(M) = M . 6. If M is bounded, then H ( M ) is also bounded.

PROOF:

( l ) is trivial.

5e

with f (xo) > supM f . By conti nuity, f(x) > sup, f in a neighborhood of xo. Therefore, H(M) is closed.

(2) If x0 rf_ H ( M ) , then there is an f E

If x0 , y0 are two points in H ( M ) , then they are contained in every convex half-space E = { x : f ( x) < sup M f } , and also the closed line segment from x0 to y0 is contained in each of these half-spaces. This shows that H(M) is convex.

( 3) We have to show that H (H( M )) C H(M) . If x

E H(H(M)) is an arbi

trary point and f an element of 2', then f (x) < supH ( M ) f < supM f , by the definition of H(M). (4) is trivial.

( 5) Let M be closed and convex. If xo rf_ M, then there is a point Yo

E M

such that d ist (xo , M) = dist(x0, Yo ) (because M is closed). Let zo be a point in the open line segment from x0 to yo. Then z0 rf_ M , and there is a function f E 5e with f(zo) = 0 and JI M < 0. Since t H f(txo + ( 1 - t)yo) is a monotone function, it follows that f(x0 ) > 0 and therefore x0 rf_ H ( M ). (6) If M is bounded, there is an R > 0 such that M is contained in the closed • convex set BR (O) . Thus H( M ) C BR (O). Remark.

H(M) is the smallest closed convex set that contains M .

5 . 2 Theor em . A domain that H(K) c c G.

G c JRN is convex if and only if K c c G implies

PROOF: Let G be a convex domain, and M c c G a subset. Then H(M) is closed and contained in the bounded set H( M ). Therefore, H(M) is compact, and it remains to show that H(M) C G. If there is a point xo E H(M) - G, then there is a function f E 5e with f (x0) = 0 and fi e < 0. It follows that sup M f < 0, and f(xo) > supM f . This is a contradiction to xo E H( M ).

5.

Holomorphic Convexity

75

On the other hand, let the criterion be fulfilled. If xo , Yo are two points of G, then K := { xo, Yo } is a relatively compact subset of G. It follows that H(K) is contained in G. Since H(K) is closed and convex, the closed line segment • from xo to Yo is also contained in G. Therefore G is convex.

H olomor phic C on vexity. Now we replace affine linear functions by holomorphic functions. Definition.

Let G C en be a domain and K c G a subset. The set

K = Kc :

{z E G

:

l f(z) l < s� I J I , for all f E

O(G) }

is called the holomorphically convex hull of K in G.

5 . 3 Pr oposition. Then

Let G c en be a dom ain, and K, K1 , K2 subsets of G.

1. K c K . 2. K is closed in G. 3. K = K . 4. If K1 C K2 , then K1 C K;. 5. If K is bounded, then K is also bounded. � �

�

�

PROOF:

( 1 )is trivial. �

(2) Let z0 be a point of G - K. Then there exists a holomorphic function f on G with l f(zo ) l > su pK i f l . By continuity, this inequality holds on an entire neighborhood U = U(zo) c G. So G - K is open. �

(3 ) SUPx l f l

tmPK i f l .

(4) is trivial.

(5) If K is bounded, it is contained in a closed polydisk pn (O, r ) . The coordi nate functions Zv are holomorphic in G. For z E K we have l zv l < supK i zv l I • r. Hence K is also bounded. �

Definition . A domain G c en is called holomorphically convex K cc G implies that K CC G. �

if

II.

76

Domains of Holomorphy

Example

In C every domain is holomorphically convex: Let K cc G be an arbitrary subset . Then K is bounded, and it remains to show that the closure of K is contained in G. If there is a point z0 E R - G, then Zo lies in aR n aG. We consider the holomorphic function f(z ) : = 1 /(z - z0) in G. If (z,_, ) is a sequence in K converging to zo, then l f ( z,_, ) l < supKifl < supKI ! I < oo . This is a contradiction. For n > 2, we will show that there are domains that are not holomorphically convex. But we have the following result. �

�

5.4

Pr op osition . If G morphically convex.

C

C" is an affine convex domain, then it is holo

PROOF: Let K be relatively compact in G. Then H ( K ) C C G. If zo is a point of G - H ( K ) , then there exists an affine linear function >. E £ with >.( zo) > supK >. . Replacing >. by >. - >. ( 0 ) we may assume that >. is a homo geneous linear function of the form >.(z)

=

2 Re(a 1 z1 +

·

·

·

+ a , z, ) .

Then f(z) := exp ( 2 . (a 1 z1 + . . . + a, z, )) is holomorphic in G, and

lf (-3 ) 1

=

exp ( >. ( z)) . Therefore, l f (zo )l > supKifl, and zo E G - k .This proves K c c • G. In general, holomorphic convexity is a much weaker property than affine convexity.

The C artan-Thullen Theor em . Let G c C" be a domain, and E > 0 a small real number. We define G, : =

{Z

E G : 8c (z) > s}.

Here are some properties of the set G,: 1 . If z E G, then there is an E > 0 such that 8c(z) > c . Therefore, G = U c: >O G,. 2. If c1 < c 2 , then G, => Gc: 2 • 3. G, is a closed subset of C" . In fact, if zo E C" - G, then 8c (zo) < c or z0 rf. G. In the latter case, the ball B, (zo) is contained in C" - Gc:. If z0 E G - Gc: and 8 := 8c(zo ) , then Bc:-o (zo) C C" - G,. So C" - G, is open.

f

a holomorphic function in G. If K C G, thenfo r any 8 with 0 < 8 < E there exists a constant C > 0 such that the following inequality holds:

5 . 5 Lemma. Let G c C" be a dom ain, K C G a compact subset, and

5.

77

Holomorphic Convexity

PROOF: For 0 < 8 < E, G := { z E G : dist(K, z) < 8} is open and relatively compact in G, and for any z E K the closed polydisk Pn (z , 6) is contained in G' c G. If T is the distinguished boundary of the polydisk and l f l < C on G', then the Cauchy inequalities yield a! . a! a f s < p < I D (Z ) I 8 1al � l f l 8 1al . C. •

5.6 Theor em (Car tan-Thullen). If G is a weak dom ain of holomorphy, then G is holomorphically convex. �

Let K CC G. We want to show that K c c G. Let dist(K, en - G) > dist (K, en - G) > 0. Clearly, K lies in G,. PROOF:

E

We assert that the holomorphically �onvex hull R lies even in G,. Suppose this is not so. Then there is a z0 E K - G,. Now let f be any holomorphic function in G. In a neighborhood U = U(z0 ) C G, f has a Taylor expansion

f(z)

= L a,_, (z

v>O

- zo) " , with

a,

=

1D" f(z0 ). v!

The function z H a,_, ( z) := ;, D " f ( z) is holomorphic in G. Therefore, l a,_, (zo) l < supK i a,_, (z) l . By the lemma, for any 6 with 0 < 8 < E there exists a C > 0 such that supK ia,_, (z) l < C/8 1 '-' l , and then

On any polydisk pn (z0 , 6) the Taylor series IS dominated by a geometric series. Therefore, it converges on P = pn ( z0 , E) to a holomorphic function f. We have f = f near z0 , and then on the connected component Q of zo in P n G. Since P meets G and en - G, it follows from Lemma 1..9 that there is a point Z l E p n 8Q n ac. Then f cannot be completely singular at z l . This is a contradiction, because f is an arbitrary holomorphic function in G, and G is a weak domain of holomorphy. • �

�

E xer ci se s

1 . Let G1 c G2 c en be domains. Assume that for every f E O(G1 ) there is a sequence of functions f,_, E O(G2 ) converging compactly on G1 to f. Show that for every compact set K c Gl it follows that Kc2 n cl = Kc l '

78

II.

Domains of Holomorphy

2. Let F : G 1 --+ G2 be a proper holomorphic map between domains in en ,

respectively em . Show that if G2 is holomorphically convex, then so is G1 . 3. Let G c en be a domain and S c G be a closed analytic disk with boundary bS. Show that S C (bS ) c· 4. Define the domain G C e 2 by�G := P 2 (0, 1 ) - P 2 (0, 1/2). Construct the holomorphically convex hull Kc for K : = { ( z1 , z2 ) : z1 = 0 and l z 2 l = 3/4}. Is Kc a relatively compact subset of G ? 5. Let F be a family of functions in the domain G. For a compact subset K c G we define -

{

K;: : = z E

}

G : lf(z ) l < s� l f l for all f E F . �

The domain G is called convex with respect to F, provided that K;: is relatively compact in G whenever K is. Prove: (a) Every bounded domain is convex with respect to the family 'l!f0 (G) of all continuous functions. (b) The unit ball B = B1 (0) is convex with respect to the family of holomorphic functions zi . z� with v, fJ, = 1 , . . . , n and k, l E No .

S in gu lar Fun ction s

6.

Nor m al Exh au stion s .Let G c en be a domain. lf G is holomorphically convex, we want to construct a holomorphic function in G that is completely singular at every boundary point. For that we use "normal exhaustions." Definition . A normal exhaustion of G is a sequence ( K, ) of compact subsets of G such that: 1 . K, C C ( Kv+ l t , for every v. 2. U::" 1 Kv = G .

6 . 1 Theor em. Any domain G in en admits a normal exhaustion. ff-G I S holomorphically convex, then there is a normal exhaustion (K, ) with K, K, fo r eve ry v. In the general case, K, := Pn (o, v) n G 1 ;v gives a normal exhaustion. 1f G is holomorphically convex, K" c c G for every v. We construct a new exhaustion by induction. PROOF:

Let Ki := K 1 . Suppose that compact sets Ki , . . . , K�-1 have been con structed, with Kj = Kj for j = 1, . . . , v - 1 , and Kj CC (JC;*+ 1 t . Then there exists a >.(v) E N such that K�_ 1 c ( K;.. ( v) t . Let K� : = K;.,(v) ·

6. Singular Functions �

It is clear that (K� ) is a normal exhaustion with K�

=

79

K� .

•

Unb ounded Holomor phic Function s . Again let G c en be a domain. �

6 . 2 Theor em. Let ( K, ) be a normal exhaustion of G with Kv = K, >. (J.L) a strictly monotonic increasing sequence of natural numbers, and (zJ.L) a sequence of points with zJ.L E K;.. ( J.L) + l - K;.. c,"l .

Then there exists a holomorphic function bounded.

f

in G such that

l f (z J.L ) I

is un

PROOF: The function f is constructed as the limit function of an infinite series f = 2:;:" 1 fw By induction we define holomorphic functions fJ.L in G such that:

2 - J.L for J.L > 1. J.L - 1 2 . I JJ.L(zJ.L) I > J.L + 1 + Ll fj (zJ.L) I for J.L > 2 . j=l 1.

I JJ.L I K>-<"l

<

:=

Let h 0. Now for J.L > 2 suppose that h , . . . , fJ.L - 1 have been constructed. Since zJ.L E K;..( J.L) + 1 - K;..( J.L) and K;.. ( J.L) = K;.. ( J.L) there exists a function g holomorphic in G such that l g(z J.L) I > q := supKA ( J.' ) lgl. By multiplication by a suitable constant we can make ,

If we set

and ( 2 ) .

fJ.L := g k

l g (zJ.L) I

> 1 > q.

with a sufficiently large k , then

fJ.L

has the properties ( 1 )

We assert that I: J.L JJ.L converges compactly in G . To prove this, first note that for K C G an arbitrary compact subset, there is a J.Lo E N such that J.L K C K;.. ( J.L ) · By construction supKifJ.L I < 2 - for J.L > J.Lo. Since the geo metric series I: 2 - J.L dominates I: JJ.L in K , the series of the JJ.L is normally convergent on K. This shows that r = 2: /L JJ.L is holomorphic in G. Moreover, o

l f(z /L ) I

>

I JJ.L(z /L ) I - L l fv(z !L ) I v # J.L

>

J.L + 1 -

(since z/L E K ( v )

(since L Tv = 1). v� 1 lf(zJ.L) I --+ oo for J.L --+ >

It follows that

L 2-v

>-.

for

v > J.L)

J.L

=·

•

1 1 . Domains of Holomorphy

80

The following is an important consequence: A

domain G is holomorphically convex if and only if fo r any infinite set D that is discrete in G there exists a function f holomorphic in G such that l f l is unbounded on D .

6 . 3 Theor em.

( l ) Let G be holomorphically convex, D C G infinite and discrete. Moreover, let ( Kv) be a normal exhaustion of G with Kv = K,. Then Kv n D is finite (or empty) for every v E N. We construct a sequence of points zJ.L E D by induction. PROOF:

�

Let z 1 E D - K1 be arbitrary, and >.(1) E N minimal with the property that z1 lies in K>- ( 1) + 1 · Now suppose the points z1 , . . . ,zJ.L- 1 and the numbers >. ( 1 ) , . . . , >. (J-L - 1 ) have been constructed such that z, E K.A ( v) +1 - K.A (v) • for v = 1 , .. . , /-L

-

1.

Then we choose zJ.L E D - K.A (J.L- 1)+1 and A(J-L) minimal with the property that zJ.L lies in K.A(J.L l + 1 . By the theorem above there is a holomorphic function f in G such that l f(zJ.L) I -+ oo for J-L -+ oo. Therefore, l f l is unbounded on D. (2) Now suppose that the criterion is satisfied, and K CC G. Then K C G, and we have to show that K is compact. Let (z,) be any sequence of points in K. Then �

�

sup{ l f(zv)l

:

v E N} <

sup l f l < oo , K

for every

f E O(G)

Therefore, { zv : v E N} cannot be discrete in G . Thus the sequence ( zv) has a cluster point z0 in G. Since K is closed, zo belongs to K. So G is • holomorphically convex. �

-

Sequences . For a domain G c en we wish to construct a sequence that

accumulates at every point of its boundary.

Let ( K, ) be a normal exhaustion of G . Then there exists a strictly monotonic increasing sequence A (J-L) of natural numbers and a se quence (zfl.) of points in G such that:

6 . 4 Theor em. 1.

zfl. E K.A(J.L) + 1 K.A(J.L) • for every f-L · 2 . If zo is a boundary point of G and U = U(z0) _

an open connected neigh borhood, then eve ry connected component of U n G contains infinitely many points of the sequence (z, ) .

PROOF: This is a purely topological result, since we make no assumption about G. The proof is carried out in several steps.

81

6 . Singular Functions

{ B" : v E N} be the countable system of balls with rational ( 1 ) Let B center and rational radius meeting 8G. Every intersection B, n G has at most countably many connected components. Thus we obtain a countable family =

C

= { CJ.L :

3

v E N such that CJ.L is a connected component of B, E B}.

(2) By induction, the sequences >-.(fl,) and (zJ.L) are constructed. Let z1 be arbitrary in CI - KI . Then there is a unique number >. (I) such that z i E K>-(1)+1 - K>-( I ) · Now suppose z i , . . . , zJ.L - I have been constructed such that Zj E cj n (KA(j)+l - K,A(j ) ), for j = 1 , . . . , f-L - 1.

We choose zJ.L E CJ.L - K>-(J.L - IJ+l and >-.(J-L) as usual. That is possible, since there is a point w E BV(J.L) n acfl. n ac if CJ.l. is a connected component of B v (J.L ) n G. Then en - KA(J.L-I ) +I is an open neighborhood of w and contains points of C,. (3) Now we show that property (2) of the theorem is satisfied, Let z0 be a point of ac, U = U(zo ) an open connected neighborhood, and Q a connected component of U n G. We assume that only finitely many zJ.L lie in Q , say ZI , . . . ,z m . Then U * := U - {z i , . . . , z m }

and

Q* : = Q - {z 1 , . . . , zm }

are open connected sets that contain no z,. Obviously, Q* is a connected component of G n U * . There is a point wo in U* n 8Q* n 8G, and a ball B, C U* with wo E B,. Then B, n G c U* n G. Moreover, B, n G must contain a point WI E Q*. The connected component C* of w1 in B, n G is a subset of the connected component of WI in U* n G. But C* is an element CJ.Lo of C. By construction it contains the point zJ.Lo . That is a contradiction. Infinitely many members of the sequence belong to Q. • 6.5 Theor em . If G is holomorphically convex, then it is a dom ain

morphy.

d

holo

�

PROOF: Let ( K, ) be a normal exhaustion of G with K = K, and choose " sequences A(J-L) E N and (z, ) in G such that z, E K>-(J.Ll+l - K>-(J.L) · We may assume that for every point z0 E ac, every open connected neighborhood U = U(z0) , and every connected component Q of U n G there are infinitely many z, , in Q.

Now let f be holomorphic in G and unbounded on D := {zJ.L clear that f is completely singular at every point Zo E 80.

:

f-L

E N} . It is •

82

II. Domains of Holomorphy

Remark. It is not necessary that a completely singular holomorphic func tion is unbounded. In 1978, D. Catlin showed in his dissertation that if G cc en is a holomorphically convex domain with smooth boundary, then there exists a function holomorphic in G and smooth in a neighborhood of G that is completely singular at every point of the boundary of G. Exer cises 1 . A domain G c c en is holomorphically convex if and only if for every z E 8G there is a neighborhood U(z) such that U n G is a domain of holomorphy. 2 . Let G1 c en and G2 c em be domains of holomorphy. Iff : G1 --+ e m is a holomorphic �pping then j - 1 ( G2) n G1 is a domain of holomorphy. 3 . Find a bounded holomorphic function on the unit disk D that is singular at every boundary point. ,

7.

E xamp les and Ap p lication s

Domains of Holomorphy 7 . 1 Pr op osition. holomorphy.

Every doma in in the complex plane e zs a domain of

PROOF: We have already shown that every domain in e is holomorphically • convex. Therefore, such a domain is also a domain of holomorphy.

7 . 2 Theor em. alent:

Th efollowing statements about domains G E en are equzv

1 . G is a weak doma in of holomorphy. 2. G is holomorphically convex. 3. For every infinite discrete subset D C G there exists a holomorphicfunc tion f in G such that I ! I is unbounded on D . 4. G is a doma in of holomorphy. The equivalences have all been proved in the preceding paragraphs. Fur thermore, we know that every domain of holomorphy is pseudoconvex. Still missing here is the proof of the Levi problem: Every pseudoconvex domain is holomorphically convex. We say more about this in Chapter V. Every affine convex open subset of en is a domain of holomorphy. The n-fold Cartesian product of plane domains is a further example. 7 . 3 Pr op osition. If G1 , . . . , Gn C e are arb itrary domains, then G G 1 x · · · x G is a domain d holomorphy. n

:=

7 . Examples and Applications

83

PROOF: Let D = {zJ.L = (zf , . . . , zf:: ) : f-L E N} be an infinite discrete subset of G. Then there is an i such that (zf) has no cluster point in Gi , and there is a holomorphic function f in Gi with limJ.L--> = I f ( zf) I = = · The function f in G, defined by [(z1 , . . . , z ) : = f (zi ) , is holomorphic in G and unbounded n on D. •

Remar k. The same proof shows that every Cartesian product of domains of holomorphy is again a domain of holomorphy.

C omplete Reinhar dt Domains . Let G c en be a complete Rein hardt domain (see Section 1 . 1 ) . We will give criteria for G to be a domain of holomorphy. For that purpose we define a map log from the absolute value space "f! to JRn by

Definition . A Reinhardt domain G is called logarithmically convex if log r(G n (C* )n) is an affine convex domain in JRn . Remark. For z = (z1 , . . . , zn ) E G we have lo g r (z ) = (loglz1 l , . . . , logiz i ) . n If z E (C* )n, then i zil > 0 for each i , and log r(z) is in fact an element of JRn . 7.4 Pr oposition. The doma in d convergence of a power series S(z) Ev > o av z " is logarithmically convex. PROOF: Let G be the domain of convergence of S(z) , and M : = log r(G n (C* )n) c JRn . We consider two points x , y E M and points z , w E G n (C* )n with log r (z) = x and log r(w) = y . If >. > 1 is small enough, >.z and :XW still belong to G n (C* )n. Since S(z) is convergent in >.z, Xw, there is a constant C > 0 such that l av l . >. 1 " 1 - l z" l < C

Thus

and

l av l . >. 1 " 1 . l w" l < C, for every v E N8

lav l . ;. l v l . l z" l t - l w" l l - t < C, for every v and 0 < t < 1 .

It follows from A bel's lemma that S(z) is convergent in a neighborhood of

This means that Zt E G and tx + ( 1 - t) y = log r (z t ) E M , for 0 < t < 1 . Therefore, M is convex. •

84

I I . Domains of Holomorphy

7 . 5 Pr op osition. Let G be a complete Reinhardt domain. If G is logarith mically convex, then it is holomorphically convex. PROOF: Let K be a relatively compact subset of G. Since G is a complete Reinhardt domain and K a compact subset of G, there are points z1 , . . . , z�,; E G n (C* )n such that

K c G'

:=

k

U P n ( O , qi )

i=l

C G,

where qi := T(z i ) ·

We consider the set Jlt = { m (z) = zv : v E N0} of monomials, which is a subset of O(G). For z E pn (o, Qi) and m E A we have Let Z

:=

{z 1 ,

•

•

,

•

l m(z) l

�

z �,; } . Then for z E K it follows that

<

sup lml K

<

sup lml G'

<

suplml, Z

for every m E A _ �

�

Suppose that K is not relatively compact in G. Then K has a cluster point Zo in aG, and it follows that l m(zo)l < SUP zlml , for every m E A _ Let h( z) : = log T ( z), for z E (C* ) n . Since G is logarithmically convex, the domain G0 := h( G n ( C* )n ) C JRn is affine convex. For the time being we assume that z0 E ( C* ) n . Then x0 := h(z0) E 8G0 , and there is a real linear function >.(x) = a1x1 + . . . + an X n such that >.(x) < >. (xo) for x E Go.

Let x = log T (z) be a point of G0, and u E JRn with Uj < Xj forj = 1 , . . . , n . Then eu1 < ex1 = l zj I , and therefore (since G is a complete Reinhardt do ,eun ) E G n (@*) " and u E Go . In particular, main) w = ( eu1 , •

•

•

>.(x) - naj = >.(x - nej ) < >.(x0), for every n E N.

Therefore, aj > 0 for j = 1 , .. . ,n. Now we choose rational numbers rj > aj and define >.(x) : = r 1 x1 + · . · + rn x n If we choose the r1 sufficiently close to aj , the inequality >. ( qi ) < >.(x0) holds for i = 1 , .. . , k, and it still holds after multiplying by the common denominator of the rj . Therefore, we may assume that the rj are natural numbers, and we can define a special monomial m0 by m0(z) := z? . . . z�n . Then l mo(zi ) l = e .A(q, ) < e >- (xo) = lmo(zo) l , for i = 1 , . . . k -

_

�

_

,

.

l mo ( zo) l > sup zlmo l , and this is a contradiction. 1f zo >t ( C * ) n , then after a permutation of the coordinates we may So

assume that ziol . . . zj0l =f. 0 and zj�1 = . . . = z�0l = 0. We can project on the space

7 . Examples and Applications

e1 and work with monomials in through as above.

variables

the

z

1

,

. . . , zz .

85

Then the proof goes

•

Now we get the following result: 7 . 6 Theorem. Let G c en be a complete Reinhardt domain. Then the following statements are equivalent: G 2. G 3. G 4. G 1.

is is is is

the dom ain of convergence of logarithmically convex. holomorphically convex. a dom ain of holomorphy.

a power se ries.

PROOF: We have only to show that if G is a complete Reinhardt domain and a domain of holomorphy, then it is the domain of convergence of a power series. By hypothesis, there is a function f that is holomorphic in G and completely singular at every boundary point. In Section 1 .5 we proved that for every holomorphic function in a proper Reinhardt domain there is a power series S' ( z ) that converges in G to f. By the identity theorem it does not • converge on any domain strictly larger than G.

An alytic Polyhed r a . Let G c en be a domain Definition . Let U C G, V1 , . . , Vk C e open subsets, and h , . . . , /k holomorphic functions in G. The set .

P := { z E U :

fi (z) E v; ,

for i =

l , ..

. k} ,

is called an analytic polyhedron in G if P CC U. If, in addition, V1 = . . . = Vk = D , then one speaks of a special analytic polyhedron in G. Remark. An analytic polyhedron P need not be connected. The set U in the definition ensures that each union of connected components of P is also an analytic polyhedron if it has a positive distance from every other connected component of P.

7 . 7 Theorem. Eve ry connected analytic polyhedron P in G is a dom ain of holomorphy. PROOF: We have only to show that P is a weak domain of holomorphy. If zo E 8P, then there is an i such that fi(zo ) E 8v; . Therefore, f(z) • (fi ( z ) - /i(z o )) - 1 is holomorphic in P and completely singular at zo .

86

II. Domains of Holomorphy

E xa mple

Let

q<

1 be a positive real number, and 2 P := { Z = (zl , z2) E C : l z1 l < l, l z2 l < l and l zl . z2 1

< q} .

Then P (see Figure 1 1.7) is clearly an analytic polyhedron, but neither affine

q

Figu r e 1 1 .7.

An

analytic polyhedron

convex nor a Cartesian product of domains. So the analytic polyhedra enrich our stock of examples of domains of holomorphy. We will show that every domain of holomorphy is "almost" an analytic poly hedron. 7. 8

Theorem. If G c en is a domain of holomorphy, then there exists a sequence (Pv) of special analytic polyhedra in G with P" CC Pv+ i and u� l p" = G . PROOF: Let (Kv ) be a normal exhaustion of G with Kv = K"" If z E 8Kv+l is an arbitrary point, then z does not lie in K, C (Kv+I t , and therefore not in Kv · Hence there exists a function f holomorphic in G for which q == supKJ f l < IJ(z) l . By multiplication by a suitable constant we obtain q < 1 < I J (z) l , and then there is an entire neighborhood U = U(z) such that I f I > 1 on U. �

Since the boundary 8Kv+I is compact, we can find finitely many open neigh borhoods Uv,j of Zv,j E aKv+l ' j = 1 , . . . , kv ' and corresponding functions !v,j holomorphic in G such that l !v,j l > 1 on Uv ,j , and 8Kv+l c U�v 1 Uv ,j · We define P" == {z E (Kv +l t : lfv,j (z)l < l for j = l , .. . , kv } · Clearly, K, C Pv C (Kv+ l t · Furthermore, M :=Kv+l - (Uv,l U . . . U Uv,kJ is a compact set with P" C M C (Kv+l t · Consequently, P" CC Kv+l · Thus P" is a special analytic polyhedron in G. It follows trivially that the sequence • (Pv) exhausts the domain G.

8. Riemann Domains over en

87

In the theory of Stein manifolds one proves the converse of this theorem. Exercises 1. If R is a domain in the real number space JRn , then

TR

= R + ilRn := {z E en : (Re(zi ), . . . , Re(z )) E R }

n is called the tube doma in associated with R . Prove that the following properties are equivalent: (a) R is convex. (b) TR is (affine) convex. (c) TR is holomorphically convex. (d) TR is pseudoconvex. Hint: To show (d) ::::::::} ( a) choose x0 , yo E R. Then the function cp ( ( ) - In 8rR (x0 + ((Yo - x0 )) is subharmonic in D. Since 8rR ( x +iy ) = 8R(x) , one concludes that t � - ln 8R ( x0 + t (y0 - x0 )) assumes its maximum at t = 0 or t = 1 . 2 . Let G c en be a domain. A domain G c en is called the envelope of holomorphy of G if every holomorphic function f in G has a holomorphic extension to G. Prove: (a) If R c JRn is a domain and H(R) its affine convex hull, then G := H(R) + ilRn is the envelope of holomorphy of the tube domain G = R + i JRn. (b) If G c en is a Reinhardt domain and G the smallest logarithmically convex complete Reinhardt domain containing G, then G is the envelope of holomorphy of G . Hint: Use the convex hull of log T(G) . 3. Construct the envelope of holomorphy of the domain Gq P2 ( 0, ( 1 , q)) u P2 (0, (q, 1)) . 4. A domain G c en is called a Runge do main if for every holomorphic function f in G there is a sequence (P v ) of polynomials converging com pactly in G to f. �

�

�

�

:=

Prove that the Cartesian product of n simply connected subdomains of e is a Runge domain in en. 5. A domain G C en is called polynomially convex if it is convex with respect to the family of all polynomials (cf. Exercise 5.5). Prove that every polynomially convex domain is a holomorphically convex Runge domain.

8.

R iem ann Dom a in s

over

en

Riemann Dom ains. It turns out that for general domains in en the envelope of holomorphy (cf. Exercise 7 .2) cannot exist in en . Therefore, we have to consider domains covering en .

II.

88

Domains of Holomorphy

Definition . A (Rzemann) domain over e n is a pair (G, 1r) with the following properties: 1. G is a connected Hausdorff space. 4 2. 7r : G -+ e n is a local homeomorphism (that is, for each point X E G and its "base point" Z := 1r( X ) E e n there exist open neighborhoods U = U(x) c X and V = V(z) c en such that 1r : U -+ V is a homeomorphism). Remar ks

1. Let ( G, 1r) be a Riemann domain. Then G is pathwise connected, and

the map 1r : G -+ e n is continuous and open. The latter means that the images ci open sets are again open. 2. If (G, 1r") are domains over e n for v = 1 , .. . , l, and x, E G, are points over the same base point z0 , then there are open neighborhoods U" = Uv (xv) c G, and a connected open neighborhood V = V(zo ) C e n such that 1l"v luv : U" -+ V is a homeomorphism for v = l , .. . , l . Examples

1.

If G is a domain in en , then (G,idc) is a Riemann domain.

2. The Riemann surface of vz (without the branch point) is the set G := { (z, w) E C* x e : w 2 = z} .

If G is provided with the topology induced from e* x e , then it is a Hausdorff space. The mapping r.p : C* -+ G defined by ( � ( ( 2 , () is continuous and bijective. Therefore, G is connected. The mapping r.p is called a uniformization ci G.

Now let 1r : G -+ e be defined by 1r(z, w ) := z . Clearly, 1r is continuous. If (z0, w0) E G is an arbitrary point, then z0 -=/=- 0, and we can find a simply connected neighborhood V(z0) c @*. Then there exists a holomorphic function f in V with f 2 (z) - z and f(zo) = wo. We denote f (z) by vz. 1 The image W == f (V) is open, and the set 1r- (V) can be written as the union ci two disjoint open sets U± := {(z, ± f (z)) : z E V } = (V x ( ± W)) n G. Let f (z) == (z, f(z)). Then f : V -+ G is continuous, and_:rr o f(z) = z. The open set U == U+ i s a neighborhood ci (z0 , w0 ) , with f(V) = U and 1r(z, w) = (z, w) on U; that is, 1rlu : U -+ V is topological. Hence (G,1r) is a Riemann domain over C . �

4

�

general topological space X is said t o be connected if it is not the union of two disjoint nonempty open sets. A space X is called pathwise connected if each two points cf X can be joined by a continuous path in X . For open sets in en these two notions are equivalent. A

8 . Riemann Domains over en

89

The space G can be visualized in the following manner: We cover e with two additional copies of e , cut both these "sheets" along the positive real axis, and paste them crosswise to one another (this is not possible in !R 3 without self intersection, but in higher dimensions, it is). This is illu strated in Figure 1 1.8. 0

Figure

1 1 . 8 . The Riemann surface of

v'z

8. 1 Pr op osition (on the uniqueness of lifting). Let ( G 1r) be a domain over en and Y a connected topological space. Let y0 E Y be a point and 'l/.;1 , $2 : Y --+ G continuous mappings with 'l/J1 (y0) = 'l/J2 ( y0) and 1r o 'l/.;1 = 1r o 'l/.;2 . Then 'l/.;1 = 'l/.;2 . ,

Let M := {y E Y : 'lj.;1 (y) = 'lj.;2 (y) } . By assumption, y0 E M , so M -=1- 0. Since G is a Hausdorff space, it follows immediately that M is closed. Now let y E Y be chosen arbitrarily, and set x := 'lj.;1 (y) = 'l/.;2 (y) and z := 1r(x) . There are open neighborhoods U = U(x) C G and V = V(z) C en such that 7r : u --+ v is topological, and there is an open neigborhood W = W(y) with '1/J>- (W) c U for >. = 1 , 2. Then PROOF:

'1/Jd w = (7r I u) -

1

0 7r 0

'l/.;1 I w = (7r I u) -

1

0 7r 0

'l/J2 1 w = 'l/J2 1 w '

and therefore W C M. Hence M is open, and since Y is connected, it follows that M = Y. • Definition. Let Z o E en be fixed. A ( R iemann) domain over en with distinguished point is a triple Q = (G,1r, x0) for which: 1. (G,1r) is a domain over en. 2. x0 is a point of G with 1r(x0) :=z0.

90

II.

Domains of Holomorphy

Let Yj = (Gj , 1rj , Xj) be domains over e n with distin guished point. We say that YI is contained in Y2 (denoted by YI -< Y2 ) if there is a continuous map r.p GI --+ G2 with the following properties: 1. 1r2 o r.p 7ri (called "r.p preserves fibers" ). 2. r.p(x! ) x 2 .

Definition.

:

=

=

8.2 Proposition. with r.p( XI) = x2 is

If YI Y2 , then the fiber preserving map r.p : GI uniquely determined. -<

--+ G2

This follows immediately from the uniqueness of lifting.

The relation

8.3 Proposition.

1 . 9 -< Q . 2. YI -< Y2

and 92

-<

Q3

==;.

YI

"-< "

-<

is a weak ordering; that is:

93 .

The proof is trivial. Two domains YI ' Y2 over en with fundamental point are called isomorphic or equivalent (symbolically YI 92 ) if YI -< Y2 and Definition.

"'

92

-<

YI ·

Two domains Yj (Gj , 1rj , Xj ) , j 1, 2 , are isomorphic if and only if there exists a topological 5fiber preserving map r.p GI --+ G 2 with r.p(x! ) x 2 . =

8.4 Proposition. =

=

:

PROOF: If we have fiber preserving mappings 'PI : GI --+ G2 and 'P2 : G2 --+ GI , with 'PI (x i ) x2 and r.p2 (x 2) XI , it follows easily from the uniqueness of fiber preserving maps that r.p2 o 'PI = idc 1 and 'PI o 'P2 = idc2 • The other direction of the proof is trivial. • =

=

A domain g = ( G, 1r, x0) with 1r(x0) = z0 is called schlicht isomorphic to a domain 90 = ( G0 , idea , z0) with Go c e n .

Definition.

if it is

8.5 Proposition. with GI , G2 c en .

Let YJ = (Gj , ide; , Xj ) , j = 1, 2, be two schlicht domains Then Y I Y2 if and only if GI C G2 . -<

Example 5

Recall that a "topological map'' is a homeomorphism!

8.

Riemann Domains over en

91

z and z -1=- 0} and 7ri (z, w ) : = z. Then Let GI := {(z, w ) E C2 : w 2 (h = (GI , 7ri , (1, 1)) is the Riemann surface of y'z, with distinguished point (1, 1). The domain (h is contained in the schlicht domain (f2 (C, ide , 1), by cp(z, w ) := z . But the two domains are not isomorphic. =

=

Union of Riemann Domains.

We begin with the definition of the union of two Riemann domains. Let Qj = (Gj , 1rj , xj ) , j = 1, 2, be two Rie mann domains over (C n with distinguished point, and Zo := 7ri (xi ) = 1r2 (x2 ). We want to glue GI , G2 in such a way that XI and x2 will also be glued. To get a rough idea of the construction, assume that we already have a Riemann domain Q = ( G, 1r, x0) that is in some sense the union of QI and Q2 . Then there should exist continuous fiber preserving maps 'P I : GI --+ G with 'PI (xi ) x0 , and 'P2 : G2 --+ G with cp2 (x2 ) := x0 . If a : [0, 1] --+ GI and (3 : [0, 1] --+ G2 are two continuous paths with a(O) = XI , (3(0) = x2 and 7ri o a = 1r2 o (3, then /'I := 'PI o a and /'2 := cp2 o (3 are continuous paths in G with 7r O ')'I = 7r 0 ')'2 and /'I (O) = /'2 (0) = x0. Due to the uniqueness of lifting, it follows that /'I = ')'2 . This means that a(t) and (J(t) have to be glued for every t E [0, 1] . Unfortunately, this is an ambiguous rule. For example, we could say that x E GI and y E G2 have to be glued if 7ri (x) 1r2 (y) . Then the desired property is fulfilled, but it may be that there are no paths a from XI to x and (3 from x2 to y with 7ri o a = 1r2 o (3. =

=

Therefore, we proceed in the following way: Start with the disjoint union on this set with the GI U G2, and take the "finest" equivalence relation following property: .

rv

1. XI "' X2 . 2. If there are continuous paths a : [0 , 1] --+ GI and (3 : [0, 1] --+ G2 with a(O) = XI , (3(0)

=

x 2 , and 7ri .

o

a = 1r2 o (3, then a(1)

rv

(3(1) .

One can equip G := ( GI U G2)/ with the structure of a Riemann domain. This will now be carried out in a more general context. rv

Let X be an arbitrary set. An equivalence relation on X is given by a partition &: = {Xv : v E N} of X into subsets with:

1. 2.

UvE N Xv = X. X" n X�' = 0 for

v -/=- f-L. The sets X" are the equivalence classes. Now let a family (� )Lei of equivalence relations on X be given with � = {X�, : vL E NL } for L E I. We set N : = fl101 NL , and

92

II.

Domains of Holomorphy

Then &: = {Xv : v E N} is again an equivalence relation (simple exercise) , and it is finer than any � . This means that for every E I and every v E N , there is a VL E NL with X" c x�, . L

.

We apply this to the disjoint union X = u .AE L G,A , for a given family (�h).AEL of Riemann domains �h = (G>-, 1r>-, x>-) over en with distinguished point. An equivalence relation on X is said to have property (P) if the following hold: 1. X>- "' xll, for >., g E L. 2. If [0, 1] --+ G>- and (3 : [0 , 1] --+ Gil are continuous paths with a(O) rv (3(0) and 7r.A o a = 1rll o (3, then a(1 ) rv (3(1). We consider the family of all equivalence relations on X with property (P) . It is not empty, as seen above in the case of two domains. Therefore we can construct an equivalence relation (as above) that is finer than any equivalence relation with property (P) . We denote it by rv p . It is clear that 7r,A (X) = 7rll(y) if x E G.A , y E Gil, and x "' P y. The relation "'P also has property (P), and the elements of an equivalence class X" all lie over the same point z = z(Xv ) · We define G := Xj rvp and ?r(Xv ) := z(Xv ) · The equivalence class of all X,A will be denoted by x. z. If 8.6 Lemma. Let y E G.A and x E Gil be given with 1rll(x) = 7r>-(Y) a :

=:

we choose open neighborhoods U = U(y) C G.A, V = V(x) C Gil, and an open connected neighborhood W = W(z) such that 1r>-I : U --+ W and 1rll : V --+ W are topological mappings, then for r.p := (1rIl l v ) - o 7r.A U --+ V the following hold: 1 . r.p(y) = X. y, then r.p(y') y' for every y' E U. 2. If x :

"'P

"'P

PROOF: The first statement is trivial. Now let a [0, 1] --+ W be a con tinuous path with a(O) = z and a(1) = 7r>-(Y') for some y' E U. Then (3 := (7r>- lu ) - 1 o a and "( := r.p o (3 are continuous paths in U and V with (3(0) y rv p X = r.p(y) = "f(O) and 7r,A 0 (3 = 7rll 0 r.p 0 (3 = 7rll 0 "!· Therefore, y1 = {3 ( 1) rv p "( (1) = r.p(y1 ). • :

=

8. 7 Theorem.

There is a topology on G such that

is a Ri�mann domain over e n with distinguished point x, and all maps 'P>- : G>- --+ G with 'P>-(x) := equivalence class of x are continuous and fiber preserving.

8.

Riemann Domains over en

93

PROOF: (1) Sets of the form 'P>. (M) for M open in G>. together with G constitute a base of a topology for G. To see this it remains to show that the intersection of two such sets is again of this form. Let M C G>. and N C G12 be open subsets. Then 'P >. (M) n cp12 ( N)

=

cp12 ( N n cp 12 I ('P >. (M))).

But cp12 I ('P>. (M)) is open in G12 • In fact, let x E cp12 I ('P>.(M)) be given, and y E M be chosen such that 'P>. (Y) = cp12 (x) (and therefore y "'P x). Let z : = 7r>. ( Y) = 1r12 (x) . Then there exist open neighborhoods U = U(y) and V V(x) and an open connected neighborhood W = W(z) such that 7r>. : U --+ W and 1r12 : V --+ W are topological mappings. Let 'P : = ( 1r12 1 v ) -I o 7r>. : U --+ V. By the lemma, cp(y) x and cp(y') "' p y' for every y' E U. So V' := cp(MnU) is a neighborhood of x in G12, and since cp12 (cp(y')) = 'P>.( Y') for every y' E U, it follows that V' C 'P 12 I ( 'P>. (M) ) . =

=

Consequently, every 'P>. is a continuous map. (2 ) Remark: Since every y E G is an equivalence class 'P>.(x), we have M = U 'P>. ( 'P>. I ( M)) for any subset M C G. AEL (3) 7r : G --+ e n is continuous: Let V c e n be an arbitrary open set, and M : = :;r-I (V). Then 'P >. I (M) = 1r>. I (V) is open in G>. , and therefore M = u ).. E L 'P>. ('P ).. I (M )) is open in G. ( 4) G is a Hausdorff space: Let YI , Y2 E G with YI -=/=- Y2 , and Z I ir(y! ) , Z 2 : = ir (y2 ) . There are two cases. If z i -=/=- z2, then there are open neighborhoods VI (z i ) and V2 (z2) with VI n V2 = 0 . Then :;r- I (VI ) and :;r- I (V2) are disjoint open neighborhoods of YI and Y2 · If Z I = z2 , then we choose elements XI E G>. , x2 E G12 with 'P>. (x! ) = YI and cp12(x2 ) = y2 , and since X I and x2 are not equivalent , the above lemma implies that there are disjoint neigborhoods of YI and Y2 · (5) G is connected: Let y = 'P>. (x) be an arbitrary point of G. Then there is a continuous path a : [0 , 1] --+ G>. that connects the distinguished point X>. to x. Then 'P >. o a connects x to y. (6) 7r is locally topological: Let y = 'P>. (x) be a point of G, and z = ir(y) 7r>. (x) . Then there exist open neighborhoods U = U(x) C G>. and W = W(z) c e n such that 7r>. : U --+ W is a topological mapping. U := 'P>. (U) _ is an open neighborho od of y, with ir(U) 7r>. (U) = W. In addition, ir l u is injective, since 7r o 'P>. = 7r>. and 7r>. l u is injective. :=

=

=

94

II.

Domains of Holomorphy -

( 7) Clearly, the maps 'P>.. : G >.. --+ G are fiber preserving, and it was already • shown that they are continuous. Now g has the following properties:

1. 2.

9>-.

-<

Q , for every

X E L.

If Q* is a domain over en with 9>-.

PROOF:

_

Q* for every >., then g

-<

-<

Q*

(of the second statement)

If Q * is given, then there exist fiber preserving mappings r.p)_ : G>.. --+ G * . We

introduce a new equivalence relation N on the disjoint union X of the G>.. by x N x1

: -<==*

x E G>.. , x' E G12 and r.p)_ (x) = 'P� (x').

It follows from the uniqueness of lifting that N has the property (P).Now we define a map r.p : G --+ G* by r.p(r.p>.. (x))

:=

r.p)_ (x) .

Since "' p is the finest equivalence relation with property ( P ) ,r.p is well defined. • Also it is clear that r.p is continuous and fiber preserving. Therefore g is the smallest Riemann domain over en that contains all do mains Y>.. · -

Definition. The domain g constructed as above is called the union of the domains Y).. , and we write g = u ).. E L Y).. · Special cases:

1. 2.

From YI -< g and Y2 -< 9 it follows that YI U Y2 From YI -< Y2 it follows that YI U Y2 "' Y2 · 3. g u g "' g . 4 . YI u 92 "' 92 u YI . 5. YI u (92 u 93 ) "' ( QI u 92 ) u Y3 ·

-<

Q.

Example Let YI = ( GI , 7ri , xi ) be the Riemann surface of JZ with distinguished point XI = (1, 1) and Y2 = (G2 , id, x2 ) the schlicht domain

1

2

< l z l < 2)

with distinguished point x2 = 1 . Then YI u g2 = (G , ir, xo ), where G = (GI U G2 )/ "'P · I Let y E 1r;- (G2) C GI . Then we can connect y to the point XI by a path I a in 7r1 (G 2 ) , and 7ri (Y) to x2 by the path 7ri o a in G 2 . But XI "'P x2 , so

8. Riemann Domains over en

y

95

1r1 (y) as well. This shows that over each point of G2 there is exactly one equivalence class. "'p

Now let z E e - {0} be arbitrary. The line through z and 0 in e contains a segment a : [0, 1] --+ @* that connects z to a point z* E G2 . There are two paths a l , a 2 in Gl with 7ri O ctl = 7rl O ct2 = a Since al (1) rv p a2 ( 1 ) , it follows that a 1 (0) "' I' a2 (0) . .

Then it follows that (f! U �h Exercises

= (e - {0}, id, 1 ) .

define q:,t : en --+ en by (e i h z1 , . . . , e i tn Zn ) . '*" t ( Z1 , . . . , Zn ) A Riemann domain g = ( G, 1r, x0) is called a Re inhardt domain over en if 1r(x0) = 0 and for every t E 11 - (e* ) n there is an isomorphism 'Pt : g --+ g with 1r o 'Pt = q:,t o 1r. Prove: (a) If G c en is a proper Reinhardt domain, then g = ( G, id , 0) is a Reinhardt domain over en . 2 (b) Let G 1 , G2 c e be defined by

1 . For t = ( t 1 , . . . , t n ) E

11

•

rt.

.-

.=

P2

(0, 1)

-

{ (z , w) E

1 { (z , w) : l z l = and lwl < 2 } ' 2 P (0, 1 ) : lwl < �} · 1

2

Gluing G1 and G2 along { ( z , w) : � < l z l < 1 and lwl < � } one obtains a Riemann domain over e2 that is a Reinhardt domain over e2 , but not schlicht. Show that this domain can be obtained as the union of 0 1 = ci , id, ( £ , D and 02 = c2 , id, ( £ , � ) . 2. Let J = {0, 1 , 2, 3, . . . } C N0 be a finite or infinite sequence of natural numbers and Pi = pn(zi, ri) , i E ] , a sequence of polydisks in en. Assume that for every pair ( i, j) E .T x J an "incidence number" Eij E {0, 1 } is given such that the following hold: (a) Eij = Eji and Cii = 1 . (b) Cij = O if Pi n Pj = 0. (c) For every i > 0 in J there is a j < i with Eij = 1. (d) If Pi n Pj n Pk -=1=- 0 and Eij = 1 , then Cik =Ejk·

(

)

(

)

Points z E Pi and w E Pj are called equivalent (z "' w ) if z = w and Cij = 1 . Prove that G := u Pi/ rv carries in a natural way the structure of a Riemann domain over en . Let 1r : G --+ en be the canonical projection and suppose that there is a point Zo E n i EJ Pi. Is there a point Xo E G such that (G,7r, Xo ) can be written as the union of the Riemann domains ( Pj , id, z 0) ?

96

9.

1 1 . Domains of Holomorphy

T h e E nvelop e of Holomor p hy

H olomorphy on Riemann Domain s Definition . Let ( G, 1r) be a domain over en . A function f : G � e is called holomorphic at a point x E G if there are open neighborhoods U = U(x) c G and V = V(1r(x)) c en such that 1rlu : U � V is topological and f o ( 1rl u) - I : V � e is holomorphic. The function f is called holomorphic o n G if f is holomorphic at every point x E G. Remar k. A holomorphic function is always continuous. For schlicht domains in @" the new notion of holomorphy agrees with the old one. Definition . Let Qj = (Gj , 1rj , xj ) , j = 1, 2, be domains over en with distinguished point, and �h -< �h by virtue of a continuous mapping r.p : GI � G2 . For every function f on G2 we define fl c1 == f o r.p.

Pr op o sition. Iff : G2 � e is holomorphic and Q I holomorphic o n GI .

9.1

PROOF:

-<

92 , then f l c 1 zs

Trivial, since r.p is a local homeomorphism with 1r2 o r.p = 7ri -

•

Definition . 1. Let ( G ,7r) be a domain over en and X E G a point. If f is a holo morphic function defined near x , then the pair (!, x) is called a local holomorphic function at x. 2. Let (GI , 7rl ) , (G2, 7r2) be domains over en , and XI E GI , X2 E G2 points with 7ri (xi) = 1r2 (x2) == z . Two local holomorphic functions ( JI , x l ) , (h , x2) are called equivalent if there exist open neighbor hoods UI (xi) C GI , U2 (x2 ) C G2 , V(z), and topological mappings

I 7ri : UI � V , 1r2 : U2 � V with fi o (7ri i ul ) - = h 0 (7r2 1 u2 )- ·

3. The equivalence class of a local holomorphic function (!, x) is denoted

by fx ·

Remark. If ( h )x 1 = ( h)x 2 , then clearly, h (xi) = h (x2 ) · In particular, if GI = G2 , 7ri = 1r2 , and xi = x2 , then it follows that h and /2 coincide in an open neighborhood of xi = x2 .

9. The Envelope of Holomorphy

97

9 . 2 Pr op o sition. Let (G 1 , 1ri ) , (G2 , 1r2) be domains over en , a.x : [0, 1] --+ G>. continuous paths with 1r1 oa1 = 1r2 oa 2 . Additionally, let !>. be holomorphic on G.x , for >. = 1 , 2 . If (h)cq (O) = ( h ) a2 (o ) , then also ( !I ) a1 ( 1 ) = ( h ) a2 (1) · PROOF: Let M := { t E [0, 1] : (fi ) a 1 ( t ) = ( h ) a 2 (t) } · Then M of 0 , since 0 E M . It is easy to see that M is open and closed in [0, 1] , because of the identity theorem for holomorphic functions. So M = [0, 1] . • Pr opo sition. Let Qj = (Gj , 1rj , xj ) , j = 1 , 2, be domains ove r en with distinguished point, and 91 --< Y2 · Thenfo r every holomorphicfunction f on G1 there is at most one holomorphicfunction F on G2 with F / c 1 = f , i.e., a possible holomorphic extension o ff is uniquely determined.

9.3

PROOF: Let F1 , F2 be holomorphic extensions of f to G2 . We choose neigh borhoods U.x (x .x ) c G.x such that the given fiber-preserving map r.p : G1 + G2 maps U1 topologically onto U2 . We have FJ o r.p / u1 = f / u1 , for j = 1 , 2, and therefore F1 l u2 = F2 l u2 • It follows that (FI )x2 = (F2)x2 • Since each point of G2 can be joined to x2 , the equality F1 = F2 follows. •

Envelopes of H olomorphy Definition . Let g = ( G,1r, Xo) be a domain over e n with distinguished point and § a nonempty set of holomorphic functions on G. Let (Q.x)>.E L be the system of all domains over e n with the following properties: 1 . Q --< 9.x for every >. E L. 2 . For every f E § and every >. E L there is a holomorphic function F.x on G.x with F.x lc = f. Then H,y; (Q) == U .xE L Y>. is called the 9 - h u l l of Q. If § = 0( G) is the set of all holomorphic functions on G, then H(Q) Ho ( G ) (Q) is called the envelope of holomorphy of Q. If § = { f) for some holomorphic function f on G, then H1 (Q) = = Hu } (Q) is called the dom ain of existence of the function f .

Let g = ( G,1r, Xo) be a dom ain over e n ' § a nonempty set of holomorphic functions on G, and Hg; (Q) = (G, 1?, x0 ) the 9 - h u l l. Then the following hold:

9.4 Theorem. 1.

Q --< Hg; (Q) .

�

2. For each function f E § there exists exactly one holomorphic function F on G with Fie = f . �

II.

98

Domains of Holomorphy

IJC:h = (G 1 , 1ri , xi ) is a dom ain over en such that 9 --< YI and everyfunc tion f E .Jk can be holomorphically extended to G1 , then Y1 --< H,y; (Q) .

3.

PROOF: Hg; (Q) is the union of all Riemann domains 9:.. = (G;.. , 7r;.. , x;.. ) to which each function f E .Jk can be extended. We have fiber-preserving maps r.p;.. : G --+ G;.. and cp;.. : G;.. --+ G . �

Let "'p be the finest equivalence relation on X == U >. E L G;.. with property (P) . 6 Then 8 is the set of equivalence classes in X relative to "' P· We define a new equivalence relation c::: on X by

x

c:::

: -¢===?-

x'

Then

N

x E G;.. , x' E G, 1r;.. (x) = 1re (x '), and for each f E F and its holomorphic extensions F1 , F2 on G;.. , respectively G, we have (F:.. ) x = (Fe)x ' ·

has property (P):

(i)For any >. we can find open neighborhoods U = U(xo ) , V = V(x;.. ) , and W = W(1r(x0)) such that all mappings in the following commutative dia gram are homeomorphisms:

Then for f E .Jk and its holomorphic extension F;.. on G;.. we have that 1 F;.. o (7r;.. / v ) - 1 = F;.. o r.p;.. o (1r/ u ) - 1 = f o (1r/ u ) - is independent of >.. Therefore, all distinguished points x;.. are equivalent. (iqf a : [0, 1] --+ G;.. and (3 : [0, 1] --+ G, are continuous paths with a(O) "' (3(0) and 1r;.. o a = 1re o (3 , then (F:.. ) a(O) = (Fe)f3(0) · It follows that (F:.. ) a(l) = (Fe)f3( I ) as well, and therefore a(1) c::: (3(1). Since g --< 9:.. and 9:.. --< Hg; (Q) , it follows that Q --< HF (Q) . Furthermore, the fiber preserving map rp : = cp;.. o r.p;.. does not depend on >.. Now let a function f E .Jk be given. We construct a holomorphic extension F on G as follows: �

If y E 8 is an arbitrary point, then there is a >. E L and a point Y>. E G>. such that y = cp;.. (y;.. ), and we define

1f y = 'Pe(Ye) as well, then y;.. "'P Ye , and therefore y;.. "' Ye as well. It follows that F;.. (y;.. ) = Fe (Ye), and F is well defined. 6

For the definition of property (P) see page

92.

9 . The Envelope of Holomorphy

99

We have F o cp = F o VJ>-. o 'P>-. = F>-. o 'P>-. = on G. This shows that F is an extension of and from the equation F o $>-. = F>-. it follows that F is holomorphic (since $>-. is locally topological).

f

f,

•

The maximality of H,y; (Q) follows by construction.

The 9-hull Hg; (Q ) is therefore the largest domain into which all functions E § can ce holomorphically extended.

f

9. 5

Iden_!ity theorem. Let YJ = (GJ , 1rj , XJ ) , j = 1 , 2, be domains over e n , and Q = (G, ?r, x) the union of Q1 and G2. Let fJ : GJ --+ e be holomor phic functions and Q = ( G,1r, x) a domain with 9 --< Yj fo r j = 1 , 2-such that Then there is a holomorphic function on G with fJ , JfO f' J = 1 , 2 .

h i e. = hi e ·

f

f

h /e

f ie = J

h/ e ,

PROOF: Let and § == { ! } . Since Y1 --< Hg; ( Q ) and = == Y2 --< Hg; ( Q ) , it follows that 91 U Y2 --< Hg; ( Q ) . �

�

�

Let be a holomorphic extension o ff to G (where Hg; (Q) = G, 1?, x) ), and = = ]115. Then

1

f

(

(fleJ i e = 11e = (J10) l e = J1e = f. fl e1

Therefore, is a holomorphic extension o ff to Gj · Due to the uniqueness = fJ for j = 1 , 2. of holomorphic extension, •

J/ ej

P seu doconvexit y. Let P" c e n be the unit polydisk, (P n , H ) a Eu clidean Hartogs figure, and : p n --+ e n an injective holomorphic map ping. Then ((P n ) , ( H )) is a generalized Hartogs figure. P = (P", @, 0 ) and 1i = (H, , 0 ) are Riemann domains with 1i --< P. We regard the pair (P.'H) as a generalized Hartogs figure. 9.6 Proposition. Let ( G, 1r) be a dom ain over e n , (P, 'H) a generalized Hartogs figure, and xo E G a point fo r which 1i --< g := (G,1r, x0 ) .

Then every holomorphic function G UP.

f o n G can be extended holomorphically to

The proposition follows immediately from the identity theorem. Definition. A domain ( G, 1r) over e n is called Hartogs convex if the fact that (P.'H) is a generalized Hartogs figure and x0 E G a point with 1i --< Q := (G,1r, xo) implies g u p "" r;; .

A domain Q = (G, 1r, xo) over en is called a dom ain of holomorphy if there exists a holomorphic function on G such that its domain ci existence is equal to Q.

f

1 00

II. Domains of Holomorphy

Remark. If G c e n is a Schlicht domain, then the new definition agrees with the old one.

9 . 7 Theor em. 1.

If Q = ( G,n, x0 ) is a dom ain over e n and § a nonempty set of holo morphic functions on G, then Hg; (Q) is Hartogs convex. 2. Every dom ain of holomorphy is Hartogs convex.

PROOF: Let (P, 'H) be a generalized Hartogs figure with 1i --< Hg; (Q). Then every function f E § has a holomorphic continuation to Hg; (Q) UP .There fore, Hg; (Q) U P + H.� ((]). On the other hand, we also have Hg; (Q) < • Hg; (Q) U P .s o Hg; (Q) U P "' Hg; (Q) . A Riemann domain (G, n) is called holomorphically convex if for every infinite discrete subset D C G there exists a holomorphic function f on G that is unbounded on D.

9.8 Theor em (Oka, 1 953). If a R iem ann dom ain ( G,n) is Hartogs pseu doconuex, it is holomorphically convex (and therefore a dom ain qf holornor phy) .

This is the solution of Levi's problem for Riemann domains over en . We cannot give the proof here. It seems possible to construct the holomorphic hull by adjoining Hartogs figures (cf. H . Langmaak, [La60] ) . It is conceivable that such a construction may be realized with the help of a computer, but until now (spring 2002) no successful attempt is known. We assume that parallel computer methods are necessary.

Boundary Point s . In the literature other notions of pseudoconvexity are used. We want to give a rough idea of these methods. Definition . Let X be a topological space. A filter (basis) on X IS a nonempty set R of subsets of X with the following properties: 1. 0 rf_ R. 2. The intersection of two elements of R contains again an element of the set R .

Example

1. If xo is a point of X , then every fundamental system of neighborhoods of xo in X is a filter, called a neighborhood filter of xo .

9 . The Envelope of Holomorphy

101

2 . Let (Xn ) be a sequence of points of X . If we define SN == { Xn n > N) , then R := {SN : N E N} is the so-called elementary filter of the sequence (x n ) . A filter is therefore the generalization of a sequence. Definition . A point x0 E X is called a cluster point of the filter R if x0 E A, for every A E R. The point x0 is called a limit of the filter R if every element of a fundamental system of neighborhoods of x0 contains an element of R. For sequences the new notions agree with the old ones.

If f : X --+ Y is a continuous map, then the image of any filter on X filter on Y , the so-called direct image.

IS

a

Definition . Let (G, 1r ) be a Riemann domain over e n . An accessible boundary point of ( G,n ) is a filter R on G with the following properties: 1 . R has no cluster point in G. 2. The direct image 1r ( R) has a limit zo E e n . 3. For every connected open neighborhood V = V(zo ) C e n there is exactly one connected component of 1r - 1 (V) that belongs to R. 4. For every element U E R there is a neighborhood V = V ( z0 ) such that U is a connected component of 1r - 1 ( V) .

Remark.

For a Hausdorff space X the following hold:

1 . A filter in X has at most one limit. 2. If a filter in X has the limit x0 , then filter .

x0

is the only cluster point of this

(for a proof see Bourbaki, [Bou66] , § 8 . 1 ) Therefore, the limit z0 in the definition above is uniquely determined. There is an equivalent description of accessible boundary points that avoids the filter concept. For this consider sequences (x , ) of points of G with the following properties: 1 . ( x , ) has nu cluster point in G. 2. The sequence of the images 1r ( x, ) has a limit zo E e n . 3. For every connected open neighborhood V = V(z0) C en there is an m E N such that for n, m > n0 the points x, and Xm can be joined by a continuous path a : [0, 1] --+ G with 1r o a([O, 1]) c V .

Two such sequences ( x, ) , (y,) are called equivalent if

7r(Yv) = Zo . 1 . limv--.oo 7r(Xv) = li mv 2. For every connected open neighborhood V = V ( z0 ) there is an n0 such that for n , m > m the points x, and Ym can be joined by a continuous path a : [0, 1] --+ G with a([O, 1]) c V. --; 00

n o

102

I I . Domains of Holomorphy

An accessible boundary point is an equivalence class of such sequences. Let 8 G be the set of all accessible boundary points of G . Even if G is schlicht, this set may be different from the topological boundary 8G. There may be points in ac that are not accessible, and it may be happen that an accessible boundary point is the limit of two inequivalent sequences. We define 6 == Gui!JG. If r0 = [xn] is an accessible boundary point, we define a neighborhood of r0 in G as follows: Take a connected open set U C G such that almost all Xn lie in U and 1r ( U) is contained in a neighborhood of z0 := limn---+ oo 1r(xn)· Then add all boundary points r = [yn] such that almost all Yn lie in U and limn-too 7r(Yn) is a cluster point of 1r ( U) . With this neighborhood definition G becomes a Hausdorff space, and * : G --+ e n with v

1r(x) :=

{

if x E G, 7r( x) nlim ---t oo 7r(xn) if x = [xn]

E i!JG,

is a continuous mapping. Definition . A boundary point r E 8G is called removable if there is a connected open neighborhood U = U(r) C G such that ( U, *) is a Schlicht Riemann domain over e n and 8 G n u is locally contained in a proper analytic subset of U. A subset M C 8G is called thin if for every ro E M there is an open neighborhood U = U(r0) C G and a nowhere identically vanishing holo morphic function f on U n G such that for every r E M n U there exists a sequence (xn) in U n G converging to such that limn-too f (x n ) = 0. v

r

Example Let G C e n be a (schlicht) domain and A C G a nowhere dense analytic subset. Then every point of A is a removable boundary point of G' := G - A. The points of the boundary of the hyperball Br (0) C e n are all not removable. Let B be a ball in the affine hyperplane H = { (zo , . . . zn ) E en+l : Zo = 1 } , and G c e n+ l - {0} the cone over B. Then every boundary point of G is not removable, since locally the boundary has real dimension 2 n + 1 . The set M := {0} is thin in the boundary, as is seen by choosing J (zo , . . . zn ) : = zo . ,

,

(G,1r) be a Riemann domain over e n . If r.p : D --+ G is a continuous mapping, 7r o r.p : D --+ en holomor phic , and (7r o r.p )' ( () -=/=- 0 for ( E D, then S := r.p(D) is called an analytic disk in G. The set bS := r.p(8D) is called its boundary.

Analytic Disk s . Let

v

9 . The Envelope of Holomorphy

l 03

Let I == [0, 1] be the unit interval. A family (St)tEI of analytic disks 'Pt (D) in G is called continuous if the mapping (z , t) H 'Pt (z ) is continuous. It is called distinguished if St c G for 0 < t < 1 and bSt c G for 0 < t < 1 . Definition . The domain G is called pseudoconvex if for every distin guished continuous family (St)tEI of analytic disks in G it follows that s1 c G.

The domain G is called pseudoconvex at r E 80 if there is a neighborhood U = U(r) C G and an c > 0 such that for every distinguished continuous family (St)tEI of analytic disks in G with 7r(St) C B10(7r(r) ) it follows that St n U c G for t E /. v

As in en one can show that a Riemann domain is pseudoconvex if and only if it is Hartogs pseudoconvex. 9.9 Theorem (Ok a). A R iemann dom ain (G, 1r) is pseudoconvex if and only if it is pseudoconvex at every point r E 80. 9. 10 Corollary. If (G, 1r) is a dom ain of holomorphy, then convex at every accessible boundary point.

0

is pseudo·

The converse theorem is Oka's solution of Levi's problem. Finally, we mention the following result: 9. 1 1 Theorem. Let (0,1r) be a Riemann dom ain over en , and

a thin set of non removable boundary points. point of 80 - M , then 0 is pseudoconvex. PROOF:

See [GrRe56] , §3, Satz 4.

If 0

M C 80 is pseudoconvex at every •

E xer cises 1 . Prove that a Reinhardt domain Q over e n must be Schlicht if it is a domain of holomorphy. 2. Prove that if ( G, 1r) is a Reinhardt domain, then for every f E 0( G) there is a power series S(z) at the origin such that f(x) = S(1r(x)) for E G. 3. Prove that the envelope of holomorphy of a Reinhardt domain is again a Reinhardt domain. 4. Prove that the Riemann surface of the function f(z) = log(z) has just one boundary point over 0 E C. 5 . Find a Schlicht Riemann domain in e 2 whose envelope of holomorphy is not schlicht . 6. Construct a Riemann domain g = (G, 1r , x 0) over e2 such that for all x,y E 1r-1 (1r(xo )) and every f E O (G) the equality f(x) = f(y) holds. X

104

II. Domains of Holomorphy �

7. Let (G, 1r ) be a Riemann domain and ( G, 7?) its envelope of holomorphy. If f is a holomorphic function on G and F its holomorphic extension to G, then f(G) F (G) 8. Consider =

G

.

{ (z, w) � < l z l < l, lwl < 1} :

ei t , w) : : < r < l , t > O and lw

Determine the envelope of holomorphy of G. 9. Let G c e n be a domain and p G --+ JR a plurisubharmonic function. {z E G p ( z ) < c} c c G, 1f zo is an accessible boundary point of B then B is pseudoconvex at zo , in the sense of the last paragraph. :

==

:

C h ap t er III

Analyt ic S ets 1.

T h e Algeb r a of P ower Ser ies

T he Banach Algebr a Bt . In this chapter we shall deal more exten sively with power series in e n . Our objective is to find a division algorithm

for power series that will facilitate our investigation of analytic sets. We denote by C[z] the ring of formal power series I: v>o avz" about the origin. Let JR.+ be the set of n-tuples of positive real numbers. Definition . Let t = (t1 , . . . , tn ) E JR.+ and f = L: v> O avz" define the "number" I I J il t by

I I J ilt : =

{ Ev>ol av lt"

E

C [zl We

if this series converges,

otherwise.

Let Bt : = { ! E C [z� : II f l it < oo }. Remark.

One can introduce a weak ordering on JR.+ if one defines

For fixed f, the function t 1---t II J il t is monotone: If t < t * ,then II J i lt < I / f li t · · Definition . A set B is called a comp lex Banach algebra if the following conditions are satisfied: 1 . There are operations +

:

B B --+ B, · : C B --+ B x

x

and

o B :

x

B --+ B

such that (a) ( B , +, · ) is a complex vector space, (b) ( B ,+,o) is a commutative ring with 1 , (c) c . ( ! g) = ( c . f) g = f ( c . g) for all f, g E B and c E C. 2. To every f E B a real number 11!11 > 0 is assigned that has the properties ci a norm: o

( a) ( b)

o

o

li e · !II = lei . 1/!11 , for c E C and f E B , 1/ f + g /1 < 1/ f /1 + 1/ g ll , for f, g E B,

106

1 1 1 . Analytic Sets

IIIII = o � J = o . II! o g il < IIIII . ll g l l , for J, g E B. (c)

3.

is complete; i.e., every sequence in B that is Cauchy with respect to the norm has a limit in B.

4. B

Bt

1 . 1 Theor em. for any t E JR+. .

= {! E

C[z] II J ilt :

< oo} is a complex Banach algeb ra

Clearly, C[z] is a commutative C-algebra with 1. Straightforward calculations show that II· . . li t satisfies the properties ( 2a) , ( 2b) , (2c) and (3). It follows that Bt is closed under the algebraic operations, and all that remains to be shown is completeness. a >-J z" . Then for every Let ( f>.. ) be a Cauchy sequence in Bt with f>.. = L v>o - � c > 0 there is an n = n (c) E N such that for all A, f-L > n ,

PROOF:

Z::: l a �>-J - a �l l t" = ll f>.. - !M ilt v> O Since t" = t�1

•

•

•

t�n #

<E

0, it follows that E

for every v E Nn0 .

I ac>-J " ] < t" " - aC>
For fixed (a�>-J ) is therefore a Cauchy sequence in C which converges to a complex number a,. Let f = L: v> O avz". This is an element of C[z]. Given S > 0, it follows that there exists an n = n ( 8) such that Y,

=

:2::: I a�>-) - a�>-+ > O

<

6

for >. > n and f-L E N

2

Let I C N0 be an arbitrary finite set. For any >. > n there exists a f-L N such that :Ev E I l a �" + >< l - a I t" < 8/2, and then "

:2::: I a�>- J v El

- a,

It"

< 8,

for >. > n .

In particular, ll f>.. - Jilt < 6. Thus f>.. - f (and then also and ( f>.. ) converges to f . Expan sion with Respect t o

additional notation: If v E N0 , t E JR+. , and E e n , write z

z1 .

= J-L( >.) E

f)

belongs to Bt,

•

For the following we require some

1 . The Algebr a of Power Series

107

(t1 , t' ) , and z = ( z1 , z1 ) . An element f = L v> O avz" E C[z] can be written in the form v = ( v1 , v1 ) ,

t

00

"""' f>. z A1 , f =�

=

with

I

f>. (z )

""

= �"'

I

I

a( A , v '){Z }" .

The series fA are formal power series in the variables z2 , . . . , zn. We call this representation of f the expansion off with respect to z1. Now the following assertions hold: 1.

2.

00

/>. E Bt ' for all A, and l: II!A IIt ' � < oo . ll zf . Jilt = t� . II J i l t , for s E No.

f E Bt

¢==;>

PROOF:

( l ) Since we are dealing with absolute convergence, it is clear that

ll!ll t

=

00

2: 1 1 />. ll t 't� . A=O

(2) We have z . f = I:r 0 fA z; + s . The right side is the unique expansion z . f with respect to z1 . Now the formula can be easily derived.

f

f

of •

Let B be a complex Banach algebra and (!>.) a sequence of elements of B. The series I::r 1 fA converges to an element f E B if the sequence Fn I:;�= 1 fA converges to f with respect to the given norm. 1 .2 Proposition. Eve ry f E B with Il l - !II < 1 is a unit in B with C onver gent Series in Banach Algebr a s . ==

f- 1

00

- """' (1

-6

-

f) A and ll f - 1 1 1

1 < - 1 - 11 1 -

111 ·

E :=

Ill - f 11 . Then 0 < c < 1 , and the convergent geometric series I::r 0 sA dominates the series I: r 0 ( 1 - ! ) A . As usual, it follows that this series converges to an element g E B. We have PROOF:

Let

n

n

( 1 - ( 1 - f)) . 2: ( 1 X=O

Jf'

t (l - !) A - I: (l - f f' A=O

As n tends to infinity we obtain

Xl

1 - ( 1 - J) n+ l f . g = 1 and 1/ g // <

I:: r 0 s A

= 1/(1

_ s) .

•

1 08

III. Analytic Sets

C onver gent Power Ser ies. A formal power series f = I:v> O avz" is called convergent if it is convergent in some polydisk P around the-origi n. In that case there exists a point t E P n JR+. , and since f converges absolutely

at t . it follows that f E Bt . On the other hand, if f E Bt , then by Abel's lemma f converges in P = pn ( o , t ) . 1 . 3 Theorem. :3 t E JR+. with II f li t < oo} is the set d conve rgent { f E C [z] 1. Hn :=

power serzes. 2. Hn is a C-algebra. 3. There is no zero divisor in H.: If f . g = 0 in H, then f = 0 or g = 0.

We have already proved the first part, and then the second part follows easily.

The last part is trivial, since C[z] contains no zero divisors. Remark. If f is convergent and f(O) = 0, then for every c > 0 there is a t E JR+. with II Jil t < c. In fact, since f (O) = 0, we have a representation f = Z1 h + · · · + zn fn . If 11 /ll t < oo, then also 11/i ll t < oo for i = 1 , _ . , n, and n

n

II Jilt = 2::.:: ti ll/ill t < max( t1 , · · · t n ) · 2::.:: 1 1/i ll t i=1 ,

This expression becomes arbitrarily small as t

--+

i=1

0.

When we go from Bt to Hn , we lose the norm and the Banach algebra structure, but we gain new algebraic properties: 1 . f E Hn is a unit ¢==;> f (O) -=1- 0. PROOF: One direction is trivial. For the other one suppose that f(O) -=1- 0. Then g == f . f(0) -1 1 is an element of Hn with g(O) = 0. So there exists a t with 1/ g//t < 1 , and f . f (0) 1 is a unit in Bt . Thus f is a unit • in Hn . 2 . The set m := {f E Hn f(O) = 0} of all nonunits in Hn is an ideal: -

-

:

(a) /I , h E m ===? h + h E m. (b) f E m and h E Hn ===? h f E m. ·

An ideal a in a ring R is called maximal if for every ideal b with a c b c R it follows that a = b or b = R. One can show that in any commutative ring with 1 -=/:- 0 there exists a maximal ideal. If a C R is maximal, then Rfa is a field. 1 .4 Theorem. The set m of nonunits is the unique maximal ideal in H, and Hn /m "'

C.

If a C Hn

is a proper ideal, then it cannot contain a unit. Therefore, it is contained in m . The homomorphism r.p Hn --+ C given by f H f( O ) is • surjective and has m as kernel. PROOF:

:

1 . The Algebra of Power Series

1 09

Distinguished Dir ection s . An element f E Hn is called zl- regular order k if there exists a power series fo ( z! ) in one variable such that:

of

1. f(z1 , 0, . . . , 0) = z f . fo (z1 ) . 2. fo (O) -1- 0. If f is zrregular of some order, f is called z1 -regular. Let f (z) = I:� 0 f>.. zr be the expansion of f with respect to z 1 . Then f is zrregular of order k if and only if fo(O' ) = . . . = fk_1 ( O') = 0 and fk (O') -1- 0. f is zl-regular if and only if f (z1 , 0, . . . , 0 ) � 0. We often need the following properties: 1. f is a unit in Hn -¢==? f is 21-regular of order 0. 2. If f>. is zl-regular of order k>. , for >. = 1 , 2 , then h . h is z1-regular of order k1 + k2 . There are elements f -1- 0 of Hn that f (0) = 0 which are not zrregular, even after exchanging the coordinates. Definition. map ac : en

Let c = ( c2 , . . . , en ) be an element of en - 1 . The linear --+ en with

is called a shear. The set I; of all shears is a subgroup of the group of linear automorphisms of en ' wilh ao = idcn . We can write ac (z) := z + z1 . (O , c ) . ln particular, we have ac (ei) = ( l ,c ) . Let f E Hn be a nonze ro e lement. Then there exists a shear a such that f o a is z1 -regular. 1 . 5 Theorem.

Assume that f converges in the polydisk P. If we had f(z1 , z') = 0 for every point (z1 , z') E P with z1 -1- 0, then by continuity we would have f = 0, which can be excluded. Therefore, there exists a point a = (a1 , a ' ) E P with a1 -1- 0 and f ( a) -1- 0. We define c := (a i ) - 1 . a ' and a : = ac . Now, PROOF:

So f o a(z1 , 01) � 0, and f o a is 21-regular.

•

Rem ark. If JI , . . . , /z are nonzero elements in Hn , then f == h . . fz -1- 0 converges on a polydisc P, and there exists a point a E P with f ( a ) -1- 0 and a1 -1- 0. As in the proof above we obtain a shear a such that h o a, . . . , fz o a are zl-regular. ·

110

1 1 1. Analytic Sets

Exercises

f(z) L: v>O avz" be a formal power series. (a) P r ve the "Cauchy estimates": f E Bt

1 . Let

=

===?

o

l av l <

l l ! l l t /t"

for

almost every v. (b) Prove that if there is a constant C with l av /s " < C , then f E Bt for t < s. (c) Let fn (z) = L: v>o av , nz" be a sequence of power series with //!n i l < C. If every sequence (a,,) converges in C to a number a, then show that (fn) is a Cauchy sequence in Bt converging to f(z) = L: v> O avz" , for every t < s . s

2. The Krull topology on Hn is defined as follows: A sequence Un) converges in Hn to f if for every k E N there is an no with f - fn E mk for n > no . What are the open sets in H,? Is Hn with the Krull topology a Hausdorff space? 3. Let B be a complex Banach algebra with 1. Show that for every f E B the series exp(f) = I::� o r fn! is convergent, and that exp(f) is a unit in B. 4. If f is a formal power series and f I:;� 0 P>. its expansion into homo geneous polynomials, then the order of f is defined to be the number =

ord(f) := min{s E No

: p8 -=f- 0).

Now let (fn) be a sequence of formal power series such that for every k E N there is an no with ord(fn) > k for n > no. Show that I:� 1 fn is a formal power series. Use this technique also for the following: If 91 , . . . , gm are elements of Hn with ord(gi) > 1 , then

:2::: afl. wfl. H :2::: afl. (g 1 (z) , . . . , 9m ( z))fl. J.L > O J.L > O defines a homomorphism rp Hm --+ Hn of complex algebras. :

2.

The Preparati o n The orem

Let a fixed element t E JR+. be chosen. When no confusion is possible we write B in place of Bt , B' in place of Bt' , and II ! I/ in place of II f 1 / t . The ring of polynomials in z 1 with coefficients in B' is denoted by B' [z1 ]. 2. 1 Weierstrass Formula in Bt . Let f and g = I:;� 0 g;.. z� be two elements of B. Assume that there exists an E No and a real number E with Division w ith Remainder in Bt .

0 < E < 1 such that

g8 is

a

s

unit in B' and ll z f

- g g-; 1 11 ·

< c t� . ·

111

2. The Preparation Theorem

Then there that

exists exactly one q E B f

=

and one r E

B'[z1 ]

with deg ( r ) <

s

such

q · g + r,

with

and

ll r ll < 1/ f l/ · 1 -1 c: Let us first try to explain the idea of the proof. If h E B, then there is a unique decomposition h qh · zf + rh , where rh E B' [z 1 ] and deg ( rh ) < s. If g is given, we define an operator T T9 : B --+ B by PROOF:

=

=

:

g ·

g; 1 . qh + rh . If T were an isomorphism, then f = T(T- 1 f) g · (g; 1 · qr-1 1 ) + rr-1 1 would T(h )

=

=

be the desired decomposition. One knows from Banach space theory that T is an isomorphism if idB - T is "small" in some sense. Since /l (id8 - T) (h) ll = ll zf - gg; 1 ll . llqh II , one can, in fact, conclude from the hypothesis of the theorem that id8 - T is "small." Now r- 1 2:r 0 (id8 - T) A . Since (id8 T ) 0 f = f and (id8 - T) 1 f (zf- g g_;- 1 )q1 , we obtain the following algorithm: =

=

Inductively we define sequences f>.. , q>.. , r>.. beginning with fo f zf q0 + ro . If f>.. zf q>.. + r>.. has been constructed for some >. > 0, then we define =

=

=

f>.. + I

:=

(z f - gg; I )q>.. ,

and obtain q>.. + I and r>.. + 1 by the unique decomposition

f>.. + 1 = zfq>.. + l + r>..+ I , r>.. + 1 E B'[z 1 ] with deg(r>.. + I ) < s. If we define q : I:: r 0 g_; 1 q>.. and r : 2:r 0 r>.. , then =

=

00

00

00

00

l::: (gg_; 1 q>.. + r>.. )

X=O

=

g · q + r.

112

1 1 1. Analytic Sets

When using this algorithm we do not need the abstract transformation T and the B anach theory of such transformations. However, it is necessary to prove the convergence of all of the series that were used. - ( zf - gg-;; 1 ) . Then ll h ll < c: . tf and gg-;; 1 = zf + h. From f>.. = zf q;.. + r;.. it follows that li rA II lll f>.. ll and 1/ q;.. /l . ll f>.. ll . Furthermore, from f>.. + I = - h · q;.. it follows that

For this let h

==

lt1 8

ll f>..+ I II

;..

.l h ll . ll q;.. ll < E .

II!AII

0 !A converges. Th us II /A II < E . II I II and Since 1 q < >-. s g 1 ll . II I II and li rA / / < c:;.. IIJII, g ll ; ;.. ll c: rt ll ; the series q = I: 0 g; 1 q;.. and r 0 r;.. also converge.

I;�

�

= I;�

The estimates for 1/ gsq/1 and IIl ii follow readily: < ll r ll

<

00

'L:0 II q;.. ll

<

l: l h ll

<

>-.=

00

>-.=0

f!S II J II ·

(X_

I >A = A=O

t1S IIJII ·

j'

1

1

-

!'"

II III · 1 - c: ·

It still remains to show uniqueness. Assuming that there are two expressions of the form it follows that

and ll (ql - q2)gs z f ll

I 1/(ql - q2)gs z f + (rr - r2 ) 1/

<

ll ( q 1 - q2 )gsh ll E · t� . II ( q1 - q2 )gs II E ll (q1 - q2 )gs zf ll· •

If

the assumptions of the theorem are satisfied and if in addition f E B'[z 1 ] , g E B' [z1] , and deg(g) = s, then q E B'[z1 ] with q = 0 or deg ( q) = deg( f) - s.

2.2 Corollary.

Let d := deg (!). For d < s we have the decomposition f Hence we have to consider only the case d > s. PROOF:

=

0 · g +f.

2. The Preparation Theorem

We assume that therefore

deg (JJ.L ) < d fo r f-L

=

1 13

0 , . . . , A . Then deg(q>-) < d - s , and

deg( f>- +d = deg (JA - r>- - gg; 1 q>-) lnax (d, s - l ,s + (d - s ) ) = d. Hence deg ( fA) < d and deg(q>-) < d - s for all A. It follows that deg( q) < d - s, and from f = g . q + r we can conclude that deg ( q) = d - s. •

The Weierstras s Condition . We use the notation from above. Definition. Let s E N0 . An element g = I:; 0 g>-z� E B satisfies the Weierstrass condition (or W-condition) at position s if 1 . gs is a unit in B'. 2. l l zf - gg; 1 ll < � tf .

�

Let R be an integral domain.' A polynomial f ( u) = fs u + fs - 1 u s- 1 + . . . + h u + fo E R[ u] is called monic or normalized if fs = 1 . A polynomial f E B'[zi] is normalized if and only if it is z 1 -regular of some order k < s. 8

lfg E B satisfies the W 2.3 Weierstrass preparation theorem in Bt · condition at position s, then there exists exactly one normalized polynomial w E B'[z 1 ] of degree s and one unit e E B such that g = e . w. PRooF: We apply the Weierstrass formula to f = zf . There are uniquely determined elements q E B and r E B'[zd with zf = q g + r and deg(r) < s (we choose an c < i such that ll zf - gg; 1 l l < stf ) . ·

But then zf - gg;1 = ( q - g;1 )g + r is a decomposition in the sense of the Weierstrass formula. Therefore, we have the estimate

llgs q - 1 11 < t� s ll zf - gg; 1 ll ·

1

� c < 1 � c < 1.

That means that g8 q and hence q is a unit in B . Let e := q - 1 and w == zf - r. Then w is a normalized polynomial of degree s, and e . w = q- 1 (zf - r) = g .

If there are two decompositions g = e 1 (zf - r!) = e2 (zf - r2 ) , then

z 1 = e1 1 . g + r1 = e2 I . g + r2 . 8

From the uniqueness condition in the Weierstrass formula it follows that e1 = e2 and r1 = r2 . • 2.4

z1 . 1

Corollary.

If g is

a

polynomial in

z1 ,

then e is also a polynomial in

An integral domain is a commutative nonzero ring in which the product of two nonzero elements is nonzero.

1 14

III. Analytic Sets

PROOF: We use the decomposition zf - gg; 1 = (q - g; 1 ) g + r . From the Weierstrass formula it follows that // r // < // z f - gg; 1 // · 1

Since w,

=

� E: < t� - 1 � c < tf .

1 , it is also true that // z f - ww; 1 // = // z f - w// = // r// < t� .

Therefore, g = e . w + 0 is a decomposition in the sense of the Weierstrass • formula, and the proposition follows from Corollary 2.2. The Weierstrass preparation theorem serves as a "preparation for the exami nation of the zeros of a holomorphic function." If the function is represented by a convergent power series g , and there exists a decomposition g = e· w with a unit e and a "pseudopolynomial" w (z1, z') = zf + A 1 ( z')z� - 1 + · · · + As (z') , then g and w have the same zeros. However, the examination of w is simpler than that of g .

Weier str ass P olyn omials. Now we turn to the proof of the Weierstrass formula and the preparation theorem for convergent power series. The ring Hn is an integral domain with l . If f E Hn and f (z) = I; 0 f>.. (z')z£ with h. = 0 for >. > s , then f is an element of the polynomial ring Hn - dz!] . If fs -=1- 0, then deg( f ) = s. lf f is normalized and f>.. (O') = 0 for >. < s, then f is zrregular exactly of order s, and f(z1 , O') = zf.

�

Definition. A normalized polynomial w E Hn - dz1 ] with and w ( z1 , O') = zf is called a Weierst rass polynomial.

deg(w)

=s

We have seen that a normalized polynomial w E Hn d z1 ] with deg(w) = s is a Weierstrass polynomial if and only if it is z1 -regular of order s . It follows easily that the product of two Weierstrass polynomials is again a Weierstrass polynomial. -

If g = e . w is the product of a unit and a Weierstrass polynomial of degree s,

then we also have that g is z 1 -regular of order s, since the unit e is z1-regular of order 0. We now show that conversely, every z 1 -regular convergent power series is the product of a unit and a Weierstrass polynomial.

2.5 Th eor em . Let g E Hn be z 1 -regular d orde r s . Then fo r every c > 0 and e v e y to E JR+. the re exists a t < to such that g lies in Bt, gs is a unit in Bt' , and //zf - gg_;-- 1 //t < c . tf .

PROOF: Let g = 0 gJ.. Z� be the expansion of g with respect to z . Then 1 g>. (O') = 0 for >. = 0 , 1 , . . . , s - 1 and g8(0') -=1- 0.

I;�

Since g is convergent, there exists a t1 < to with l / g / / t 1 < oo . Then g>. E Be1 for all >., and in particular,

2 . The Preparation Theorem

/(z ')

=

1 15

g8 ( 0') - 1 gs (z' ) - 1 E Bti .

Now, f(O') = 0. Thus there exists a t2 < t1 such that 1 1 ! / l t ' < 1 for all t < t2 . Therefore, g8 is a unit in Bt ' , and g an element of Bt . Let h := zf - gg; 1 . Then h E Bt for all t < t 2 , and we have an expansion h = 2::: � 0 h;.. z � with h. = 0 , h;.. = - g;.. g; 1 for >. -=/:- s, and h;.. (O') = 0 for >. = 0, 1 , . . . , s - 1 . 1f t1 > 0 is sufficiently small, then 00

E t1 . < t2 2' s

for all t = ( t 1 , t ' ) < t2. And since h;.. (O') = 0 for >. = 0 , . . . , s - 1 . for every small t 1 there exists a suitable t ' such that

Consequently, 1/ h//t < E . tf.

•

Remar k . In a similar manner one can show that if g 1 , . . . ,gN E C [z] are convergent power series and each gi is zrregular of order si , then for every E > 0 there is an arbitrary small t E JR+ for which gi E Bt , (g; )s, is a unit in Bt' and llzt' - gi (gi ) s/ 11 < E . t�'

Weier str ass Pr ep ar ation Theor em

2.6 Theorem (Weier str ass division for mula) . Let g E Hn be z 1 - regula r of order s. Thenfo r every f E Hn the re are uniquely dete rmined e lements q E Hn and r E Hn - dz1 ] with deg(r) < s such that f=

q · g + r.

Iff and g are polynomials in z1 with deg(g)

=

s, then q is also a polynomial.

PROOF: There exists a t E JR� and an E with 0 < E < 1 such that f and g lie in Bt., g8 is a unit in Bt ' , and ll z f - gg; 1 ll t < E . tf . It then follows from the division formula in Bt that there exist q and r with f = q . g + r .

Let two decompositions of f be given:

1 16

III. Analytic Sets

We can find a t E JR� such that f, q1 , q2 , r1 , r2 lie in Bt and g satisfies the vV-condition in Bt . From the Weierstrass formula in Bt it follows that q1 = q2 • and r 1 = r2 .

Let g E Hn be z1 regular of orde r s . Then the re exists a uniquely dete rmined unit e E Hn and a We ierst rass polynomial w E Hn - dzi] of degree s such that

2.7

Theorem (Weierstrass preparation theorem) .

g=e

·

w.

If g is a polynomial in z1 , then e is also a polynomial in z 1 . PRooF: There exists a t E JR� such that g satisfies the W-condition in Bt . The existence of the decomposition g = e . w with a unit e and a normalized polynomial w of degree s therefore follows directly from the preparation the orem in Bt . Since g is z1-regular of order s, the same is true for w. So w is a Weierstrass polynomial.

Now, w has the form w = zf - r, where r E Hn - 1 [z1] and deg ( r) < s. Thus, if there exist two representations g = e1 ( zf - r1 ) = e2 ( zf r2 ), it follows that 1 1 zf = e;- · g + r1 = e2 .g t r2 . The Weierstrass formula implies that e1 = e2 , • r1 = r2 and therefore w 1 = w2 -

.

Exercises

1 . Write a computer program to do the following: Given two polynomials f (w, x,y) (of degree n in w and degree m in x and y) and g(w, x, y) with g(w, 0, 0) = w 8 , the program uses the Weierstrass algorithm to determine y and r (up to order m in x and y) such that f = y . g + r. 2. Let f : p n- I x D --+ C be a holomorphic function andn 01< r < 1 be a real number such that ( H f(z' , () has no zero for z' E p - and r < I CI < 1 . 1 Then prove that there is a number k such that for every z' E p n the function ( H f(z' , () has exactly k zeros (with multiplicity) in D . Use this statement to give an alternative proof for the uniqueness in the Weierstrass preparation theorem. 3. Show that the implicit function theorem for a holomorphic function f : n e X (1: --+ (1: with f (O) = 0 and fzn (0) # 0 follows from the Weierstrass preparation theorem.

3.

Prim e Factorization

Unique Factorization. Let I be an arbitrary integral domain with 1 . Then I * : = I { 0} is a commutative monoid with respect to the ring multiplication, and the set JX of units of I is an abelian group . -

3.

Prime Factorization

1 17

Let a, b be elements of I*. We say that a divides b (symbolically a I b) if there exists c E I * with b = a . c. We can also allow the case b = 0. Then every element of I* divides 0, and a unit divides every element of I . Definition. Consider an element a E I * - Ix . 1 . a is called irreducible (or indecomposable) if from a = a1 . a2 (with a 1 , a2 E / * ) it follows that a 1 E / x or a2 E J x . 2. a is called p rime if a I a1a2 implies that a I a1 or a I a2 . Irreducible and prime elements can be defined in an arbitrary commutative monoid. In / * every prime element is irreducible, and in some rings (for example, in Z or in JR[X]) it is also the case that every irreducible element is prime. In Z[H] one can find irreducible elements that are not prime. Definition. I is called a unique facto rization domain (UFD) if every element a E J X can be written as a product of finitely many primes. One can show that the decomposition into primes is uniquely determined up to order and multiplication by units. In a UFD every irreducible element is prime and any two elements have a greatest common divisor (gcd). Every principal ideal domain2 is a UFD, and in this case the greatest common divisor of two elements a and b can be written as a linear combination of a and b. For example, Z and K[X] (with an arbitrary field K ) are principal ideal domains. So in particular, C [X] is a UFD .

Gaus s ' s Lemma. Let I be an integral domain. Two pairs ( a ,b), ( c ,d) E I x I * are called equivalent if ad = be. The equivalence class of a pair ( a,b) is called a fraction and is denoted by a fb. The set of all fractions has the structure of a field and is denoted by Q(I). We call it the quotient field of I. The set of polynomials f ( u) = a +a 1 u+ . + an un in u with coefficientsai E I 0 constitutes the polynomial ring J [ u] . The set J0[u] of monic polynomials in J[u] is a commutative monoid. Therefore, we can speak of factorization and irreducibility in J 0 [u] . 3 . 1 Gauss' s lemma) . Let I be a unique facto rization domain and Q = Q(I) . lf w 1 , w2 are ele ments of Q0 [u] with w1 w2 E J0 [u] , then W>, E JO[u] fo r >. = 1 , 2. PROOF: For >. = 1, 2, W>, = a>,,o + a;>,, I u + . . . + a>,,s,_ _ 1 us ,.. - l + u8" with ax,, E Q . Therefore, there exist elements d>- E I such that d>, . w>, E J [u] . We can choose d>, in such a way that the coefficients of d>, . W>, have no common divisor (such polynomials are called p rimitive). 2 A principal ideal domain is an integral domain in which every ideal is generated

by a single element.

1 1 1. Analytic Sets

1 18

We define d := d1d2 and assume that there is a prime element p with p I d. Then p doesn't divide all coefficients d>.a:>.,v of d>. . W>. . Let J-l>. be minimal such that p f d>.a>.,JL>- . Then JL ( d 1 wi) ( d2w2 ) = . . . + u �' ' + 2 ( da1 ,JL, a2,JL2 + something divisible by p ) + . . . .

Since I is a UFD, p doesn't divide (d1a1,JL, ) (d2a2,JL2 ) . So the coefficient of u�' ' + JL2 is not divisible by p. But since w1w2 has coefficients in I, every divi sor of d must divide every coefficient of d . w1 w2 = ( d1wi ) ( d2w2 ) . This is a contradiction ! 3 When d has no prime divisor, it must be a unit. But then d1 and d2 are also 1 units, and W>. = d>. (d>.w>.) belongs to I0 [u] . • 3 .2 Corollary. 1.

2.

3.

4.

Let I

re

a unique factorization domain.

If a E J0 [u] is prime in Q[u] , then it is also prime in I0 [u] . If a E J0 [u] is reducible in Q[u] , then it is reducible in J0 [u] . Every element of I0 [u] is a product offinitely many prime elements. If E J0 [u] is irreducible, it is also prime. a

PROOF: l . Let a E J0 [u] be a prime element in Q[u] . 1f a divides a product a' a" in I0[u] , then it does so in Q [u]. Therefore, it divides one of the factors in Q[u] . Assume that there is an element b E Q[u] with a' = ab. By Gauss's lemma b E J0 [u]. This shows that a is prime in J0 [u] .

E J0 [u] be a product of nonunits a 1 , a2 E Q[u) . If ci E Q is the 1 1 highest coefficient of ai , then c1 c2 = l ci 1 ai E Q0 [u) and a = ( c1 a1 ) ( c;- a2) . 1 By Gauss c; ai E I0 [u] , and these elements cannot be units there. So a is reducible in I0 [u] .

2. Let

a

,

3 . Every element a E J0 [u) is a finite product a = a 1 . . . az of prime elements of Q[u]. One can choose the ai monic, as in (2). Using Gauss's lemma several times one shows that the ai belong to I0 [u] . By (1) they arc also prime in JO [u].

4. Let a E J0 [u] be irreducible. Since it is a product of prime elements, it • must be prime itself. Remark. In the proof we didn't use that I is a unique factorization do main. We needed only the fact that Q[u] is a UFD (since Q is a field) and the statement of Gauss's lemma: If a 1 , a2 E Q0 [u) and a1 a2 E I0 [u] , then ai E I0 [u) for i = 1, 2. 3

The original version of Gauss's lemma states that the product of primitive poly nomials is again primitive. The reader may convince himself that this fact can be derived from our proof.

3. Prime Factorization

Factorization in Hn . I = H, .

1 19

Now the above results will be applied to the case

Definition. Let f E Hn, f = L� o P>. be the expansion of f as a series of homogeneous polynomials. One defines the order of f by the number

ord(f) : = min{>. E N0

:

P>. -=1- 0}

and

ord(O) := oo.

(See also Exercise 1 .4 in this chapter) Then the following hold:

ord(f) > 0 for every f E H,. 2. ord(f) = 0 -<==* f is a unit. 3. ord(h . h) = ord(h) t ord(f2). 1.

3.3

Theorem.

PROOF:

Hn

is

a

unique factorization domain.

We proceed by induction on n .

For n = 0, Hn = C is a field, and every nonzero element is a unit. In this case there is nothing to show. Now suppose that the theorem has been proved for n 1 . Let f E Hn be a no nu nit, f -=1- 0. Iff is decomposable and f = h · h is a proper decomposition, then ord(f) = ord(h ) + ord(/2), and the orders of the factors are strictly smaller than the order of f. Therefore, f can be decomposed into a finite number of irreducible factors. -

It remains to show that an irreducible f is prime. Assume that f I h /2, with f>. E (Hn )* for >. = 1 , 2. There exists a shear a such that h o a, h o a and f o a are zi-regular. If we can show that f o a divides one of the f>. o a , then the same is true for f and f>. . Therefore, we may assume that h, /2, and f are ZI-regular. By the preparation theorem there are units ei, e2, e and Weierstrass polyno mials WI , w2 , w such that h = e i -WI , h = e2 ·w2, and f = e · w. Then w divides wi w2. If wi w2 = q . w with q E Hn , then the division formula says that q is uniquely determined and a polynomial in zi . So w divides wiw2 in H�_ dzi] . Since w is irreducible in H,, it must also be irreducible in H�- I [zi] · By the induction hypothesis Hn- I is a UFD, and therefore w is prime in H�- I [zi] . It follows that w I WI or w I w2 in H�_dzi] and consequently in Hn. This means • that f I h or f I h in Hn ·

Hensel' s Lemma. Let w E Hn[u] be a monic polynomial of degree s. There is a polydisk P around 0 E en where all the coefficients of w converge to

1 20

III.

Analytic Sets

holomorphic functions. Therefore, we can look at w as a parametrized family of polynomials in one variable u . By the fundamental theorem of algebra w(O, u) splits into linear factors, and the same is true for every w(z, u) with z E P. We now show that such splittings are coherently induced by some splitting of w in H� [u] , at least in a neighborhood of 0 . 3.4

Let w(O, u) = TI�= I ( u - C>.)8" be the decomposition into linearfacto rs (with c, -=/=- cJL for v -=/=- f-L and si + · . . + s z = s). Then there are uniquely determined polynomials WI, . . . ,wz E H� [u] with the following prope rties: Hensel's lemma.

deg(w>.) = S>., fo r ).. = 2. W>. (O, u) = (u - C>.)8" . 3 . W = WI · · · · · Wz . 1.

1 , .. . , l .

PROOF: We proceed by induction on the number l. The case l We assume that the theorem has been proved for l - 1 .

=

1 is trivial.

First consider the case w(O, 0) = 0 . Without loss of generality we can assume that c1 = 0. Then w(O, u) = u8' h( u ) , where h is a polynomial over C with deg(h) = s - SI and h(O) # 0. So w is u-regular of order SI , and there exists a unit e E H� [u] and a Weierstrass polynomial wi with w = e . WI · Since wi (O, u) = u8' , it follows that •

l

e(O, u) = h(u) = IT (u - c>.)8" >.=2 By induction there are elements w2, . . . , wz E H�[u] with deg(w>.) = S>., W>. (O, u) = (u - C>.) 8" and e = w2 · · · w1 . Then w = wiw2 · · · wz is the de sired decomposition.

If w(O, O) -=/=- 0, then we replace w by w'(z, u) := w(z, u + ci ) and obtain a

decomposition w'

= wi . . . wf

as above. Define

This gives a decomposition w = w1 . . . wz in the sense of the theorem. The uniqueness statement also follows by induction. •

The Noetherian Property. Let R be a commutative ring with 1 . An R-module is an abelian group M (additively written) together with a composition R x M -+ M that satisfies the following rules:

r(xi + 2 2 ) = rxi + rx2 for r E R and xr, x2 E M . 2. (ri + r2)x = ri x + r2x for ri , r2 E R and x E M . 3 . ri (r2x) = (rir2)x for ri , r2 E R and x E M . 1.

3. Prime Factorization

4. 1

· x

121

= x for x E M.

These are the same rules as those for vector spaces (and the elements of a module are sometimes also called vectors). However, it may happen that rx = 0 even if r -=!=- 0 and x -=1=- 0. Therefore, in general, an R-module has no basis. So-called free modules have bases by definition. An example is the free module RQ := R x . . . x R (q times), with a basis of unit vectors. An example of a nonfree module is the Z-module M : = Z/6Z, where 2 · 3 = 2 · 3 = 0.

If M is an R-module, then a submodule of M is a subset following properties: 1. 2.

N

C M with the

x,

r

y E N =? x + y E N . E R and x E N =? r x E N .

A submodule of an R-module is itself an R-module. Example The ring R is also an R-module. The composition is the ordinary ring mul tiplication. In this case the submodules of R are exactly the ideals in R. An R-module M is called finite if there is a finite set { x 1 , . . . , xn } C M such that every x E M is a linear combination of the xi with coefficients in R. The free module Rq is obviously finite. But Z/6Z is also finite, being generated by the class I. Definition. An R-module M is called noethe rian if every submodule N C M is finite. A ring R is called noetherian if it is a noetherian R-module. This means that every ideal in R is finitely generated (in the sense of a module). 3.5

of

Proposition. ideals

becomes stationary,

Let R be a noetherian 1ing. Then any ascending chain Io

i.e.,

c

/1

c Jl c · · · c R

there is a ko such that h

=

ho fo r k > ko .

PROOF: The set J : = u� 0 h is obviously an ideal. Since R is noetherian, J is generated by finitely many elements h , . . . , fN . Each f" lies in an ideal h., . If ko = max ( k1 , . . . , kN ) , then all f" are elements of ho . So h = ho for

k > ko.

•

3.6 Theorem. PROOF:

IfR is

a

noethe rian ring, then Rq is a noetherian R-module.

We proceed by induction on q.

122

III.

Analytic Sets

The case q = 1 is trivial. Assume that q > 2 and the theorem has been proved for q - 1 . Let M c Rq be an R-submodule. Then I : = { r E R : 3 r' E Rq- I with (r, r' ) E M ) is an ideal in R and as such is finitely generated by elements ri , . . . , rz . For every r;.. there is an element r� E Rq I such that r;.. : = (r;.. , r� ) lies in M . The set M ' := M n ( {0} x Rq - I ) can be identified with an R-submodule of Rq - r , and by the induction assumption it is finite. Let r ;.. = ( 0, r�. ) , >. = I + 1 , .. . , p, be generators of M '. An arbitrary element x E M can be written in the form x = (xi , x' ) with J

XI E I. Then XI = I:>-. = l a;.. r;.. , a;.. E R, and

x-

l

2.:: a;.. r ;..

X=l

=

(0, x' -

l

2.:: a;.. r �)

X1

E M' .

That is, there are elements az + I , . . . , ap E R such that x

1

- 2.:: a;.. r ;.. = X1

p

2.::

>-.= 1 + 1

a;.. r;.. .

Hence { ri , . . . ,rp} is a system of generators for M . 3 .7 Riickert b a sis theorem.

noetherian.

•

The ring Hn of convergent power series is

PROOF: We proceed by induction on n. For n = 0, Hn = C, and the statement is trivial. We now assume that n > 1 and that the theorem has been proved for n - 1 . Let I C Hn be a nonzero ideal and g -=/=- 0 an element of I. Without loss of generality we can further assume that g is 2 1-regular of order s .

Let = <1>9 : Hn -+ ( Hn- I ) 8 be the Weierstrass homomorphism, which is defined in the following manner: For every f E Hn there are uniquely defined elements q E Hn and r = ro + r1zi + · · · + rs - I zf I E Hn- dzi] such that f = q · g + r. Let (f) : = (ro , . . . , rs- d ·

Now, is an Hn- rmodule homomorphism. By the induction hypothesis Hn- I is noetherian, and so (Hn- ! ) 8 is a noetherian Hn- rmodule. Since M : = @ ( Ijs an Hn-I -submodule, it is finitely generated. Let r;.. = (r6"l , . . . , r�?:) I ) , >. = 1 , .. . , l , be generators of M . If f E I is arbitrary, th en J = q · g + r with r = ro + ri zi + · · · + rs -I zf - I , and there are elements ai , · . . , az E Hn- I such that (ro, ri , · . . , rs - 1 ) = <1>9 ( ! ) L�= l a;.. r;.. . Hence we obtain the representation

4.

f

Branched Coverings

1 23

l

= q.

g + z:::a ).. . (rg'' ) + riA) Z1 + · + r�A\ zf- 1 ) , .>..= 1

The set {g, r<1l , . . . , rUl } with r< A) of generators of I .

·

=

·

rbA) + riA) z1 +

·

·

1 · + r�A) 1 zf - is a system •

Exercises 1 . Prove that O(q is not a UFD . 2 . Let M be a finite Hn-module, and m C Hn the maximal ideal. If M m . M , then M = 0 . 3. Let f n: pn - l x D --+ C be a holomorphic function such that for every z' E p - 1 there is a unique solution Zn = cp(z') E D of the equation f(z', zn ) = 0 . Use function theory of one variable to show that 'P is continuous, and use Hensel's lemma to show that 'P is holomorphic. 4. Let f E Hn be 2 1-regular, f = e · w with a unit e and a Weierstrass polynomial w E Hn - dz1 ] . Prove that f is irreducible in Hn if and only if w is irreducible in Hn - dz1] . 5. Show that f( z , w) : = z 2 - w 2 ( 1 - w) is irreducible in the polynomial ring
4.

Br anched C over ings

Germs. Let B c en be an open set and Zo E B a fixed point. A local holomorphir: function at z0 is a pair (U, f) consisting of an arbitrary neigh borhood U = U(z0) C B and a holomorphic function f on U. Two such functions f : U --+ C and g : V --+ C are called equivalent if there is a neigh borhood W = W (z0) c U n V such that f i W = g i W . The equivalence class of a local holomorphic function (U, f) at z0 is called a g e rm (of holomorphic functions) and is denoted by fzo . The value f (z0) as well as all derivatives of f at z0 (and therefore the Taylor series of f at zo) are uniquely determined by the germ. On the other hand, if a convergent power series at z0 is given, then this series converges in an open neighborhood of z0 to a holomorphic function f , and the germ o ff determines the given power series. So the set Ozo of all germs of holomorphic functions at z0 can be identified with the @ algebra of all convergent power series of the form L:v > o ( z - z0 ) v . This algebra is isomorphic to the algebra Hn and has the same algebraic properties. Let fzo # 0 be any element of Ozo with f (z0) = 0. Then there are a neigh borhood U(z0) c B , a neighborhood W (O) C @+ a holomorphic function e on U, and holomorphic functions a1 , . . , a, on W such that after a suitable change of coordinates the following hold: .

1 24

III. Analytic Sets

e (z ) -!= for every z E U. 2. f(z) = e(z) . w (z - zo) for w (wi , w') = wi + ai (w')zf- I + . . . + as (w') .

1.

0

P seu d op olynomials. A pseudopolynomial of degree s over a domain G c en is a holomorphic function w in G X c that is given by an expression I w(u, z) = u s + h1 (z)u s - + · . . + hs (z) , with h i , . . . , hs E 0, where 0 = 0 (G) denotes the ring of holomorphic functions on G. The set of pseudopolynomials of any degree over G will be written as 0°[u] . We begin with several remarks on the algebraic structure.

4. 1 P r op o sition . If rmg 0

=

G zs a domarn, z. e., a connected open set, then the

0( G) zs an integral domarn.

PROOF: We need to show only that 0 has no zero divisors. Assume that JI , h are two holomorphic functions on G with both fi =/=- 0. Since G is a domain, their zero sets are both nowhere dense in G, and there is a point • z E G with h (z) . f2 (z) -!= 0. So h

. h =/=- 0.

It also follows that 0° [u] is free of zero divisors. We denote by Q the quo tient field of 0. Then the group Q[u] x of units in the integral domain Q[u] consists of the nonzero polynomials of degree 0. If 0* C 0 is the multiplica tive subgroup of not identically vanishing holomorphic functions on G, then Q [u] x n 0 = 0* .

4.2 P rop o sition .

If w� , w2 E Q0[u] are pseudopolynomials with WI . w2 E 0°[u] , then WI , W2 E 0° [u] .

PROOF: If w = U s + (JI/gi)us - I + . . . + Us /gs ) is an arbitrary element of Q0 [u] , then for all z E G the germs gi,z are not 0. For a moment we omit the i . If the quotient of fz and gz is holomorphic, i. e . , fz = h, . gz with h, E Oz , then h, is uniquely determined and there is a ball B around z in G such that hz comes from a holomorphic function h on B and the equation f = h . g is valid in B . If we take another point z' E B, the germ of h at this point is the quotient of the germs of f and g at this point. So z f--t hz (z) defines a global holomorphic function h on G. We write h = f j g. Thus, if w, := u s + ( (h )z/(g l )z)u s - I + · · · + (( Js )z/(gs )z) lies in O� [u] for every z E G, then w E 0° [u] . Now we apply Gauss's lemma in the unique factorization domain Oz Hn Let w := W I . w2 • Then (wi )z . ( w2)z = w, E O� [u] for every z E G. Conse quently, the coefficients of (wi )z are holomorphic, and by the remarks above • this means that w, E 0° [u] . "'

4 . Branched Coverings

1 25

The following is an immediate consequence of the above two propositions:

4.3 Theorem. Let G E en i. e., every element

is

be a domain. Then 0°[u] is a factorial monoid; a product cl finitely many primes.

Euclidean Domains Defin ition . An integral domain I is called a Euclidean domain if there is a function N : I * -+ N0 with the following property (division with remainder) : For all a, b E I , b i= 0, there exist q, r E I with 1 . a = q · b + r, 2. r = 0 or N (r) < N(b) . The function N is called the norm of the Euclidean domain. Examples

1. Z is a Euclidean domain with N(a) := Ia!.

2. lf k is a field, then k[x] is a Euclidean domain, by N( J ) : = deg ( J ) . Every Euclidean domain I is a principal ideal domain and thus factorial. lf a, b are elements of I, then the set of all linear combinations

r · a + s · b i= 0 ,

r, s E I ,

has an element d with N (d) minimal. The element d generates the ideal a = { ra + sb : r, s E I } and is a greatest common divisor of a and b. It is determined up to multiplication by a unit. Now assume again that G is a domain in en , 0 = 0( G) , and Q = Q( 0) . Then Q[u] is a Euclidean domain. 1f W I , W2 are pseudopolynomials in 0°[u] , there is a linear combination in w = r iwi + r2w2 =/=. 0 in Q[u] with minimal degree. It can be multiplied by the product of the denominators of the coefficients in r i and r2 . Then ri , r2 , and w are in O[u] , and w is a greatest common divisor of wi , w2.

The Algeb r aic Derivative. Let 0 and Q be as above. lf w E 0° [u] has positive degree, then it has a unique prime decomposition w = W I · . . w1 . The degree of each wi is positive. We say that w is (a pseudopolynomial) without multiple factors if all the w, are distinct. The (algebraic) derivative of a pseudopolynomial is defined as follows. lf w = l:�=O a v u v , then D(w) : = l:�= I v · a, . u v - I . Thus D (wi t w2) D (wi . w2)

_

D(wi) + D(w2 ) , D(w!) . w2 + W I . D (w2)

1 26

1 1 1. Analytic

Sets

4 . 4 Theorem. An element w E 0° [u] is without multiple factors if and o n ly zf a greatest common divisor of w and D (w) is a function h E 0* .

has the irreducible wi as a multiple factor, then D(w) is also divisible by wi · This is also true in Q[n] . So a greatest common divisor is certainly not a function h E 0*. Assume now that w = ni Wi has no multiple factor. Then PROOF:

If w

If the degree of the greatest common divisor 1 of w

and D(w) is positive, then 1 is a product of certain w, So at least one w, divides both w and D( w ) . Then wi divides w 1 . . . D (wi ) . . . w1 and hence D(wi ) · This is not possible, because D(wi ) has lower degree. So the degree of the greatest common divisor is 0, • and therefore it is a function h E 0*. Symmetr ic Polynomial s.

Definition . A polynomial p E Z[u 1, . . . ,us ] is called symmetric if for all i , j we have p (u1, . . . ,u i , . . . uj , . . . ,us) = p(u l , · . . , uj , . . . , u i , . . . ,us)·

There are the elementary !>ymmetric polynomials a1 , . . . , a, defined as follows: a 1 (u 1 , . . . , us ) - U1 + · · · + Us , a2 (u 1 , . . . , us ) - u l (u2 + . . . + us ) + u 2 (u3 + . . . + us) + . . · + us - 1Us ,

The following result is proved, e . g . , in the book [vdW66] 4.5 Theorem. lfp E Z[u1 , . . . , us] is symmetric, then there is exactly one polynomial Q(y 1 , . . . , Y s) E Z[y 1 , . . . , Ys ] such that p = Q( a 1 , . . . , a, ). Consider the special symmetric polynomial pv(u1 , . . . , us) = IJ (u i - Uj ) 2

The Discriminant.

i <j

(square of the Vandermonde determinant). Since it is symmetric, there is a uniquely determined polynomial Qv ( y1 , . . . , y,) with integral coefficients such that

4. Branched Coverings

1 27

Definition . If w u8 +h1 (z)u s - l + · . + hs(z) is a pseudopolynomial in 0° [u] , then A, = Q v (- h1 , h2 , . . . ( - 1 ) 8 hs ) is called the discriminant of w. It is a holomorphic function in G, and we denote its zero set by D,. .

=

,

It is well known from the theory of polynomials that where w 1 , . . . , w , are the zeros of the polynomial if and only if there is a pair i # j with w; = w1 .

u

f-t

w(u, z) . So �w (z)

=

0

Assume now that w is without multiple factors. Then there is a linear combi nation of w and D(w) that is a function h E 0* . We restrict to a point z E G with h(z) # 0. Then the greatest common divisor of w(u, z) and D(w) (u, z) is 1 . This means that w( u, z) has no multiple factors; i.e., the zeros of w( u, z) are all distinct. So �w (z) # 0, and D, is nowhere dense. Example Let G c en be a domain, a, b holomorphic functions in G, and u2 - a(z) . u + b(z) . In this case

w(u, z)

pv(u 1 , u 2 ) = IJ (u ; - u1 ) 2 = (u1 - u 2 ) 2 = (u1 + u2 ) 2 - 4 · u l u 2 . i <j

·

�w (z) = Q v ( a(z) , b(z)) = a(z) 2 - 4b(z). If z E G and �w (z) # 0, there are two different solutions of w(u, z) = 0. Hyper sur fa ces. We use the theory of pseudopolynomials to study ana lytic hypersurfaces. Such analytic sets are locally the zero set of one holomor phic function. Assume that f is a holomorphic function in a connected neigh borhood of the origin in en+ I that is not identically 0. Without loss of gener ality we may assume that A = N( f) contains the origin. Then a generic com plex line £ through 0 meets A in a neighborhood of 0 only at the origin. After a linear coordinate transformation, £ = { ( u ;z) : z = 0 } is the first coordinate axis. By the Weierstrass preparation theorem fco,o) = e (o,o) · wco,o) in the ring Hn+I , where e co, O ) is a unit in Hn+l , and W(o,o) E Hn [u] a Weierstrass poly nomial. We can represent the germs locally by holomorphic functions. Thus there is a domain G C e n containing 0 , and a disk D = { u E e : lui < r} such that in U : = D x G there are a holomorphic function e that does not vanish in U and a pseudopolynomial w over G with f = e . w in U. We may assume that A n (8 D x G) = 0 . Therefore, the zero set of f in U is that of W.

128

1 1 1. Analytic Sets

We can decompose w into prime factors. Using the fact that any power of a prime factor vanishes at the same points as the prime factor does, we may assume that w is without multiple factors. Then the discriminant A, is not identically zero in G. We set D, = {z E G : 6w (z) = 0 } .

4 . 6 Theorem (on branched cover ings). If z o E G - D, there are a neigborhood W = W(zo) C G - D, and holornorphic functions ft , . . . ,fs zn W with fi ( z ) -=/= J1 (z) fo r i -=/= j and z E W such that

w(n, z ) = (n - JI (z) ) · . . (n - fs (z)) in C W There are fewer than s points over any point zo E D, (see Figure III. l ) . X

·u

A

G

Figure 1 1 1. 1 . A branched covering over G

A point z E G above which there are fewer than s points is called a branch point. All points of the discriminant set Dw are branch points. Over all other points our set A is locally the union of disjoint graphs of holomorphic func tions, and is therefore regular. PROOF: For zo E G - D, the polynomial w(n, zo) has s distinct roots. We write w(n, zo) = (n - ci ) · · · (n - c8 ), where the ci all are distinct. If w(n, z ) = '1L3 + h1(z)n 8 - 1 + . . . + hs (z ) , then the germ

Wz0 := n 2 + (hi )z01L 8 - 1 + + (hs)z0 Hn- By Hensel's lemma it has is a polynomial over Oz0 w, , = w1 .zo . . . ws ,zo with the following properties: ·

"'

·

·

a

decomposition

4.

1 . w,,z0 (u, zo ) = U 2. deg (wi,zo ) = 1 .

- C;

Branched Coverings

1 29

for i = 1 , . . . , s .

We have w;,z0 u - ri , with ri E H,. There are a connected neighborhood W(zo) C G - Dw and holomorphic functions h , . . . , is in W such that the power series ri converge to j;. Since the germs of w and ( u - h) . . . ( u - is ) coincide at z0 , it follows from the identity theorem that =

wkxw

=

(u - h ) . . . ( u - is ) ,

and since W c G- D, it also follows that ii (z) -=!= i1 ( )for i -=!= j and z

z

E W.

•

Examples 1. Let G = e and w = zf - z2 . Then the discriminant is given by A, (z2) = 422, and Dw = {0} c C. If z2 E @*, there is a neighborhood W C e - D, where VZ2 is well defined. There we have w = (z1 - VZ2) . (z1 + VZ2) · This gives a surface above e that is a connected unbranched 2-sheeted covering over e- {0}. The point 0 is a branch point. This is the (branched) R iemann surface of y'Z. The unbranched part was already discussed in Section 1 1.8. 2. A completely different situation is obtained if we take w = zf - z� (z1 - z2 ) (z1 + z2 ) . The discriminant is 4z� in this case, and the discrim inant set D, is again the origin in C. The set A consists of two distinct sheets, which intersect above 0, and both are projected biholomorphically onto C. The set A - { 0 } , i.e., A without the branch point, is no longer connected. 3. In higher dimensions the situation is even more complicated. Let us con sider the analytic set A = N (f) , where f ( z1 , . . . , Zn ) = z�' + . ' . + z�n with si > 2 for i = 1 , . . . , n. This is a very simple holomorphic function. The derivatives are iz, = si . zf'- 1 , and their joint zero set consists only of 0 E e n . So all other points of A are regular. Every line £ through the origin lies completely in A, or i has a zero of order s with s > min( s 1 , . . . , s,) on £ at the origin. Therefore, there is no line that intersects A in 0 transversally. From this one can conclude that 0 is in fact a singular point of A (see, for example, Exercise 8.2 in Chapter I). Now we look on f : en -+ e as a fibration with general fiber ·

At = {z E e n : z�' + · · · + z�n = t} .

Then A = Ao has an isolated singularity, while all other sets At are regular everywhere. We call the family (At)tEIC a deformation of A.

130

III. Analytic Sets

The Unbranched Part . We assume that

w(u, z) a pseudopolynomial over A = { (u, z)

G

c en is a domain and

G of degree s. We set G' = G - Dw , E e x G : w(u, z) = 0} ,

A' = AIG' the part of A over G'. Then A' is an unbranched covering of G'. It is an n-dimensional submanifold of e x G', and we have the canonical projection 1r A --+ G. If ( u0 , z0 ) E A' is a point , there is a small neighbor hood and

:

B = B ( u0 , z0) c A' that is mapped by 1r holomorphically and topologically onto a ball around z0 in G'. We also call B a ball. The holomorphic map (7r i B ) - 1 : 7r(B) --+ en+l is a local parametrization of A'. A complex function f in B is called holomorphic if f o (7r i B ) - 1 is holomorphic. In particular, the components of 1r itself are holomorphic functions on B.

For holomorphic functions in B we have the same properties as for holomor phic functions in a domain of en . For example, the identity theorem remains valid, and we obtain the following results: 4. 7 Proposition. Assume that A1 is a connected component of A' and that M is an analytic subset of A1 . Then M = A1, or M is nowhere dense in A1 . 4.8 Proposition. If f is a holomorphic function on A', and A1 a connected component of A', then either f vanishes identically on A1 or its zero set is nowhere dense in A1 . 4.9 Proposition. Let A1 again be a connected component of A'. Assume that M is a nowhere dense analytic set in A1 and that f is a holomorphic function in A1 - M that is bounded along M . Then f has a unique holomor phic extension to A 1 . Decompositions. We consider the interaction between the decomposi

tion of a pseudopolynomial into irreducible factors and the decomposition of its zero set into "irreducible" components. 4.1 0 Proposition. Let G c en be a domain and w(u, z) a pseudopolyno mial over G without multiple factors. Then w is irreducible if and only if the intersection of its zero set A with e x ( G - E) is connected for every nowhere dense analytic subset E C G which contains the discriminant set Dw . Since locally over G' = G - Dw the set A' = AIG' looks like a domain in en , a nowhere dense analytic set does not disconnect A' , locally and globally. Therefore, we may assume that E = Dw·

PROOF:

If w is not irreducible, every factor wi defines an analytic set Ai over G - Dw . The intersection of different Ai is empty. So AI(G - Dw) is not connected.

4.

Branched Coverings

131

If, on the other hand, AI G1 has the connected components A;, i = 1 , . . . , s , with s > 0, then for any point z E G-Dw there is a ball B C G - Dw around z such that Ai lE splits into graphs of holomorphic functions f1 : j = 1 , . . . , si. In each case we form the pseudopolynomial Wi = ( u - h ) · · · (u - is , ) . The zero set of this wi is exactly Ai I B, and it determines w; and vice versa. So over the intersection of two different balls the pseudopolynomials must be the same, and thus we obtain global holomorphic functions wi in G - Dw . If z E Dw , then there is a neighborhood W of z such that Ai i (W - Dw ) C AIW is a bounded set. S o the coefficients of w; are bounded over this neighborhood and extend holomorphically to G. We also denote this extension by Wi , and • for reasons of continuity it follows that w = WI · · · Ws. If the w; are the irreducible factors of w, we call their zero sets A; the irre ducible components of A. The sets A� = A; IG' are the connected components of AIG'. 4.11 Proposition. Assume that w * , w are pseudopolynomials without mul tiple factors over a domain G and that A * = { w * = 0} C A = { w = 0}. Then w * is a factor of w. Let D denote the union of the discriminants of w * and w. It is a nowhere dense analytic set in G. Over G - D we decompose the two unbranched coverings into connected components. There we have A * = AI U · · · U As• and A = AI U · · · U As with s * < s. This yields pseudopolyno mials over G - D that extend to pseudopolynomials WI , . . . , Ws over G, with • w * = WI · · · Ws• and w = WI · · · Ws. This implies the result.

PROOF:

The following result is proved analogously. 4.12 Proposition. Assume that w is free of multiple factors and that A = {w = 0} is the disjoint union of two nonempty sets M', M" that are closed in C x G. Then there are pseudopolynomials w', w" over G with M' = {w' = 0}, M" = { w" = 0} , and w' · w" = w. The construction is first carried out outside Dw . \Ve set G' = G - Dw and use the fact that every nonempty open subset of A' = AIG' must be a union of connected components of A'. If w = WI · · · ws is the decomposition into irreducible factors, then we may assume that there is an s * with 0 < s * < s such that for w' = WI · · · Ws • and w" = Ws• + I · · · Ws we have M' IG' = { (u, z) E C x G' : w '( u , z) = 0} and M" IG' = {( u , z) E C x G' : w" ( u z) = 0} .

PROOF:

,

It is now essential that in a continuous family f ( u, z) of holomorphic functions of one variable u the zeros depend continuously on the family parameter z ( "continuity of roots" ) . We omit the proof here. If we apply this fact ( and the equations A = M' U M". w = w' · w" ) , we get that the sets M'IG' and

132

III.

Analytic Sets

M " I G' are not empty (i.e., 0 < s * < s) and that their closures in e x G are M', respectively M". • Project ions. In the next section we will investigate zero sets of several

holomorphic functions. Here we begin with the simplest case, the common zero set N of a pseudopolynomial w over G C en and an additional holomor phic function f in a neighborhood of A = { (u, z) E e x G : w(u, z) = 0}. Our method involves the projection of N to G.

4.13 Proposition. Assume that f = f(u, z) is a continuous function on A that is holomorphic outside of e x Dw and does not vanish identically in a neighborhood of any point of A. Then the projection of N = {! = w = 0} to G is an analytic set N' = {! = 0}, where f is a holomorphic function in G that does not vanish identically.

PROOF: If z E G - Dw , we have a ball B C G - Dw around it such that

over B our w has the form w(u, z) = (u - fi (z)) · · · (u - fs (z) ) . The function f does not vanish identically on any graph u = fi · Consequently,

f(z)

:=

f( JI (z) , z) · · · f( Js (z) , z)

does not vanish identically. In the usual way we obtain the holomorphic function f in the entire set G - Dw · It is bounded along Dw . So it extends to a holomorphic function in G. • Now consider the following situation: Assume that G is a domain in e n and that w is a pseudopolynomial over G without multiple factors. Let f be a holomorphic function in a neighborhood of A = {w(u, z) = 0} C e x G that does not vanish identically on any open subset of A and define

N

:=

{ (u, z) E e x G : w(u, z)

=

f(u, z)

=

0}.

Denote by N' the projection of N to G. We want to give a definition for "unbranched points" of N. The difficulty is that there may exist such unbranched points of N lying in the set of branch points of w.

4.14 Proposition. For any point z0 E N' there is an arbitrarily small linear coordinate change in z1 , . . . , Zn such that thereafter the line parallel to the z1 -axis through zo intersects N' in zo as an isolated point. In such coordinates there is a neighborhood U(zo) C G, a domain G' in the space en - l of the Variables Z1 = (z2 , . . . , Zn ) , and a pseudopolynomial w'(z1 , z') over G' such that { (z1 , z') E e x G' : w'(z1 , z') = 0} = N' n U. A "small" linear change of the coordinates z1 , . . . , Zn means here that the transformation is very near to the identity. Since f does not vanish

PROOF:

4.

Branched Coverings

133

identically, after a small generic coordinate transformation the line parallel to the zra.xis through zo intersects N' in z0 as an isolated point . And then • it is also clear that U, G' , and w ' with the desired properties exist. Let us now assume that we have chosen a point zo E N' and suitable coordi nates as above, and that U, G' , and w ' have also been chosen.

Definition. In the given situation, a point (u, z ) E N n (C x U) is called an unbranched point of N if z E N' - C x Dw' and there is a neighborhood V = V(z) C N' - C x Dw' with a holomorphic function g on V such that N n (C x V) is the graph { u = g(w) : w E V}. (Figure III.2 shows the situation. )

u

A

..... . :--...

o

examples of unbranched points �

Figure 111.2.

C

X

Dw'

Branched and unbranched p oints of N

4.15 Theorem. In the given situation, in every neighborhood of an arbi trary point ( u 1 z i ) of N n (C x U) there are unbranched points of N. ,

PROOF: We may assume that

z1

E N' - (C x Dw' ) . Then we take a small

neighborhood w = E X ul of (u l , z i ) such that the following hold: ul c u - (C X Dw' ) . 2 . ( u � , zi ) is the only point of A above

1.

z1

in W.

134

3.

III. Analytic Sets

The set A n W is defined by a pseudopolynomial w * over

Setting N� c N' n projection 1r : W --+

U� to be the image set of W n N under U1 , take Ut so small that N� = N' n U1 .

U1 . the canonical

We restrict w * to N� and replace possible multiple factors by one factor at a time. So we obtain a new pseudopolynomial w1 without multiple factors over N� such that

{ (u, z) E E x N� : w1 (u, z) = 0 } , E a suitable disk. near to (u1 , z1) we can find points (u z , z2) E N n W lying over

NnW=

Arbitrarily N{ - Dw, . All of these points are unbranched points of N, since we can find neighborhoods W2 (u 2 , z2 ) c W and U2 (z 2 ) C U1 with the same properties as W and U1 • Now choose U2 so small that it contains no point of Dw, and that Wt i N' n u2 decomposes into linear factors. Then every sheet of A I (N' n U2) • with the property that it contains points of N is a graph over N' n U2 . Exercises 1 . Prove that every symmetric polynomial in u1, . . . , Us can be written as a polynomial in the power SUmS Sk : = U � + . . . + u�. 2 . Calculate the discriminant of a cubic polynomial. 3. Let D : = Dr (O) C C and f be a holomorphic function in an open neigh borhood of D without zeros in 8D. 1f f has in D the zeros c1, . . . ,c8 (some of them may be equal), then

_ 1 r 21r i len

f u, z1 ,

:

4. Let

D

Zn -

("'' ! '(( ) d( = t em . f(( ) j=l 1

x P" --+ C be a holomorphic function in the variables

Assume that f is u-regular of order s and that for every z E P" the function u f--t J(u, z) has exactly s zeros u1 (z) , . . . , u 8 (z) (with multiplicity) in D. Show that the coefficients of the "pseudopoly nomial" w(u, z) : = n;=l (u - Uj (z)) are holomorphic. 5. Prove the "continuity of roots": Let f(u, z) be u-regular at the origin. Show that there is an r > 0 such that if g(z) is a function defined in a neighborhood of 0 with lg(z) I < r and f(g(z), z ) = 0 ,then g is continuous at 0. 6. A complex function f on an analytic set A in a domain G C en is called holomorphic if it is locally the restriction of a holomorphic function in the ambient space. (a) Let A : = {(w, z) E C 2 : w2 - z3 } be the Neil parabola. There is a bijective holomorphic parametrization of A given by w = t3 and z = t2 Describe the holomorphic functions on A as functions of the parameter t. Is there a meromorphic function on A that has a pole at oo with main part tm ? •

•

•

,

•

5.

135

Irreducible Components

(b) Show that the analytic set A = {(w, z 1 , z2 ) E C 3 : w2 = z1 z2 } is not regular at the origin. Consider the holomorphic map (t 1 , t2 ) �--+ (t 1 t2 , ti, t§). It is a "two-to-one" map. Describe the local holomorphic functions on A in t1 , t2 • 7. Prove that there is a topological holomorphic map from @* onto A := { (w;) E C2 : W 2 = Z1 z2 }. 8. Prove that there is no topological holomorphic map from @* onto the "elliptic surface" A := { (� ) E C 2 : w 2 = (z 2 - 1)(z 2 - 4)}. 9. Define the pseudopolynomial w E O(C 2 )[u] by w(u, z 1 , z2 ) := u 2 - u . z 1 and determine the discriminant set and the irreducible components of A := {w = 0}. Let f on A be defined by f(u, z 1 , z2 ) := z2 . (u 1 ) . Consider N := {f = w = 0} c C 3 and determine the projection N' c C 2 and the set of unbranched points of N. -

5.

I r r edu cib le C omp on en t s

Embedded -Analytic S et s . We wish to study general analytic sets. Since it is easier to work with pseudopolynomials than with arbitrary holo morphic functions, we introduce the notion of "embedded-analytic sets." These are subsets of the common zero set of finitely many pseudopolyno mials, and they are not a priori analytic by definition. But later on, it will turn out that they are indeed analytic. Assume that G c e n- d = { z ' = (zd+ I , . . . , zn )} is a domain and that wi (zi; z ' ) ,i = 1 , .. . , d, are pseudopolynomials over G without multiple fac tors. The zero sets of the single Wi intersect transversally4 in cd X G. We denote by D c G the union of the d discriminant sets belonging to the wi. We call it the union discriminant set. We put

A := {(z1 , . . . , zd , z ' ) : wi (zi; z') = O, for i =

1 , .. . , d and

z' E G}.

�

Over any ball B C G - D the set A lB consists of finitely many disjoint holomorphic graphs. Every graph is contained in a connected component Z of AI ( G - D ) . We call the closure of Z in A an irreducible embedded-analytic component of A. �

�

�

�

Definition . If A is defined as above, any union of finitely many irreducible embedded-analytic components of A is called an embeddedanalytic set of dimension n - d. Two submanifolds M, N C en intersect transversally at a point z E M n N if the entire space is spanned by vectors that are tangent to M or N at z. fu our case the common zero set of the pseudopolynomials w; contains enough vectors to span en. Therefore, � say that these zero sets intersect transversally. �

4

136

1 1 1. Analytic Sets

By definition any embedded-analytic set A can be decomposed into finitely many irreducible components.

The surrounding set A :J A is not uniquely determined. Sometimes we can make A smaller by throwing away those irreducible factors of w; that do not vanish identically on A . Then the wi are uniquely determined by A . �

Assume that A is an embedded-analytic set in A c r,cd xG and that f is a holomorphic function in a neighborhood of A that does not vanish identically on any open subset of A. If N = {Z E A : f(z) = 0 } , then fo r any point z0 E N there is an arbitrarily small linear change cl the coordinates z' such that the affine space parallel to the (z1 , . . . , zd+l)- "axis " through z0 intersects N in a n isolated po int. If z1 E N is any point near z0, then there are unbranched po ints qf N arbitrarily near z1 . A t all of these points N is a submanifold of dimension n - d - 1 . 5 . 1 P r opo sition .

PROOF:

We proceed as in the proof for the last theorem of the previous section. The procedure to find unbranched points will be denoted by ( * ) .

First we construct the projection f of j, which is holomorphic in G. For this observe that if D is the union discriminant set of the polynomials w 1 , . . . ,wd and z' E G - D, we always have the same number of points z�, . . . ,zs in A over z'. We set f(z') = f(z1) . . . f(zs) and obtain j, which is holomorphic on G - D. Since it is bounded along D , we can extend it holomorphically to G. Therefore, the projection set is N' = {J-(z') = 0}. Assume now that z� E N' . Then, after an arbitrarily small linear coordinate change in the variables z', the line L parallel to the Zd+ 1-axis through z� intersects N' in an isolated point. Then by Weierstrass's theorem we can find a neighborhood U = U(z�) C G, a domain G', and a pseudopolynomial w'(zd+ l , z") over G' without multiple factors such that U n N ' is equal to the set {z' = (zd+ l • z") E C x G' : w'(zd+ I , z") = 0}, with z" = (zd+2 • . . . , zn ) ·

Since the space r,c d x L intersects N at z0 in an isolated point, it remains to prove the existence of unbranched points. We apply ( * ) to N n (Cd x U) and prove in the same way as before that for points z1 E N n (C d x U) there are unbranched points of N arbitrarily near to z1 . Of course, at these points N • is a submanifold of dimension n - d - 1 . For the following we use the same notation and hypotheses as above. 5 . 2 Theorem.

Let N' C G be the projection cl N . For every point zo x G, after a suitable linear change of the coordinates z' (z0, z�) E N there is a neigborhood U U(z�) C G, a domain G' in the space of the variables Zd+2 , . . . , Zn , and a pseudopolynomial w' over G' without multiple factors such that:

C r,c d

I. N'

n

U=

{w' = 0} .

=

5.

2. N n (e d

x

Irreducible Components

U) is an embedded-analytic set of dimension

n -

d

-

137

1.

PROOF: We use the notation and results from the proof above. Thus the first statement is clear. We set �+ I :=w'. Restricting the Wi to ed X (N' n U) and projecting them down to e d X G' we get pseudopolynomials w i ( Zi; z") ' over G'. Then N n (e d x U) is in the joint zero set of w 1 , . . . , w d+I · Let A be the union of those irreducible components of this set that contain the unbranched points of N. Since N is the closure of unbranched points, it +I follows that N n (e d x G) C A . By the mapping theorem that we prove in the next paragraph, every irreducible component of A is in N. So we have the desired equality. •

Images of Embedded -Analytic Set s . Assume that G = e d X G' c en is a domain and A c G an irr educible embedded-analytic set over G'.

Let G1 = e dl X Gi c en, be a domain, A I c G 1 an embedded-analytic set over Gi , F a holomorphic mapping from a neigh borhood of A C G into GI such that - C A I for some nonempty open subset U c A. Then F(A) c 4 5 . 3 M apping theo r em .

We denote by D C G the union of the discriminant sets of the wi (zi, z' ) that define the surrounding set A for A . It is sufficient to prove that F (A n (ed x (G ' - D ) ) ) C A I . Since we can connect two points of A n (ed x (G' - D ) )by a chain of arbitrarily small balls, it is enough to give the proof for such a ball. So we may replace A by a ball in en- d and may assume that F is defined in a neighborhood of B . PROOF:

�

Let A I be an embedded-analytic set in

� (WI , W' ) = · · · = W � d 1 ( Wd 1 , W ) = 0 } A 1 -- { WI 0 . So - c 4 Then w; o Flu 0, and by the identity theorem wi o FJB �

I

_

For an arbitrary point v E G� we choose a small transformation of the coordinates in ed , such that AI is also embedded in a set Ay = { wY = . . . = wd, = 0} . The transformation can be made arbitrarily small, and we can do it so that

A1 n .Ar n (ed,

x

{v}) = A I n (ed ,

x

{v}) .

Then AI is given by the infinite set of holomorphic equations wi = 0, w'[ = 0, v E G� . If F maps a nonempty open part of B into A 1 , then by the identity theorem wi o F = w; o II 0, and consequently - c A I · This completes the proof. • _

Remark. Assume that A is an analytic set in a domain G c en and that z0 is a point of A . If zo is a regular point of dimension n - d, then there is a neighborhood U = U(zo) c G such that A n U is an embedded-analytic set.

138

llL Analytic Sets

In fact, there is a neighborhood U with holomorphic functions JI , . . . , jd such that N(fi , . . . , fd ) = A n u and the rank of the Jacobian is d everywhere. We may assume that i_

1, . . . , d J - 1, . . . , d

) # 0.

Then the transformation F ( z 1 , . . . , zn ) = (h (z) , . . . !d ( z) , zd+ l , . . . , zn ) maps a neighborhood of z0 biholomorphically onto a neighborhood of the image point. If the inverse is given by ,

then A is given by the equations

Zd

=

gd ( O, . . . , 0, Zd+ l , . . . , Zn ) ·

So A n U is an embedded-analytic set.

Local Decomposition. We use embedded-analytic sets to show that an arbitrary intersection of analytic sets is again an analytic set. First we consider the following situation. Let G c e n be a domain and Zo E G a point. Assume that at z0 a set 9of local analytic functions f is given such that 1 . For every f E 9 there is a connected open neighborhood U ( z0) C G with f E O(U) and f =/=- 0. 2. f(zo) = 0. We want to construct a "maximal" analytic set S* in a neighborhood U* ( zo ) c G such that for each zero set N of finitely many elements f E :7 there is a neighborhood V = V(z0 ) c G with S* n V c N n V. Then S* is uniquely determined near z0 and can be considered as the common zero set of the functions f E 9 Lt may be nontrivial even if the domains of definition of the functions f tend to the point z0 . For example, if :7 is the set of the func tions fn (z) : = z!/ (1 - nzl ) , defined on Un := {z E C n : Re(z l ) < 1/n}, then S* is the analytic set { z1 = 0} in an arbitrary neighborhood of 0 , whereas the intersection of the Un does not contain any neighborhood of the origin. We employ the results from the beginning of this section several times and carry out an induction on the codimension of the embedded-analytic sets obtained from the functions f E Y' .

5.

Irreducible Components

139

(A) We begin with one arbitrarily chosen function f E Y'. The equation f 0 gives an analytic set5 S of codimension 1 . We decompose S into irreducible components si in a neighborhood U(zo) (given by pseudopolynomials in c X G'), and we choose the neighborhood U so small that the Si stay irreducible in the whole neighborhood. =

(B) Next we try to obtain codimension 2. If every function f

E .'/'

vanishes identically near z0 on Si, we leave Si unchanged (and have it as a codimension ! component for our S* ) . Otherwise, there is an f' E :7 that does not vanish identically in any small neighborhood of z0 on Si. We apply Theorem 5.2: After an arbitarily small linear change of the coordinates z' the set Si n { f' 0} is a finite union of irreducible embedded-analytic sets Sij of codimension two, which stay irreducible if we pass to some smaller neighborhood of z0 . The Sij are embedded in the zero set of two pseudopolynomials wi1 ( z1 ; z3 , . . . , z,) and w�(z2 ; z3 , . . . , zn )· =

(C) Now codimension 3 follows. For this we need consider only the Sij . Leave Sij unchanged if every f vanishes on Sij (and get codimension 2 components for S* ). Otherwise, find an f" E :7 not vanishing identically on Sij , and (after an arbitrarily small coordinate change of the variables z" (z3 , . . . , z,)) the set Sij n { ! " 0} is the union of a finite set of irreducible embedded-analytic sets Sij k , given in the zero set of three pseu dopolynomials w� k (z;.. ; z4 , . . . , zn ), >. 1 , 2 , 3. =

=

=

(D) Continuing, it is possible to obtain components of codimension 1 , 2, . . . , n 1 , and finally one reaches dimension 0. If there is a ! -dimensional component S Si, .. . in - l such that not every f E :7 vanishes on S , we have to replace S by the one-point set {z0 } . Then the procedure stops. Only finitely mar,.y steps were necessary. -

=

We obtained a finite system :70 of local holomorphic functions j, f ', J " , . . . and a finite system A of irreducible embedded-analytic sets Si, Si1, Sij k , . . . and may assume that they all are defined in one neighborhood U(z0) C G, that every s E A of dimension d is embedded in a set e n- d X Gd, and that the union discriminant sets Ds C Gd C <e d belong to the embedding of S. The necessary linear coordinate change in z' can be made at the beginning of the procedure, i.e., once for all steps of the procedure. If S E A is an irreducible embedded-analytic set that has an open part in the union of the other sets of A , then it also has an open part in an irreducible S' E A, S' i= S . It follows by the mapping theorem that it is completely contained in S '. Then we simply throw it away and denote the new system again by A . After finitely many steps we have that the intersection of every 5

An exact definition of dimension and codimension of analytic sets will be given later. Here we use embedded-analytic sets, for which the dimension has already been defined. An analytic hypersurface is obviously embedded-analytic.

140

111.

Analytic Sets

S with the union of the rest of A is nowhere dense in S . Moreover, the points of S over Ds are nowhere dense in S . We denote by S* Usu t S the union of all remaining components. Then S* is given by the finitely many holomorphic functions f E Y0 . Therefore, it is an analytic set. 1f Y c Y is an arbitrary finite subset, then in a small neighborhood of z0 every f E Y vanishes at every z E S * .So S* c N(Y') . Obviously, S* is uniquely determined by this property. =

A

A

Finally, we want to show that the decomposition into irreducible embedded analytic sets is unique. For that we use the notion of regularity for points of embedded-analytic sets just as in the analytic case. Clearly, the intersection of two different S, E A contains no regular point. So the points of every S E A are regular if they are not in such an intersection and not over DA . We denote the set of regular points of S* by S* and set S S n S* for S E A . Then for S E A the sets S are the connected components of S * . Since the set S* is uniquely determined in a neighborhood of zo , this is also true for its connected components. And since the closure of S is S, the irreducible embedded-analytic components S are also uniquely determined near zo . .

.

.

=

.

.

5 .4 Theor em.

The intersection of (even infinitely m any) analytic sets is an analytic set and is locally afinite union of components that are irreducible embedded-analytic sets. This decomposition is locally uniquely determined.

Let {A ' : L E I } be a family of analytic sets in a domain G c e n , and Zo E A == n,E/ A ' an arbitrary point. We consider the system y of all local holomorphic functions f such that:

PROOF:

1 . f is defined in an open neighborhood U of z0 (depending on f). 2. f =j:. 0 near zo. 3. There is an L E I such that f vanishes near z0 on A , . As above, z0 is contained in an analytic set S* that is the union of irreducible embedded-analytic sets S and that is given by a finite subsystem Yo c Y.

If z is a point of A that is sufficiently near z0 , then every f E Yo is defined at z and vanishes on some A , and consequently at z . This shows that A c S * in a neighborhood of z0 . On the other hand, let z be a point in the intersection of S* with a small neighborhood of z0. Any analytic set A , is given by finitely many holomorphic functions fL . . . , J'N that belong to the system Y. Then by construction every ft vanishes on every embedded-analytic component S of S * ,in particular at z . Therefore, S* C A , for all L. Thus S * is contained S* near zo . in the intersection A of the A , and we have the equality A Since S* is an analytic set that has a unique decomposition into irreducible s embedded-analytic sets, this completes the proof. =

5.

Irreducible Components

141

Analyticit y . Now we are able to prove the following result, which we announced at the beginning of the section: 5.5 P ropo sition . Every embedded-analytic set A in a domain red X G c e n is an analytic set. PROOF: As in the last part of the proof of the mapping theorem, it follows that the embedded-analytic set A is given as the joint zero set of infinitely many holomorphic functions. Theorem 5 .4 shows that A is an analytic set. • Consequently, every analytic set has locally a unique decomposition into ir reducible analytic components.

The Zariski T op ology. We prove that the system of all analytic sets has the properties of the system of closed sets of a topology. 5 . 6 Theor em.

The system A the following properties:

ci

all analytic sets in a domain G

c en

and the empty set belong to A . 2. If A 1 , . . . , At E A, then also A = Ut= l A i E A . 3. If I is an index set and {A , : L E I } a collection cl analytic sets in then A = n, E J A, is also an analytic set in G .

has

1. G

G,

PROOF:

( 1 ) G is defined by the zero function, and 0 by the constant function 1 .

(2) Let z E A = A 1 U · · · U At . Then in a neighborhood U(z) there are holomorphic functions fi,J : i = 1 , . . . , l j = 1 , .. . , d, such that for all i we have U n Ai = N ( fi , l , . . . , !i, dJ · It follows that U n A = N ( h ,1, . . . ft,J1 :ji = 1 , . . . , di) ·

3) This is Theorem 5 .4.

8

So the analytic sets are the closed sets of a topology in G. We call this topology the (analytic) Zariski topology of G. It plays an important rule in complex algebraic geometry.

Global Decomp ositions . Assume that G c e n is a domain and A c G . an analytic subset. We call A irreducible if the set of regular points A c A is connected. It follows that A has the same dimension d at all regular points. This number d is called the dimension of A and is denoted by dim( A ). Every irreducible embedded-analytic set is also an irreducible analytic set. 5 . 7 Theor em.

Every analytic set A has a unique decomposition into count ably m any irreducible analytic subsets A, . The covering .9/ = { A i : i = 1 , 2, 3, . . . } is locally finite.

142

III. Analytic Sets

We decompose A into connected components. Let A' be such a component. It has dimension d in all of its points. PROOF:

We consider a point z 0 E A that lies in A' . In a neighborhood U = U(zo) C G we have a decomposition of A into finitely many irreducible embedded A analytic components 1 ' . . . , A,. By Af we denote the set of points of Ai that are not over the union discriminant set. Some d-dimensional Af meet A '. Their union A* is contained in A' and dense in A' n U . Hence, the closure of A* in U is equal to A' n U. But A * is an analytic set. From this it follows that A' is an analy!ic set, that only finitely many A' locally). Since the intersect U and that the union of all A ' is A (as it istopology of G is countable, it follows that the set of the A' is countable. • '

If A is irreducible and A = A1 arbitrary analytic sets, then A = A 1 or A = A2.

5 . 8 Corollary.

Sometimes this condition is used

as

U

A2 , where A1 , A2 are

the definition of irreducibility.

5 . 9 Proposition. Let A, B C G be irreducible analytic sets. If there is an open set U C G such that A n U -=/= 0 and A n U C B n U, then A C B .

P ROOF:

This is an immediate consequence of the mapping theorem.

•

Another corollary is the following: 5 . 10 Identity theorem (for analytic set s ) . Let A, B C G be irreducible analytic sets. If there is a point zo E A n B and an open neighborhood U = U(zo) C G with A n U = B n U, then A = B . 5 . 11 Prop osition. Let A , B c G be analytic subsets with A C B. If A is irreducible, then A is contained in some irreducible component of B . U .x. EA B.x. be the unique decomposition into irreducible Let B components. We can choose an open set U C G and a finite set {>..1 , . . . , >- t } C A such that U n A i= 0 is irreducible and U n B = (U n B.x., ) U · · · U (U n B.x.1 ) Then U n A = U n A n B = (U n A n u . . . U ( U n A n B.x.1 ) . Thus there is an index j such that U n A = U n A n Aj , so U n A c U n B.x.j . It follows • that A is contained in B.x.j .

PROOF:

=

B.x.tJ

Now we can generalize the notion of the dimension to arbitrary analytic sets. Definition . If A C G is an analytic set with irreducible components Ai, then dim( A) := supi dim( A i) is called the (complex) dimension of A.

6. Regular

and

Singular Points

143

In general, the dimension of an analytic set can be oo. But if C1 cc G is a rel atively compact subdomain, then only finitely many irreducible components intersect C1 . So the dimension of A n C 1 is finite. An analytic set is called pure-dimensional of dimension d if all its irreducible components have the same dimension d. Exercises 1. Let A be an analytic set near the origin in e n . Assume that every ir reducible component of A has dimension > 1. Show that there exists a neighborhood U = U(O) such that A n U is the union of irreducible one-dimensional analytic sets containing 0. 2 . Consider A := {(z1 , z2 ) E e 2 : z� = zr + zD. Show that A is irreducible, but A has a nontrivial decomposition into irreducible components in a small neighborhood of the origin. 3. Let A1 1 A2 c e n be analytic sets. Prove that A1--=-A2· is analytic. 4. Let {A; : i E N} be a locally finite family of irreducible analytic sets in a domain G c en . Suppose that Ai � A . for i # j and prove that the Ai are the irreducible components of theif union. Regu lar and Sin gu lar P oin t s

6.

Comp a ct Analytic Set s .

Our goal is to prove the following simple

proposition. 6 . 1 Prop osition. If G c e n is a domain and A c G a n irreducible compact analytic set, then A consists of a single point. We first prove a lemma. 6 . 2 Lemma. Iff is a holomorphic function in a neighborhood of A, then

f !A

is constant.

PROOF: We assume that the dimension of A is n - d. Since A is compact, there is a point z0 E A where Iii takes its maximum. After a linear coordinate change there is a neighborhood U = U(z0) c G and a domain G' c e n-d such that A n U is an embedded-analytic set over G'. Denote by D c G' the union discriminant set. Over every z" E G' - D our A n U has points. The point z0 lies over some z0 E C' and we may assume that it is the only point of A n U over z0. It remains to construct the elementary symmetric functions associated with f !Anu over G' - D. For this, if z1, . . . ,z 8 are mapped onto z", we define fi(z") = ai(J(zi ) , . . . ,f(zs)). These functions are holomorphic on G' - D and bounded along D. So they extend to holomorphic functions in G'. The s

:

144

III.

Analytic Sets

absolute value of every extension takes on its maximum at z�. By the max imum principle each such is constant in G'. Since the values of f can be reconstructed from the values of the f; , it follows that f is constant over G' , in particular in some open subset of A. Since A is connected, it follows that • f is constant on A and then by continuity also on A. PROOF of the proposition: All coordinate functions z; must be constant on • A . So A is a single point. A consequence is that every compact analytic subset A C G consists of only finitely many points.

Embeddin g of Analytic S et s . Assume that A c G is an analytic set in a domain G c e n , that 0 E A, and that the plane P = {zd+l = · · · = zn = 0} intersects A in an isolated point.

In a neighborhood U(O) the set A is an analytic subset d a n embedded-analytic set of dimension n - d that is defined over a domain G' in the space d variables z' = (zd+l , . . . , zn ) . If the set d ( n- d)- dimensional regular po ints is dense in A , then A is itself a n embedded-analytic set.

6 . 3 Theorem .

Remark.

No coordinate transformation is necessary for this statement !

PROOF: By definition, A is the zero set N(JI , . . . , fN ) of finitely many holomorphic functions in a neighborhood of 0. Since A n P = { 0} is isolated, there is an i such that j; does not vanish identically in any neighborhood of 0 on the z1-axis. Consequently, f; is z1-regular, and we can apply the Weierstrass preparation theorem, which implies that A is locally contained in the zero set of a pseudopolynomial w(z1 ; zz , . . . , zn ) · Now we proceed by induction on d. In the case d = 1 there is nothing t o prove. 1f d > 1 we consider the projection 7r : e n -+ en -l with Z f--t Z1 = (zz , · . . , Zn ) · To h , . . . , fN there are associated as usual holomorphic functions f 2 , . . . , f N of z' such that

1r({w = h = . . · = fN = 0} ) = A := {z' : j 2 (z') = . . . = J N (z') = 0) in some neighborhood of 0 . The intersection of P' = {z' : Zd+ I = . . . = zn = 0} with A contains 0 as an isolated point. So we have for A the same situation, but with one dimension fewer. By the induction hypothesis it follows that there are pseudopolynomi als wz , . . . , wd such that A locally is contained in the set {wz (zz ; z ") = . . . = wd (zd ; z") = 0} . By exchanging z1 with Zz, we obtain a pseudopolynomial w1 (z1 ; z") such that A is contained in {w 1 = Wz = . . . = Wd = 0} .

6 . Regular and Singular Points

1 45

- d are dense in A , we can take the union A* of those irreducible components of the embedded analytic set that contain such a regular point. Then A * and A are identical. This completes the proof. If the regular points of dimension

n

•

Remark. This result is also known as the "embeddingtheorem of Remmert and S Lein."

Again we consider a domain G c en, an analytic set A c G, and a domain G' c en- d such that 7r(G) c G' (where 1f : z M z' (zd+ I , . . . , z )). Suppose n that there exists a domain G* C G such that for every z' E G' there exists a neighborhood U U(z') c c G' with (e d x U) n G * c c (e d x U) n G and (ed X U) n A c G * . Then the following holds. =

=

If the set d regular (n - d) - dimensional po ints of A is dense in A , then A is a n embedded-analytic set over G'.

6 .4 Pr op osition .

PROOF: We take an arbitrary point zb E G'. It follows from the hypotheses that the set (ed x {z6}) n A is compact and analytic. Therefore, it consists of finitely many points. Each of these points has a neighborhood such that the restriction of the set A to this neighborhood is an embedded-analytic set over a neighborhood of zb . By multiplying the pseudopolynomials by the same distinguished variable belonging to our various points we obtain d uniquely determined pseudopolynomials over a neighborhood U'(zb) C G' such that A n (e d x U') is just their joint zero set. But these pseudopolynomials glue together to form global pseudopolynomials over G' . •

Regular P oints of an Analytic Set . Assume again that G c en is a domain, and A C G an analytic set. 6.5 Theorem .

For any z0 E A there is a fixed neighborhood U(z0 ) c G with finitely m a ny holomorphic functions JI , . . . , JN whose joint zero set is A n U such that fo r all d at every regular point z E A n U of dimension n - d the rank of their Jacobian at z is equal to d. It is remarkable that this statement is also true for a singular point z0 of A . PROOF: After applying a linear coordinate transformation in en we can find a neighborhood U U(zo) C G such that A n U is a finite union of irreducible embedded-analytic components. To give these in the canonical form a further coordinate transformation is not necessary. We denote by A' the union of all ( n - d)-dimensional irreducible components of A n U and choose pseudopolynomials w1 , . . . , wd of minimal degree such that A' is contained in the common zero set of the wi . =

�

Let D c A' be the set of points that lie over the union discriminant set DA' . Its dimension is equal to n - d - 1 . Now we carry out the proof in

146

III.

Analytic Sets

several steps and construct sets D t , D2 ,.... . . . , D n- d+ I = 0 with D i + I c Di and dim (Di ) = n - 4- i . We begin with DI :=D. The Jacobian of w1 , . . . , Wrt has rank d in A' - D1 (implicit function theorem). �

�

Next we decompose D1 into irreducible components C>.. and choose, if possi ble, in each C>.. a point z ;-, where A' is regular. We can make U so small that only finitely many C>.. occur in U. Then we apply another linear transforma tion in en that is near the identity such that in a small neighborhood of any Z>.. the set A' can be written as an ( n - d)-dimensional holomorphic graph {z

:

Zi

=

J>.. , i (Zd+l , . . . , zn )

for i = 1, . . . , d} .

Finally, we apply a linear transformation that is very near the identity to the variables ZI , . . . , Zrt such that for every >. and every point z in A' above (z�+ l ' . . . ,z�) the first d coordinates of z are distinct. Now we use Proposition 6.4. In the new coordinates (and in a slightly smaller neighborhood, which we again denote by U) the set A' is again an embedded-analytic set contained in the common zero set of pseudopolynomi als WI , . . . ,wd. We choose wi with minimal degree. We can assume that the components of D1 are still irreducible in U, and that the points z ;-, are still in U. In a neighborhood of z;-. we have a decomposition �

with w;_ i (z>.. ) -=/= 0. So the Jacobian determinant of WI , . . . ,wd with respect to the �ariables .z t ' . . . , z,t does not vanish at any Z>.. - We denote the zero set of this Jacobian in DI n U by D2 . It-has dimension n - d - 2, and WI ' . . . , Wrt , WI ' . . . ,wd have rank d on A' - D2 · �

�

Now apply the same procedure to D2 and ob.t ain an ( n- d D3 and continue in this way until reaching D n- d+ I = 0. �

- 3)-dimensional

By putting all of the pseudopolynomials together, in a small neighborhood U of zo we get holomorphic functions Jr , . . . , jd, fd+ I , . . . . fN (with N = ( n - d + 1) . d ) whose rank is d in every regular point of A' n U. Since the pseudopolynomials always were chosen with minimal degree, it follows that A' = N(h , . . . , JN ) near zo .

Now set A + = A n U - A' . It is the union of the remaining irreducible com ponents of A n U. We may assume that U is so small that A + is the common zero set of finitely many holomorphic functions gi , . . . , gs in U. Multiplying the fi by the g1 yields finitely many holomorphic functions in U that de scribe the set A n U. No point of A' n A + is a regular point of A. For every z E A' - A + there is a gj that does not vanish there. So the rank of the Jacobian of the fi . gj is equal to d at every nonsingular point of A' - A + .

6.

Regular and Singular Points

147

The same procedure can be used for every d and the corresponding A', and consequently, in finitely many steps we obtain a neighborhood of z0 and holomorphic functions in this neighborhood with the desired properties. •

The Singular L o cus. From the preceding theorem we conclude the following: 6 . 6 Theorem.

Th e set Sing(A) again an analytic set.

singular points

d

an analytic set A is

The intersection of two irreducible components of A belongs to Sing( A) . The union S of all these intersections is an analytic set.

PROOF:

Assume that z0 is a point of an irreducible component A' of A and dim(A') = n - d. Then there is a neighborhood U = U (z0 ) C G with holomorphic functions h , . . . N vanishing exactly on A' n U such that their Jacobian has rank d in each of the regular points. Let S* be the analytic set of all points of A' n U where all d x d minors of the Jacobian vanish. Clearly, S* is contained in Sing( A') n U. On the other hand, at any point of Sing( A') n U the Jacobian of h , . . . N cannot have rank d. So Sing(A') n U = S *, and Sing( A') is analytic in G.

,f

,f

The union of S and the sets Sing(A') for all irreducible components set Sing(A) . It is analytic, since the union is locally finite.

A'

is the •

The set Sing(A) is called the singular locus of A.

Extending Analytic Set s . Let G c

cc n

be a domain.

6 . 7 Lemma. Let Zo = (ziO} ' . . . ' z�0)) c e n be a n arbitrary point and E = { z : Zi = zi0l for i = 1 , .. . , d} a n affine plane of codimension d containing zo . IfA C G is a n irreducible analytic set of positive dimension that is n o t a

subset

of

E, then there is a n open dense subset C C ed such that

fc( zl , . . . , zn ) : = c1 ( z1

- zi0> ) + . . . + cd (zd - zA0> )

does not vanish identically o n Afo r every c = (c1 , . . . ,cd ) E C . In particular, fo r any hyperplane Ho C en containing E there is a hyperplane H arbitrarily close t o H0 and also containing E such that dim(Ai ) < dim( A) - l fo r every irreducible component A, of A n H . We define cp : )!A contradiction, and consequently, V must be a proper subspace of ed. For any does not vanish identically on A. c in the open dense subset C := e d - V, 0} is a hyperplane containing E . Thus He := { z : • PROOF:

:= fc·

: !ciA

fc(z)

fc

=

148

llL Analytic Sets

Our main tool for extending analytic sets is the following.

If E = {z E en : Z l = . . . = Zd = 0} is a n ( n - d) dimensional plane and A an analytic set in G - E , whose irreducible compo nents all have d imens ion n - d + l with 0 < 1 < d , then the closure A of A in G is an analytic set in G . 6. 8 P r op osition .

The proposition is of a local nature. We may assume that A is irreducible and 0 E E n G. It is enough to construct a continuation of A into a neighborhood of 0 . PROOF:

Let c = (O, . . . , O, ed+l , . . . , c,) be an arbitrary point of E n G. We consider the following family of ( d - 1 )-dimensional planes through c : For

i = 1, . . . , n - d + l j = I, . . . , d - l we have the linear map LA : e d - l -+ e n - d+ l and define P (c, A) := c t { (w', LA (w' )) : w' E So P(c, A) consists of vectors w

ed - 1}.

= (WI , · . . , wd - t , Wd - l +l • · . . , wn ) with

d-l for i = 1 , . . . , 1, :2::: aij . wj d- l ed -l + i + Z::: aij . w, for i = l + 1 , . . . , n - d + l . ,=1 Then every P( c, A ) meets E exactly in c = ( 0 , . . . ,O,ed+l , . . . ,en )· If 0 is the zero matrix, then P( c) == P( c, 0) is the plane ed - l { ( 0 , . . . , 0, ed+l ' . . . ,en )} ' �1

X

In the next chapter we will introduce Grassmannian manifolds and a topology on the set of linear subspaces (with fixed dimension) of a given vector space. In our case it follows that a neighborhood of c + Po is given by the set of all planes P = c + P with P ffi (Ox e n - d+ l ) = e n . This shows that every ( d - l ) dimensional plane through c that is near P(c, O)is of the form P ( c, A ) . We choose real numbers 0 < r1 < r2 and r > 0 so small that the "shell" S = {z = (

'

z ,z

"

) E ed - l e n- d+ l . r1 x

" ' < li z II < r2 and i z l < r }

is a relatively compact open subset of G. Only finitely many irreducible com ponents A , of A enter S . We can find a hyperplane H1 containing E that intersects the A, not at all or in codimension 1. Let A , be the finitely many irreducible components of H1 n A , that enter S. We can find another hyper plane H2 containing E that intersects all A , at most in codimension 2. We continue this procedure running through the irreducible components of the A, n H2 . After finitely many steps we have n - d + l hyperplanes such that their intersection P is a plane P( c, A ) that meets A n S in at most finitely

6. Regular and Singular Points

149

many points. By the above lemma we can choose this plane arbitrarily near

Po := P(c).

Now apply a linear coordinate transformation very near to the identity that leaves E invariant and maps our plane P (c, A) onto Po and replace the transformed shell S by a new shell S' (in the new coordinates) that is a little bit smaller such that S' is contained in the old (tran sformed) S. Thi s can be done so that 8S' n P0 n A 0. =

Earlier we proved that A is an embedded-analytic set in a neighborhood of the points of the intersection A n P0 n S' over a domain G' in the space of variables Zd- l + I , . . . , Zn . So there is a small closed ball B c cn- d+ l around the origin such that (cd - l x B) n 8S' n A remains empty, every irreducible component of (<e d- l x B) n S' n A enters P0 n S', and every plane through a point of B and parallel to P0 intersects A n S' in at most finitely many points.

Zd+ I , . . .

,

Z

n

S'

E

c

I I . . . .

0

A n Bo

Figure

11 1.3. Intersecting A

with B0 and Po

Zl , . . . , Zd-l

Now, the set B0 {z' E <e d- l : ll z' ll < r2} x B is a neighborhood of the origin in e n , and each of our parallel planes through points of B E meets A n B0 in a compact analytic set and therefore in at most finitely many points (cf. Figure 111.3). At all of these points the set A is locally an embedded-analytic set over B. By multiplying the pseudopolynomials by the same distinguished variable that we obtained for the different intersection points over the same base point in B - E, we have that An ( cd-l X (B-E) ) is -

150

Ill.

Analytic Sets

an embedded-analytic set over B - E . The coefficients of the corresponding pseudopolynomials over B - E are bounded along E . Hence, they can be analytically extended to B. This means that A n (B0 - E ) has a unique analytic continuation to B0 • • The proposition just proved is also true if A c G - E is an analytic set, whose irreducible components all have dimension greater than n - d, since we can write A as a finite union of pure-dimensional analytic sets. As a consequence we have the following theorem.

Assume that G c en is a domain, K c G an ( n - d)- dimensional analytic subset, and A an analytic subset of G - K all components of which have d imension > n - d. Th en the closure A of A in G is a n analytic set in G. 6 . 9 Theorem of Remmert-Stein .

PROOF : If zo E K is a regular point, then K can be transformed in a neighborhood of z o to a plane E. So A is analytic in a neighborhood of zo and therefore in all regular points of K. We can replace K by the set K1 of singular points of K, which is analytic again and has lower dimension. By the same argument we show that A is analytic at all regular points of K1 . • Continuing in this way we prove that A is analytic in G.

This theorem first was proved by

R.

Remmert and

K.

Stein; see [ReSt53] .

The Local Dimen sion . We show how our results are linked with the classical dimension theory of analytic sets.

Let G c en he a domain, A c G an analytic set, and Zo E A a point. There is an open neighborhood U = U(z0) c G such that U n A is a finite union of irreducible analytic components A 1 , . . . , At . If we choose U small enough, then the A, are uniquely determined. Definition .

In the given situation the uniquely determined number dimz0 (A) := max dim(A.>-) .\ = 1 ,

.

.. , 1

is called the (local) dimension of A at z 0. The set A has dimension 0 at z0 E A if and only if there is an open neigh borhood U = U(z0) c G such that U n A = {z0}.

Let k : = dimz0 (A) be positive. Th en k is the smallest number with the property that there are holomo rphic functions . . . , ik in a small neighborhood U of z0 such that z0 is isolated in A n N(JI , . . . , !k ) ·

6 . 1 0 Pr oposition .

h,

PROOF: If dimz0 (A) = k, then there must be at least one k-dimensional irreducible component Pt of A at z0 .

6 . Regular and Singular Points

151

If f is any holomorphic function near z0 , then either !l A' 0 (and therefore A n N ( f) still k-dimensional) or A' n N(J) is ( k - I)-dimensional. So at least k functions are required. _

On the other hand, by Lemma 6 . 7 we can find a holomorphic function h near z0 that does not vanish identically on any irreducible component A' of dimension k at z0• It follows that A' n N(h ) has dimension k - 1 for all such components A '. We can repeat this process, and after k steps we reach dimension zero, so that z0 is isolated in A n N ( h , . . . , fk ) . • Definition . If A has dimension k at z0 , then any system {h , . . . , fk } of holomorphic functions with AnN( h , . . . , fk ) = { z0} is called a parameter system for A in z0 .

6. 1 1 Ritt' s lemma. Let B C A be closed analytic sets in a domain G C en . Th en B is nowhere dense in A zj and only if dimz (B) < dimz (A) for every z E B . PROOF:

Let the criterion be fulfilled, and z0 be an arbitrary point of B. Then there exists an open neighborhood U of z0 in G and a parameter system { it , . . . , fk } on U for B at z0• Since dimz0 (A) > k, it is not possible that z0 is isolated in AnN( h , . . . , fk ) · This means that (A-B) nN(h , . . . , fk )nW -=1= 0 for every neighborhood W = W ( z0) . So B is nowhere dense in A. On the other hand, let B be nowhere dense in A, and z0 a point of B . In a small neighborhood U of z0 we have unique decompositions into irreducible components: BnU

=

B1

U

...

U Bm

and

A = A1

u

· · .

u

A1 .

Every component B; is contained in a component Aj(i ) , and for any open neighborhood W = W(zo ) we have (Aj (i) B; ) n W -=1= 0, because otherwis e, there would exist points z E W n B; where B is dense in A. So dim( B;) < dim(Aj(i ) ) for all i . It follows that dimz0 (B) < dimz0 (A) . • _

Let G c en be a domain, A c G an analytic set, and Z o E A a point. 1f dimz0 (A) = k , then the number n - k is called the codimension of A at z0. 6 . 1 2 Second Riemann exten sion theorem . Suppose that n > 2 and that the analytic set A C G has everywhere at least codimension 2 . Th en any holomo rphic function f on G - A has a holomorphic extension to G . PROOF:

We may assume that A is irreducible of codimension d > 2. If z0 is a regular point of A, then there is a neighborhood U of z0 such that U n A

152

III.

Analytic Sets

is biholomorphic ally equivalent to an open subset of a linear subspace E of codimension d. By the theorem on removable singularities (see Section II. l) f can be holomorphically extended to zo . We repeat this procedure. Beginning with the set Sing(A), which has codi mension d + 1 , after finitely many steps only a set of isolated points remains. Since f can also be extended to these points, we obtain the desired result. • Exercises 1 . Let A 1 , A2 c en be analytic sets. Show that dim(A 1 n A2 ) > dim( A! ) +

dim(A2 ) - n. 2. Assume that G c en is a domain and j a nonconstant holomorphic function on G. Prove that there is an at most countably infinite set Z c e such that A, := {z E G : f(z) = c ) is regular for c E e - Z. 3. Let G c en be a domain and A c G an analytic set. Prove that for any k > 0 the closure of the set A k : {z E A : dimz (A) = k } is either empty or is a pure k-dimensional analytic subset of G. 4. Let G c en be a domain and fl , . . . , fm holomorphic functions on G. Denote by N the common zero set N(fi , . . . , fm ) · Show that if the rank of the Jacobian of h , . . . , fm at some point zo E N is equal to r, then there is a neigborhood U U(z0) c G and a closed submanifold M c U of dimension less than or equal to n - r such that U n A c M . 5 . Let A be an analytic set in a domain G c en. For every point Zo E A the set lz0 (A) := { (J)z0 E Oz0 : f \ A 0} is an ideal in Oza · Show that there are a neighborhood U of z0 and holomorphic functions h , . . . , fk on U such that: (a) A n U = N ( h , . . . , fk ) · (b) lz0 (A) is generated by the germs (JI )z0 , • • • • (Jk )z0 • Show that the vector space n Tzo (A) = w E en l::: w vfzv (zo ) = 0 for every f E lz0 (A) =

=

{

v=l

has dimension n - rkz ( h ' . . . , fk ) . It is called the Zariski tangential space of A at zo. Let A be defined as in Exercise 5, and suppose that z0 is a regular point of A. Show that Tz0 (A) is the set of tangent vectors a(O) (see Section I. 7) , where a : I -+ en is differentiable, a(I) c A, and a ( O ) Zo . Let A be an analytic set in a domain G c en ' and let Zo E A be an arbitrary point. The embeddinf? dimension of A at zo is the smallest integer e such that there is an open neighborhood U = U(zo) and a closed submanifold M c U of dimension e with A n U C M . It is denoted by embdimz0 (A). Prove that z f--t embdimz (A) is upper semicontinuous on A, and that embdimz0 (A) dimc (Tzo (A)). Consider the analytic set A { w exp ( l /z ) } c {( w , z ) E e 2 : z -=!= 0}. Determine the closure of A in e 2 . 0

6.

}

=

7.

a.

=

=

=

C h ap ter IV

C omp lex M an ifold s 1.

T h e C omp lex Str u ctu r e

C omplex C oor din ates . Let X be a Hausdorff space, i. e . a topological space satisfying the Hausdorff separation axiom. Sometimes such a space is also called a separated space or a 2'2-space. Hausdorff spaces are the most common in topology (for example, every metric space is a Hausdorff space), but non-Hausdorff spaces do arise, in particular in algebraic geometry. The space en with the Zariski topology is not Hausdorff. ,

We think that a space X is too big if there exists a discrete subset with the cardinality of the continuum. Therefore, we demand that the topology of X have a countable base. In this case X is said to satisfy the second axiom
Vv C Ur(v) for every v E N. The refinement map is not uniquely determined, but we can fix it once and for all. A covering Y {Vv : v E N } is called locally finite if each x E X has a neighborhood U U(x) such that U meets only finitely many Vv. =

=

Definition . A Hausdorff space X is called paracompact if every open covering f/ of X has a locally finite open refinement W. Every compact space is paracompact. Furthermore, every locally compact space with countable basis is paracompact.

154

IV. Complex Manifolds

For the moment we only assume that X is a Hausdorff space. Definition. An n-dimensional complex coordinate system ( U,cp) in X consists of an open set U C X and a topological map cp from U onto an open set B c en . If p E X is a point, then every coordinate system ( U, cp) in X with p E U is called a coordinate system at p . The entries z; in z = cp(p) are called the complex coordinates of p (with respect to ( U, cp) ) .

If j is a complex function in U, we can consider it as a function of the complex coordinates zt , . . . , Zn , by Two (n-dimensional) complex coordinate systems ( U,cp) and (V,1/;) in X are called (holomorphica lly) compatible if either U n V = 0 or the map

cp o 1/J - 1 : 1/J ( U n V) --+ cp(U n V) is biholomorphic (see Figure IV . I ) .

u

Figure IV. l .

v

Change of coordinates

The sets B,p : = 1/J ( U n V ) and B'P := cp(U n V ) are open subsets of e n . 1f z; (respectively Wj ) are the complex coordinates with respect to cp (respectively '!(;) , then compatibility of the coordinate systems means that the functions z; = z;(W t , . . , w,) and wj = Wj (zt , . . . , zn ) are holomorphic. .

1 . The

Complex Structure

155

A covering of X with pairwise compatible n-dimensional complex coordinate systems is called an n-dimensional complex atlas on X . Two such atlases A 1 and A2 are called equivalent if any two coordinate systems ( U,cp) E A 1 and (V, 1/;) E A2 are compatible. An equivalence class of en-dimensional) complex atlases on X is called an n-dimensional complex structure on X. It contains a maximal atlas that is the union of all atlases in the equivalence class. Definition. An n-dimensional complex manifold is a Hausdorff space X with countable basis, equipped with an n-dimensional complex struc ture. Every complex manifold is locally compact and paracompact. Examples 1 . The complex n-space en is an n-dimensional complex manifold. The

complex structure is given by the coordinate system (en , id) . 2. If X is an arbitrary n-dimensional complex manifold, then any nonempty open subset B c X is again an n-dimensional complex manifold. For p E B there is a coordinate system ( U,cp ) in X at p . Then (Un B, cp l uns ) is a coordinate system in B at p. All of these coordinate systems are compatible. 3. Let G c en be a domain and X c G a k-dimensional complex sub manifold. Of course, X is a Hausdorff space (in the relative topol ogy) with countable basis. For z0 E X there are open neighborhoods W W(z0) c G and B B(O) c en and a biholomorphic map F : W --+ B such that =

=

F(W n X)

=

{(w 1 , . . . , wn ) E B : wk + I

=

·

· · =

W n

=

0} .

Let pr' : en --+ e k be the projection (w l , . . . ' W ) M (w l , . . . ' wk ) · We n define U : W n X and cp := pr' o F : U --+ e k . Then ( U,cp) is a k dimensional complex coordinate system in X at z0. If ( V,'lj; ) is another coordinate system, with 1/J pr' o E , then =

=

is holomorphic. So we get a complex structure on X . 4. Finally, let ( G,1r) be a Riemann domain (over C " ). Then G is a connected Hausdorff space, and for every p E G there is an open neighborhood U = U(p ) such that B == 1r (U) is open and cp : 1r l u : U --+ B is topological. Then ( U, 'P) is a complex coordinate system. If 1j; = 1r l v is another coordinate system, then for x E U n V we have cp(x ) 1/; (x) = 1r(x) : z and =

=

=

156

IV. Complex Manifolds

Therefore, the coordinate systems are compatible. We get a complex structure on G. One can prove that G has a countable basis (cf. [Gr55] , §2] ). So every Riemann domain over en is an n-dimensional complex manifold.

H olomorp hic Fun ctions. Let X be an n-dimensional complex mani fold. Definition . A complex function f on an open subset B c X is called holomorphic if for each p E B there is a coordinate system ( U, cp) at p such that f o cp- 1 : cp(U n B ) --+ e is holomorphic. If z1 , . . . , zn are the complex coordinates with respect to ( U,cp) , then 1 (z1 , . . . , zn ) f--t f o cp - (z1 , . . · , Zn )

is a holomorphic function in the ordinary sense. If zv = zv ( w1 , . . . , Wn ), where w1 , . . . , w, are the complex coordinates with respect to a coordinate system ( V, 7/!), then f 0 7/•- 1 ( W1 , . . . , Wn ) = f 0 \0- 1 ( Z1 ( Wt , . . . , Wn ) , . . . , Z n ( W1 , . . . , Wn ) ) is also holomorphic. So the definition of holomorphy is independent of the coordinate system. We denote the set of holomorphic functions on B by O(B). It is a @-algebra with unit element. Examp le Let G c en be a domain and X c G a k-dimensional complex submanifold. We consider a complex coordinate system ( U,cp) in X , where U is the inter section of X with an open set W c G and cp pr' o F , with a biholomorphic 0}. map F : W --+ B C en such that F(U) {w E B : Wk +1 = . . . w, If f is a holomorphic function on G, then =

=

=

fix o cp - 1 (w1 , · . . , wk )

=

.f o F- 1(w1 , . . . , wk , O, .

=

. . , 0)

is holomorphic. Therefore, fix is a holomorphic function on the complex manifold X . 1 . 1 Identity theorem.

Let X be connected. If f, g are two holomo rphic functions o n X that coincide in a nonempty open subset U C X , then f = g. PROOF: Let W {x E X : f(x) g (x)}. Then wo -=/= 0 , since U C W. Assume that there exists a boundary point xo of W" in X and let ( U,cp) be a coordinate system at x0 with cp(x0 ) = 0. Then all derivatives of f o cp - 1 and g o cp - 1 must coincide at 0. It follows that the power series of these functions around the origin are equal. But then f g in a whole neighborhood of x0 • and this is a contradiction. =

=

=

1. The Complex Structure

157

If there were a point x E M : = X - W" that was not an interior point of M , then x would be a boundary point of W " . This shows that M must be open. Since X is connected, M has to be empty. • 1 . 2 M aximum pr inciple.

Let X be connected, f E O(X ) , and x0 E X a point such that I i i has a local m aximum at x0 . Th en f is constant. The functions f and g := f(x0) are both holomorphic on X . If (U, rp) is a coordinate system at x0 and B : = rp(U) , then fo := f o rp -1 is holomorphic on B, and l fo l has a local maximum at z0 := rp(x0 ). Thus there is an open neighborhood B' = B'(z0) c B such that fo is constant on B' and f is constant on U' := rp-1 (B'). So f l u = g l u , and by the identity theorem • f = g; i . e . , f is constant.

PROOF:

1 . 3 Cor ollary.

If X is compact and connected, then every holomo rphic

function o n X is constant. The conlinuous funclion I J I Lakes ils maximum al some poinl of • X . Now the corollary follows from the maximum principle. PROOF:

1 . 4 Cor ollary. mension in e n .

There zs no compact complex submanifold of positive di

Let X c e n be a compact connected submanifold. Then the stan dard coordinate functions zv lx must be constant, for v = 1 , .. . , n . This means that X is a single point. If X is not connected, it is a finite set of • points. PROOF:

Remark.

Another proof is given in Section 1 1 1 .6.

Riemann Sur fa ces. An (abstract) R iemann surface is by definition a ! -dimensional connected complex manifold. Example The complex plane e and every domain in e are Riemann surfaces. Recall the Riemann surface of .,fZ, X

= { ( w, z ) E e 2 : w2 = z, z -/= 0} .

Since X is a Riemann domain over e, it is a !-dimensional connected complex manifold. From the projection 1r := pr 2 1 x : X -+ e we get complex coordinate systems (U, rp) with rp : = 7r l u and sufficiently small U .

158

IV. Complex Manifolds

The function f : X -+ C with j ( w z) with f 2 = z. ,

:=

w is a global holomorphic function

Example

The Riemann sphere C C U { oo} is a compact connected Hausdorff space. We have two coordinate systems (C, rp) and (C - { 0 }, 1/;) with rp(z) = z and 1/J (z) = ljz. On @* = C n (C - { 0 }) we have rp o 1j;- 1 (z) = 1/z, and this is holomorphic. So C is a compact Riemann surface. Every global holomorphic function on C is constant, but there are nontrivial meromorphic functions, for example f(z) = z (with one pole at oo ) . Here a me romorphic fun ction on X is a function f that is holomorphic outside a discrete subset P c X and satisfies JiQJ, I f (x) l 00 for every p E P. =

=

The points of P are called the poles of f

H olomor phic M appings . Let F : X -+ Y be a continuous map be tween complex manifolds. Definition. The map F defined above is called holomorphic if for any p E X there is a coordinate system ( U, rp) in X at p and a coordinate system ( V, 1/; ) in Y at F(p) with F(U) c V such that 1/J o F o rp- 1 : rp(U) -+ 1/;(V)

is a holomorphic map .

The map F : X -+ Y is holomorphic if and only if for any open subset V C Y and any f E O(V) it foll ows that f o F E O(F- 1 ( U) ) . 1 .5 Proposition .

The proof is an easy exercise. The category of complex manifolds consists of a class of objects, the complex manifolds, and a class of sets such that to any pair ( X , Y ) of objects there is assigned a set O(X, Y ) (which may be empty) , the set of holomorphic maps between X and Y. In a general category this set would be called the set of morphisms from X to Y . For ( G F ) E O(Y, Z) x O(X, Y ) we always have the composition G o F E O(X, Z) such that the following axioms hold: ,

1 . If H o G and G o F are defined, then ( H o G) o F = H o ( G o F ) . 2. For any manifold X we have the identity map idx E O(X , X) such that idy o F F and F o idx F , if the compositions are defined. =

=

l . The Complex Structure

1 59

Another example for a category is given by the topological spaces and contin uous mappings. If we replace in our definitions above the field e by lR and the word "holomorphic" by "differentiable", we get the category of differentiable manifolds and differentiable mappings. From every n-dimensional complex manifold we obtain a 2n-dimensional differentiable manifold by "forgetting the complex structure." A holomorphic function f : X --+ e is obviously a holomorphic mapping. More generally, in the case of a Riemann surface a meromorphic function f on X may be viewed as a holomorphic mapping f : X --+ C. Definition. A biholomorphic m ap F : X --+ Y is a topological map such that F and F- 1 are holomorphic. If there exists a biholomorphic map between X and Y , then the manifolds are called isomorphic or biholomorphically equivalent, and we write X Y . "'

Remar k. 1f X is a complex manifold and ( U, cp ) a complex coordinate system with cp ( U) = B c en , then cp : U --+ B is a biholomorphic map.

C ar tesian P r o d u ct s . Assume that X1 , . . . , X , are complex manifolds of dimension n 1 , , n,. Then the set X = X1 x . . . x Xm carries a natural topology generated by the sets U = U1 x · · · x Um , U; c X; open for i = 1 , . , m. One sees immediately that X is a Hausdorff space with countable basis. .

.

.

.

.

Given complex coordinate systems (U; , cp; ) in X; , for i = 1 , a coordinate system ( U, cp ) for X by

.

.

.

, m,

one defines

It is clear that two such coordinate systems are compatible. So we obtain an n-dimensional complex atlas and a complex structure on X . The projections p; : X --+ X; are holomorphic maps for i = 1 , . . , m. .

A simple example is en

=

e

X

•

.

•

X

c.

n times The Cartesian product of two complex manifolds X 1 , X2 satisfies the follow ing unive rsal property:

Given any complex manifold Y and any two holomorphic maps F : Y --+ X1 and G : Y --+ X2 , there exists exactly one holomorphic map H : Y --+ X1 x X2 with F = p1 o H and G = P2 o H . Although trivial in our case (we set H important in more general categories.

-

(F, G) ) , this property becomes

1 60

IV. Complex Manifolds

Analytic Sub sets . Let X be an n-dimensional complex manifold. Definition . A subset A c X is called analytic if for each point p E X there are a (connected) open neighborhood U = U(p) and finitely many holomorphic functions !I , . . . , fm on U such that

UnA

= { q E U : fi (q) = 0 for i =

1 , . . . , m}.

We call A an analytic hypersurface if we can always take m = 1 . From the definition it follows that A is a closed subset of X . Locally, an analytic set in X is the same as an analytic set in an open set B c e n . So most properties of analytic sets in e n can be transferred. 1 . 6 Propositio n . If X is connected and A C X analytic, then either A = X or A is nowhere dense and X - A is connected.

Assume that A f= X . If A is somewhere dense in X , then A contains interior points (because it is closed in X ) . Since X is connected, the interior of A has a boundary point p E X - A (same argument as in the proof of the identity theorem). We take a connected neighborhood U = U(p) such that A n U = {q E U : fi (q) = · · · - fm (q) = 0}. Then U contains an open subset V (consisting of interior points of A ) where !I , . . . , fm vanish identically. By the identity theorem they vanish on the whole set U, and p cannot be a boundary point of the interior of A . This is a contradiction, and it follows that A is nowhere dense. PROOF:

If X - A is not connected, it can be decomposed into two nonempty open subsets U1 , U2 . The function f : X - A -t e with f(x) = O on U1 and j(x) = 1 on U2 is holomorphic and bounded. By Riemann's extension theorem (which can be applied locally) there exists a holomorphic function f on X that coincides with f outside A . Since f can take only the values 0 and 1 , it is locally constant. But on the connected manifold X every locally constant • function is constant. This is a contradiction.

-

-

Let JI , . . . , fm be holomorphic functions that are defined on an open subset U c X . Let p E U be a point and (V, 1/J) a complex coordinate system in X at p . The mapping f (!I , . . . , fm) : U -+ e m is holomorphic, and we define =

i

J _

-

_

This is something like a Jacobian matrix of f at coordinate system 1/J. Since

1, . . . , m 1, . . . , n

p,

).

but it depends on the

1.

161

The Complex S tructure

we have This shows that

is independent of the chosen coordinate system. Definition . An analytic set A c X is called regular (of codimension d) at a point p E A if there are an open neighborhood U = U (p) c X and holomorphic functions h, . . , fd on U such that: 1 . A n u = { q E u : fi ( q ) = . . . = fd ( q ) = o } . .

2.

rkp (fi , . . . , fd )

=

d.

The number n - d is called the dimension of A at p .

If A is regular at every point, A is called a complex submanifold of X 1 .7

Propositio n . An analytic set A is regular of co dimension d at p E A if and only if there is a complex coordinate system ( U, cp) in X at p with cp(U) = B C en and cp(U n A ) = {W E B : Wn- d+ l = . . = w, 0} .

=

.

If A is a complex sub manifold cl X , then A itself is a complex manifold.

PROOF: Let (U, 1j;) be an arbitrary coordinate system at p and W := 1/; ( U) . Then A == 1/; (A n U) is an analytic subset of W that is regular of codimension d at z0 := 1/J(p), and there exists a biholomorphic map f!rom W onto an open { w : Wn- d+ l 0 and f ( A ) neighborhood B B( O ) c en with f(zo ) . . . w. 0). We take cp : f 1/;. _

=

=

=

=

=

=

o

If A c X is a submanifold, then A inherits a natural complex structure from • X . This can be demonstrated in the same way as in the case X en . =

Example Let F : X -+ Y be a holomorphic map from an n-dimensional manifold into an m-dimensional manifold. Then GF

is called the graph of F.

:=

{(x , y) E X

x

Y : y

=

F(x) )

162

IV. Complex Manifolds

Let Po E X be a point and qo := F(p0) E Y. We choose coordinate systems ( U,cp ) in X at p0 and ( V,?,b) in Y at q0 , with F(U) C V. Then ( U x V,cp x ?,b) is a coordinate system in X x Y at (p0 , q0 ) E G F · Writing 1,b o F = (!I , . . . , fm ) we get 1 GF n (U x V ) = { ( cp x 1,b) - ( z , w ) : j;

a

1 cp - ( z ) - w; = 0 for i = 1 , . . . , m}.

So GF is locally defined by the functions gi (p, q ) := j; (p) - w; o 1,/J( q ) , for i = 1 , . . . , m. Since rk (po ,qo) (g1 , . . . ,gm ) = m, we see that G F is an n-dimensional submanifold. The diagonal �x c X x X is a special case, which is given as the graph of the identity: �x = { (x , x') E X x X : x = X'}. Example

Let A = { (w, z1 , z2) E C3 : w2 = Z1 Z2 } · The projection p : (w , z1 , z2 ) M 2 (z1 , z2 ) realizes A as a branched covering over C that is the zero set of the pseudopolynomial w( w ; z1 , z2 ) = w2 - z 1 z2 . Outside the discriminant set Dw = {(z1 , z2 ) : z1 z 2 = 0} it always has two regular leaves over C2 . So A is everywhere 2-dimensional and regular outside Dw . It is even regular outside the origin, since V'w(w, z1 , z2 ) vanishes only at ( 0 , 0 , 0 ). One can show that 0 is, in fact, a singular point; e . g . , by using Exercise 8.2 in Chapter I. 2 The map cp : C -+ A with cp(t1 , t2 ) : = ( t1 t2, ti , t§ ) is surjective. We call it a unifo rmization. The Jacobian

vanishes exactly at (0, 0) . The image point (0, 0 , 0) = cp(O, 0 ) is then called a nonuniformizable point. Analogously to the situation in en one proves that the set S ing ( A ) of singular points of an analytic set A is a nowhere dense analytic subset. The set A is called irreducible if A - Sing (A) is connected. To every analytic set A C X there is a uniquely determined locally finite system of irreducible analytic sets (A>.) >.cA such that A is the union of all these irreducible components A>. .

Differ entiab le Function s. Let X be an n-dimensional complex man ifold, B c X an open set. A function f : B + C is called differentiable (respectively smo oth), if for every complex coordinate system ( U,cp) with U n B -!= 0 the function j o cp - 1 is differentiable (respectively infinitely dif ferentiable) on cp(U n B ) . We denote the IR-algebra of real-valued smooth

1 . The Complex Structure

163

functions on B by g(B ) and the «:>algebra of complex-valued smooth fuc tions by g ( B, q.

1 . 8 Pr op osition. In every complex manifold X there is a sequence of com

pact subsets (K, ) with Kn C (Kn + 1 t and U:' 1 Kn = X .

PROOF: Since X is locally compact with countable basis, we can find a countable basis ( Bv ) v E N of the topology of X such that each Bv is compact. We take K1 := B 1 . lf n 1 is the minimal number such that K1 C B 1 U · · · UB nu then k 1 > 2, and we take K2 := B 1 U . . . U Bn 1 , and so on. •

We call ( Kn ) n EN a compact exhaustion of X

1 .9 Pr oposition. Let an open covering fl = { U, : L E I } if X be given, and two real numbers r, r' with 0 < r' < r . Th en there is a locally finite open

refinement Y = { V, : A E L } of fl such that the following hold:

For each A E L there exists a complex coordinate system (V.>- , 'P>- ) in X with 'P .>- (V.>-) = Br ( O ) . 2. Th e open sets 'P.>- 1 (Br' ( 0 ) ) also cover X . 1.

PROOF: We use a compact exhaustion ( K, ) and define M1 := K1 and Mn == Kn (Kn - 1 t for n > 2. Then (Mn ) is a covering of X by compact sets. -

We consider a fixed M = Mn . For each x E M there is an index L = L (x) E I and an opcn ncighborhood V = V(x) c U, ll ((Kn + l t - Kn - 2 ) · Wc can makc V so small that there is a complex coordinate system rp : V --+ Br (O ) with rp(x ) = 0, and we define V' : = rp- 1 ( Br' ( 0) ) . The set M is covered by finitely many neighborhoods V� 1 , , V� m like our V'. Then . n .

•

.

.

Y := { Vn i : n E N, i = ,

l , ..

.

, mn } 8

is the desired covering.

Definition . A (smo oth) partition of unity on X is a family (rp, ) ,EI of smooth real-valued functions such that: 1 . rp, > 0 everywhere. 2. The system of the sets supp( rp, ) is locally finite. 3. E, Ef rp, = 1.

1 . 10 Theor em. For any open cowering fl = { U,

pa rtition

d

unity ( rp, ) with supp(rp , ) C U, .

:

L

E I } of X there is a

164

IV. Complex Manifolds

PROOF : We have a locally finite refinement Y = { VA : A E L } of f/ and complex coordinates cpA : VA -+ Br (O) as in Proposition 1 .9. If 11, : e n -+ lR is a smooth function with 0 < 1/;(z) < 1 , 1/; (z) E 1 on Br' (O) and 1/;(z) = 0 on e n - B, ( 0 ) ,we define a smooth function 1/;A on X by 1/;A = 1/J 0 cpA on VA and 1/;A (x ) = 0 otherwise.

Let T : L -+ I be a refinement map (with VA c U (A ) ) · Then 1f/ = { W, : L E I } with W, := U AET -1 (, VA is an open refinement of f/ with W, C U, . ) In addition it is locally finite, because for 17 E X there is a neighborhood P = P(x) such that P n VA -1= a only for A E La, La c L finite. But then P n W, -1= 0 only for L = T(A ) , A E La. 7

We define cp, : = 'l: AET - 1 ( ,) 1/JA . The sum is finite at every point. So cp, is c A 1 ( Br' (0)) , smooth and has its support in W, . Every 11 E X lies in a set p where 1/;A is positive. Therefore, cp == '2: , cp, is well defined and everywhere • positive. Now we can define the partition of unity by cp, == cp, j cp. 1 . 1 1 Cor ollar y. Let U C X be an open set and V CC U a n open subset. Then there exists a function f E G"(X) with f lv = 0 and f l cx-u) = 1 . PROOF: The system { U, X - V} is an open covering of X . Let { cp1, cp2 } be a partition of unity for this covering. Then supp(cpl ) c U, supp(cp2 ) c X - V, • and cp1 + cp2 = 1 . We take f == 'P2 ·

Tangent Vector s . Let X be an n-dimensional manifold and a E X an arbitrary point. Defin ition . such that

A derivation on X at a is an R-linear map

v[f · g]

=

v[f] · g (a) + f(a) · v[g]

v

:

G"(X) -+ lR

for f, g E G"(X)

If c is constant, then v[c] = 0 for every derivation v .

1 . 1 2 Pr op osition . Iff E G"(X) and f l u = 0 fo r some open neighborhood U = U(a) C X , then v[f] = 0 fo r every derivation v at a. PROOF: We choose a function g E G"(X) such that g l v = 0 for some open neighborhood V = V (a) CC U and g l cx - u ) = 1 . Then g . f = f, and from the derivation rule it follows that

v [ f] = v [g . f] = v [g] . f(a) + g (a) . v[f] = 0 . •

1.

The Complex S tructure

165

1 . 1 3 C or ollar y. Iff,g are twofunctions of g(X) with f l u = g l u fo r some open neighborhood U = U( a ) C X, then v[f] = v[g] fo r every derivation v at the point a . It follows from the corollary that we can restrict derivations to locally defined functiOnS and WOrk With COOrdinateS. If cp = ( Z 1 , . - - , Zn ) : U -+ en iS a coordinate system at a and zv = xv + i y, then we define partial derivatives at a by

( a[}y, ) a [!] : = (f

0 cp- 1 )y, (cp(a) ) ,

for i = 1 , . . . , n. The partial derivatives depend on the chosen coordinate system, but once we have made our choice, every derivation v at a has a unique representation'

v=

b

t a ( a� ) a + t bi ( a8y, ) a ' i= 1 i= 1 ;

'

with ai = v[xi] and i = v[yi] for i = 1 , . . , n. .

In en the space of derivations is isomorphic in a natural way to the space of tangent vectors. But what is a tangent vector on a complex manifold X ? We start with a differentiable path a : I -+ X , where I c lR is an interval with 0 E I, and a(O) = a . Let (U, cp) be a coordinate system in X at a. Then we a,) and get the tangent vector can write cp o a = ( a 1 , .

.

.

•

( cp o a)' ( 0 ) = (a � ( 0 ), . . . , a� (0)).

Unfortunately, this vector depends on the coordinate system. But a tangent vector at a should somehow be completely determined by a pair (p ,c ), where cp is a coordinate system at a and c = ( c1 , . . . , c,) E en an arbitrary vector. In this sense the tangent vector to a is given by the pair (cp, (cp o a)'(O ) ) . If we take another coordinate system 1/;, then (1/; o a) ' (O ) = (cp o a) ' (O) . J'I/J o
Therefore, we call two pairs ( cp, c ) and ( 1/J, c' ) equivalent if the Jacobian of 1/J o cp- 1 at cp( a) transforms c into c' , i.e., if

c = c' . Jcpo'I/J ' ( 1/; (a )) t . An equivalence class is called a tangent vector at a . The set Ta (X) of all tangent vectors at a is called the tangent space. It carries the structure of a complex vector space, which can be defined on representatives: 1 For the proof use the fact that every smooth function f on a domain G c en has near Zo E G a unique representation f(z) = L�= l gv (z) (xv - xe) + L �= l hv (z) (Yv ye) with gv(zo ) = fxv (zo ) and hv (zo ) = fYv ( zo ) -

IV. Complex Manifolds

1 66

( rp, c i ) + ( rp, c2 ) .A · (rp, c ) .

-

(rp, c1 + c2 ) , (rp, .\ - c ) .

The C omp lex Str u cture on the Sp ace of Der ivation s. If f = g + ih is a complex-valued smooth function on the open set B c X and v a derivation at a E B, then we defme v[f] : =v[g ]

+ i v[h]

.

1 . 1 4 Pr op osition . For every c E en and every coordinate system rp at a there is a unique derivation v at a such that v[f] = = c . V(j

o

rp- 1 ) (rp(a)) t fo r every holomorphicfunction f

The derivation v depends only o n the equivalence class of ( rp, c ) . P ROOF: If a coordinate system rp = (z 1 , . . . , zn ) with zv = x v vector c = a + ib are given, then v can be defmed by

+ i yv and a

t (�) t ( )

8 av a + bv a v a v= Yv a v= 1 1 If f is a holomorphic function, then fYv = ifxv and fxv = fzv · Consequently, v

:=

n

v[j] = :l) a v + i bv)(J o rp- 1 ) xv (rp(a)) = C · '\! (! o rp- 1 ) (rp(a)) t . v 1 o=

The uniqueness follows from the equations v[x v ]

+ i v [y v]

= v[zv] = Cv

If the pair ( rp, c) is equivalent to ( ?,b, c '), then c'

. '\1

( f o 1,b - 1 ) ( 1,b (a) ) t

c ' · Jcpo'I/J (1,b (a) ) t · '\1 (J o rp - 1 ) ( rp( a)) t . V ( fo rp - 1 ) ( rp( a)) t . -1

-

c

Therefore, v is determined by the equivalence class of ( rp, c ) .

•

The assignment ( rp, c ) f--t v induces a real vector space isomorphism between the tangent space Ta (X) and the space of derivations at a. It follows that the tangent space has complex dimension n . The pair ( rp, ev ) is mapped onto the derivation (8/ox v )a , and (rp, ie,) onto (o jo yv ) a 1 . 1 5 Pr op osition . A complex structure o n the space cl derivations at a is

given by

J(v ) [j] = i . v [f], fo r every holomorphicfunctio n f .

1 . The Complex Structure Obviously, J is lR-linear, and J o J(v) = -v.

P ROOF:

167

•

If v corresponds to the tangent vector given by ( cp, c ) , then J ( v) corresponds to ( cp, ic) . One must distinguish carefully between the real number J (v) [ !] and the complex number i . v [ !] if f is a real-valued smooth function. The differential operators ( a I azv ) and ( aI az v ) are not real-valued deriva tions, and therefore they do not correspond to tangent vectors. But they are nevertheless useful. If v is a derivation, then a

a

n

n

u=l

v=l

2::= av u o cp - 1 )x v + � b v ( f o cp- 1 )y v

v [ Jl

n

L av . ( (f 0 cp- 1 ) zv + ( J 0 cp- 1 )zv )

V= [ n

v= 1 n

n

u=l

if Cv := av

v

=1

+ ib, for v = 1 , . . . , n.

Therefore, every (real-valued) derivation can be written in the form

v=

I:C v (a� ) + t cv ( � ) v Zv v=l v=l a

a

·

The Indu ced M apping. Let F : X --+ Y be a holomorphic map be tween complex manifolds. Let x E X be an arbitrary point, y := F( x) € Y. Defin ition . defmed by

The tangential map F* = (F* )x : Tx (X) --+ Ty (Y) 1s

( F* v) [g ] = = v [go F] , for derivations v and functions g E G"(Y) .

The map F* is linear, acting on tangent vectors as follows:

if cp is a coordinate system at x, and 1/J is a coordinate system at

y.

Now we have an assignment between the category of complex manifolds (with a distinguished point) and the category of vector spaces. To any manifold

168

IV.

Complex Manifolds

X and any point a E X there is associated the tangent space Ta (X) . To any holomorphic map F : X -+ Y with F(a) = there is associated the homomorphism F* : Ta ( X ) -+ Tb (Y ) . This assignment has the following

b

properties:

(idx ) * (G o F) ,

-

i drx (X) ' G * o F (if G *

:

Y -+ Z is another

holomorphic map).

Such an assignment is called a covariant functo r. If it interchanged the order of the maps, it would be called a contravariant functor. Remark. For historical reasons the elements of the tangent space are called contravariant vectors and the elements of its dual space covariant vectors. But the tangent functor behaves covariantly on the tangent vectors and con travariantly on the covariant tangent vectors in Ta (X ) '· One should keep this in mind.

Immer sion s and Sub mer sion s. We are particularly interested in the case where the (local) Jacobian of a holomorphic map F : X -+ Y has maximal rank. If n = dim (X) and m = dim(Y) , then the rank is bounded by min(n, m ) . Only two cases are possible: Definition . The holomorphic map F is called an immers ion at x if rk(F* ) = n < m, and F is called a s ubmersio n at x if rk(F* ) = m > n. In the first case ( F* ) is injective; in the second case it is surjective. x

We call F an immersion (respectively submersion) if it is an immersion (re spectively submersion) at every point x g X. Remark. If F : X -+ Y is an inj ective immersion, then for every x E X there are neighborhoods U(x) c X and V(F(x)) c Y such that F(U) is a submanifold of V . In addition, if X is compact, then F(X) is a submanifold of Y. We omit the proof here. 1 . 1 6 Theorem .

Let tio ns are equivalent:

xo E X

be a point,

y0

== F( xo) . The following condi

1 . F is a submersio n at xa. V(yo) C Y with 2. There are neighborhoods U U(x0) C X and V F(U) c V , a manifo ld Z, and a holomorphic map G : U -+ Z such that x f--t (F(x) , G(x)) defines a biholomorphic map fro m U to an open subset d V X Z. 3 . There is an open neigborhood V = V( Yo) C Y and a holomorphic map s : V -+ X with s (y0) = x0 and F o s = idv . ( Then s is called a local section for F. ) =

=

1.

The Complex Structure

1 69

PROOF: ( 1 ) ===? ( 2 ) : We can restrict ourselves to a local situation and assume that U = U(O) c en and V = V(O) c em are open neighborhoods, and F : U -+ V a holomorphic map with F(O) = 0 and rk(Jp (O)) = m.

We write Jp (O) = (J;, ( o ) , J� (O) ) , with J;, (O) E Mm,m (C) and J� (O) E Mm , n-m (
JF (O) =

( J;, (O) 0

J� (O)

En-m

),

for z' E

e m , z" E en-m ,

and therefore det JF (O) -=/= 0.

By the inverse function theorem there are neighborhoods U(O) c U and W(O) c en such that F : U -+ W is biholomorphic.

Z := e n-m

We observe that is a complex manifold, and G : = pr, : U -+ Z with (z' , z" ) f--t z" is a holomorphic map such that ( F,G) = F is biholomor phic near 0 , �

( 2 ) ===? (3) : If U, V , Z, and G are given such that F(U) c V and (F, G) U -+ W c V x Z is biholomorphic, then s : V -+ X can be defined by

s(y) : = (F, G) - 1 (y, G(x0)). Then (F, G) (s(y0 )) = (y0, G(x0)) = (F, G)(x0 ) , and therefore s(y0) = x0 . Furthermore, (F,G) o s(y) = (F,g) o (FG)- 1 (y, G(x0 )) = (y, G(x0)) . Thus F o s(y) = Y · (3) ===? ( l ) : If s is a local section for F, with s(y0 ) = x0, then F* o s * (v) = V for every v E Ty 0 (Y) , Thus it follows immediately that F* is surjective. • L

If F : X -+ Y is a submersio n, then fo r each y E Y the fiber F- 1 (y) is empty or an ( n - m)-dimensiona l submanifold if X , In the latter case Tx (F - 1 (y )) = Ker((F* )x) for all x E F - 1 (y) . 1 7 Cor ollary,

PROOF : We consider a point x0 E X , Let M := p - 1 (yo ) be the fiber over y0 : = F(x0). Then we can find neighborhoods U = U(x0) C X , V = V(y0 ) c Y , a manifold Z, and a holomorphic map G : U -+ Z such that ( F,G) : U -+ W C Vx Z is biholomorphic. It follows that M n U = (F, G)- 1 ( {y0} x Z) n U is a manifold of dimension n m,

-

Since FIM is constant, we have F* ITx0 (M) = 0. This means that Txo (M) c Ker(F* ). Since these spaces have the same dimension, they must be equaL •

1 70

IV. Complex Manifolds

G luing. Assume that X is a set that is the union of a countable collection (Xv)v EN of subsets such that the following hold: 1 . For every Y

E N there is a bijection cp" : X" -+ Mv , where M" is an

n-dimensional complex manifold. 2. For every pair ( YJL) E N x N the subset cpv (Xv n X, ) is open in Mv , and the map

is biholomorphic. 3. For every pair of points a E X , and b E X , with a -=/= b there are open neighborhoods U(cpv (a)) C Mv and V(cpiL (b)) C MIL with cp" 1 (U) n 1 cpiL (V) = o _

1 . 1 8 Pr op osition .

Under the above conditions there is a unique complex structure on X such that the X" are open in X and the cp" : X" -+ Mv are biholomorphic. PROOF: We give only a sketch of the proof and leave the details as an exercise for the reader.

A subset U C X is called open if cpv (U n Xv) is open in M" for every Y. Then the collection of open sets has the properties of a topology on X . In addition, for every open set W c M" the set cp" 1 (W) is open in X . Consequently, the maps cp" : X" -+ Mv are homeomorphisms. From the last hypothesis it follows that the topology on X is Hausdorff, and since the collection of the X" is countable, it has a countable basis.

If U c A1" is open and 1/J : U -+ e n a coordinate system, then 1/J := 1/J o cp" : cp" 1 (U) -+ en is a coordinate system for X . One checks easily that two such coordinate systems are biholomorphically compatible. So we obtain a • complex structure on X . One says that X is obtained by gluing the manifolds M" . Another way to describe this process is the following. Let there be given a collection of complex manifolds Mv , open subsets M"IL c Mv , and biholomorphic maps cpv11. : MIL" -+ M"IL (including 'Pw = idMJ · Consider pairs (x, v) with x E M" . Then (x, Y ) is called equivalent to (y,f-1) if X

E M"IL '

y E MIL" and x = cp"IL (y).

The set X of equivalence classes is the result of the gluing process. one has to add a condition that ensures the Hausdorff property.

Of

course,

Exer cises 1 . Let M be a compact connected complex manifold with dim(M) > 2 and N c M a closed submanifold of codimension greater or equal to 2. Show that every holomorphic function f : M - N -+ e is constant.

2.

2. Let

Complex Fiber Bundles

171

X be a Riemann surface.

(a) If f is a meromorphic function on X , and P the set of poles, show that f(x) for x E X - P, i(x) := oo for x E P

{

3.

4.

5.

6.

defines a holomorphic map j : X -+ C. (b) Prove that every holomorphic map f : X -+ C. that is not identically oo defines a meromorphic function on X . Let f : X -+ Y be a nonconstant holomorphic map between Riemann surfaces, x0 E X , and y0 : = f(x0) E Y . Prove that there is a k > 1 such that there are complex coordinates cp : U ( x0 ) -+ C. and 1/.J : V (y0) -+ C. with: (a) cp (x0 ) = 0, 1/.J(yo) = 0. (b) f (U) c V. (c) 1/.J o f o cp - 1 ( z ) = z k . The general linear group GL n (
X xz Y

:=

{ ( x , y) E X

x

Y

:

f(x) = f(y)}

is a complex submanifold of X x Y .

2.

C omp lex Fib er B u n d le s

Lie Gr oup s and Tr an sformation Gr oup s. Assume that G is a set that has the structure of a group and at the same time that of an n dimensional complex manifold. The inverse of g E G will be denoted by 9 - 1 , the identity element by e, and the composition of two elements 9 1 , g2 E G by 9192 · Defin ition . erties hold:

We call G a complex Lze group if the following two

prop

IV. Complex Manifolds

172

1 . The mapping g f--t g - I (from G to G) is holomorphic. 2. The mapping (gi , g2 ) f--t gig2 (from G x G to G) is holomorphic. There are many examples of complex Lie groups. The simplest one is the space en , where the composition is vector addition. Another example is the group e* with respect to ordinary multiplication of complex numbers. The most important example is the general linear group GL n (C) : = {A E Mn (
Its complex structure is obtained by considering it as an open subset of e n . The multiplication of matrices is bilinear, and the determinants appearing in the calculation of the inverse of a matrix A are polynomials in the coefficients of A . Every matrix A E GL n (q defines a linear and therefore holomorphic map
Then
·

=

=

·

·

=

=

=

We want t o generalize this situation. Let X be a complex manifold and G a complex Lie group. Definition. We say that G acts analytically on X (or is a complex Lie transformation group on X ) if there is a holomorphic mapping
The holomorphic map x f--t
Often we write gx instead of
r

:=

Zwi + . . . t zw2n

is a subgroup of the (additive) group en generated by WI ' . . . , W2n · The group r acts on en by
2.

Complex Fiber Bundles

173

Fiber Bundles . Let X and F be complex manifolds and G a complex Lie group acting analytically and faithfully on F . Definition. A topological (respectively holomorphic)fiber bundle over X with structure group G and typical fiber F is given by a topological space (respectively a complex manifold) P and a continuous (respectively holomorphic) map 1r : P -+ X , together with 1 . an open covering � { U< : L E I } of X , 2. for any L E I a topological (respectively biholomorphic) map =

'P<

: 1f -

l (U<) -+ UL X F

with pr, o 'P< 1r, 3. for any pair of indices 1 L , K) E I x I a continuous (respectively holo morphic) map g<,. : U< n U,. -+ G with =

for x E U<,.

:=

U< n U,.

and p E F .

The maps 'P< are called local trivializations and the maps g,,_ a system qf transition functions. Since G is acting faithfully, we get the following compatibility condition:

Now let a system of transition functions (g<,.) be given such that the com patibility condition is satisfied. Using the gluing techniques mentioned at the end of Section 1 , a suitable bundle space P, a projection 1r : P -+ X , and local trivializations can be constructed as follows: Identifying (x, p) and (x, g<,_(x)p), we can glue together the Cartesian prod ucts U,. x F and U< x F over U<><· Due to the compatibility condition this works in a unique way over U<><>-. . The obvious projection from P to X is continuous. Therefore, P is a fiber bundle over X with structure group G and typical fiber F . If the transition functions g <>< are holomorphic, then P carries the structure of a complex manifold, and the projection and the local trivializations are holomorphic. So P becomes an analytic fiber bundle in that case. Examp l e

Let X be an n-dimensional complex manifold. There is an open covering � = { U< : L E I } ,together with complex coordinate systems

IV. Complex Manifolds

1 74

We take C as typical fiber and @* as structure group, acting on C by multi plication. Then we can define transition functions g<,. : U<,. -+ C* by

The compatibility conditions are satisfied because of the chain rule and the determinant product theorem. So by the gluing procedure described above we get a holomorphic fiber bundle over X that is called the canonical bundle and is denoted by Kx .

Eq uivalence. Let 7rp : P --+ X and 7rQ : Q --+ X be two topological or holomorphic fiber bundles over the same manifold X , with the same fiber F and the same structure group G. We assume that there is an open covering {U< : L E I } of X such that there are trivializations 'P< : 1r/ (U<) --+ f/ U< x F and 1/;< : 1rQ 1 (UJ --+ U< x F . =

Defin ition . A fiber bundle isomorphism between P and Q is a topo logical (respectively biholomorphic) map h : P --+ Q with 7rQ o h 7rp such that for any L E I there is a continuous (respectively holomorphic) map h, : U< --+ G with =

The bundles are called equivalent in this case. We give a description of bundle equivalence in the context of transition func tions. Let (g: ,. ) be the system of transition functions for P with respect to f/ , and (g:�) the corresponding system for Q. Then we have 1/J< o h o cp � 1 (x, g ;,_ (x)p) 1/J< o h o cp,_ 1 (x, p) 1/J< o 1/;;; 1 (x, h,. (x)p)

(x, h< (x)g;" (x )p)

_

(x , g;� ( x)h ,. (x)p).

Since G is acting faithfully, it follows that ( C)

Two systems of transition functions (g;,_) and (g:�) with respect to the same covering are called topologically (respectively analytically) equivalent or cohomologous if there are continuous (respectively holomorphic) maps h< : U< --+ G satisfying condition ( C ) . Equivalent fiber bundles have equivalent systems of transition functions. On the other hand, it is easy to see that fiber bundles constructed from equiv alent systems of transition functions are themselves equivalent. Now we will demonstrate that the latter remains valid in passing to finer coverings.

2.

Complex Fiber Bundles

1 75

Let us assume that there are given systems of transition functions g' = (g:,.J and g" = (g:� ) with respect to a covering fl = { U< : L E I} . We call them equivalent as well if there is a refinement Y = { V, : v E N } of fl (with refinement map T : N -+ I ) and a collection of maps h, : Vv -+ G such that

To show that the systems are equivalent in the old sense, we define h< : U< -+ G by In fact, h, is well defined, since on U< n V,, we have

an d therefore g :� (v ) hv (g ;T (v ) ) - l get

The bundles are equivalent! If two bundles are given with respect to two different coverings, then they are

called equivalent if they are equivalent with respect to a common refinement, for example, the intersection of the coverings. Then everything works as above.

C omplex Vector Bundles . manifold.

Let X be an n-dimensional complex

Defin ition . A complex topological (respectively holomorphic) vector bundle of rank r over X is a topological (respectively holomorphic) fiber bundle V over X with
1 76

IV. Complex Manifolds

section (or simply section) in V over U is a continuous (differentiable, holomorphic) map s : U -+ V with 1r o s = idu . We denote by r(U, V) (or O(U, V)) the vector space of holomorphic cross sections in V over U , by tf:(U, V) the space of differentiable sections, and by �0 (U, V) the space of continuous sections. The vector bundle V is called globally generated if the canonical map r(X, V) -+ Vx with s r-+ s(x) is surjective for every x E X . Let (UL , <1\)<El be a collection of vector bundle charts
0

(X ' z . gL t< (X ) t )

for

( X ' z ) E ULt< X e r .

If s is a holomorphic section in V, then defines a system of holomorphic maps SL : UL -+ er ' and we obtain the compatibility condition s< (x) = s ,. (x) , g<,_(x) t

on

U<>< ·

On the other hand, any such system (s< ) defines a global section s . Example We can define vector bundles by giving a system of transition functions. The construction of the bundle space is carried out with the same gluing technique as for general fiber bundles. If X is an arbitrary n-dimensional complex manifold, and (U< , 'P< )<El a com plex atlas for X , then { Un � E I} . defines a system of transition functions with respect to � The corresponding vector bundle T(X ) is called the tangent bundle of X . It results from gluing (x, c) E u,. X e n with (x, c . gL,_(x) t ) E UL X e n , for x E U<,. . Therefore, we can identify the fiber (T(X) ) with the tangent space Tx (X). The local trivializations
x

For a E U< the trivialization
A holomorphic section in T (X ) is also called a holomorphic vector field.

2.

Complex Fiber Bundles

1 77

Defin ition . Let V, W be two holomorphic vector bundles over X . A vector bundle homomorphism between V and W is a fiber preserving holomorphic map TJ : V -+ W such that for any x E X a linear map TJx : Vx -+ Wx is induced. The map TJ is called a vector bundle isomorphism if TJ is bijective and TJ, 1)- 1 both are vector bundle homomorphisms.

A holomorphic vector bundle V of rank r over X is called trivial if it is isomorphic to the bundle X X c r . This is equivalent to the existence of a frame { 6 , . . . , �r } of holomorphic sections �i E f( X, V ) such that for every x E X the elements 6 ( x ) , . . . , �r (x) E Vx are linearly independent. In this case V is globally generated. But there are also nontrivial bundles that are globally generated.

A holomorphic map TJ : V -+ W is a vector bun dle homomorphism if and only if for each pair of vector bundle charts
W- l 0 1) 0
=

( 2 ,Z .

h(x) t ) .

We omit the elementary proof.

Standar d C on str u ction s . Let X be an n-dimensional complex man ifold. We can think of a vector bundle over X as a parametrized family of vector spaces. Therefore, numerous constructions from linear algebra carry over to the theory of vector bundles. 1 . The dir ect sum : If V, W are two vector bundles over X , then the direct sum, or Whitney sum, V EB W : = V x x W carries a vector bundle structure that is defined as follows:

Let "1£ = { U< : � E I } be an open covering of X such that there are vector bundle charts
can be defined by

If g<,. , respectively h,, are transition functions for V , respectively W , then the matrices

178

IV. Complex Manifolds

are transition functions for V EB W. The fiber of the Whitney sum is given by (V EB W)x = Vx x Wx · 2. The dual bun dle: Let 1r : V -+ X be a holomorphic vector bundle of rank r with tri vializations
We will show that for every x E X there is a natural isomorphism ( V' ) x -+ (Vx ) ' = HomiC ( Vx , C). Given elements x

E

U< , v

E

Vx , and A

E

' (V )x , we define

using the vector space isomorphisms (
cr .

For

X

E UL n u" we obtain (
I
X

(
This shows that the definition of A(v) is independent of the trivializations. 3 . Ten sor power s of a line bundle: Let 1r : F -+ X be a line bundle with transition functions g<" : U<" -+ @ * . Defin ition . For k E N, the tensor power p k is the line bundle defined by the transition functions l". We give an interpretation of p k using the dual bundle 7r1 : F' -+ X . As sume that there is an open subset U c X and a holomorphic function f : (1r' )- 1 (U) -+ C. If 1/;< : (F' ) I u, -+ U< x C are trivializations (with ¢< o 1/J" 1 ( x, z ) = (x, z · g<�< (x ) - 1 ) ) , then we have a power series expansion J o 1j;-; 1 (x, z ) = on

00

l:::a v ,< (x) z v

v =O

1/JL(F'I u, n u ) with holomorphic functions av,L on u n UL . Over UL n u" n u the following holds:

2.

Complex Fiber Bundles

1 79

00

f o 1/J" 1 ( x, z) = J o 1/;;- 1 (x, z · g< �< (x ) - 1 ) = :2::: (av, < (x)g<�< ( x ) - v ) z v , v =O

and therefore av,< = av, �< . g�" . This means that av = ( av,J is a cross section in p k over U. So every holomorphic function ! on F ' that is homogeneous of degree k on the fibers is a section in p k . In particular, F/: can be identified with the space L k ( F� , q of k-linear functions f : F� x . . . x F� -+ C. For e l, .

.

.

, e k E Fx the tensor product e 1 ® . . . ® e k E ( F k )x is defined by

The tensor power e ® · . . ® e of an element e E Fx is denoted by ek for short. Finally, we define p- k : = ( F') k "' ( F k )'. 4 . The ten sor pr oduct: Let p : V -+ X be a vector bundle of rank r with transition functions G, : U<" -+ GL r (q , and 1r : F -+ X a line bundle with transition functions g<" (with respect to the covering f/ = { U< : L E I} ) . Defin ition . The tensor product V ® F is the vector bundle of rank given by the transition functions

r

Let
and on the other hand,

It follows that Consequently, 17 = (17<) is an isomorphism

a

cross section in V ® F, and we obtain for every x

(V ® F)x "' HomiC ( F� , V,).

For v E Vx and e E Fx the tensor product

v

®e

E ( V ® F) x is defined by

(v ® e) (.\) := .\ (e) . v.

180

IV.

Complex Manifolds

Lifting of Bundles. Let f : X -+ Y be a holomorphic map between complex manifolds and p : V -+ Y a vector bundle of rank T. Definition .

The

j* V : = X

lifted

bundle. or pullback. f*V over X is defined by

x y V :=

{(x , v ) E X

x V :

J (x) = p (v)}.

The bundle projection p : f* V -+ X is given by p(x, v ) : = x. The fiber of j * V over x E X is given by (f* V)x = Vf(x) · Therefore, the lifted bundle is trivial over the preimage sets f- 1 (y). One has the following commutative diagram: v

j*V -1-

p

X

�

-1-

y

p

If f/ = { U, E I } is an open covering of Y such that V is trivial over U, , then f/ := { U, = rl (U, ) : E I } is an open covering of X such that !* v �

: L �

L

�

is trivial over U, . If
iP,(x, v )

:=

(x, (
1f G, are transition functions for V with
i.e., f* V is given by the transition functions G, o f . If � E r(U, V ) is a holomofJZ_hic se�tion over some open subset U C Y, then � can be lifted to a section � E r(U, j*V) given by

-

�(x) : = (x, �( J (x) ) ) .

Subbundles and Quotien t s .

Defin ition . Let 1r : V -+ X be a vector bundle of rank T. A subset W c V is called a sub bundle (of rank p ) of V if there is a pdimensional linear subspace E c cr ' and for any X E X an open neighborhood u = U(x) and a trivialization
One sees immediately that:

Complex Fiber Bundles

2.

1. 2. 3.

181

Wx W n Vx is always a pdimensional linear subspace of Vx. W is a submanifold of V . W is itself a vector bundle. =

Example

Let V be a vector bundle of rank r on X and Y c X a submanifold. If we denote the natural injection Y '---+ X by j, then Vly := j *V is a vector bundle of rank r on Y. We apply this to the tangent bundle T( X). Choose an open covering %' = (U,),EI such that there are complex coordinate systems cp, = (zL . . . ,z�) for X in U, with the following properties: 1 . U, n Y = {z:t + 1 = . . . = z�. = 0}. 2. zJ: , . . . , z:t are complex coordinates for Y A trivialization

If?,

: j *T (X ) I u, nY -+ (U, n

(y, [cp" c] )

r+

Y )x

en is given by

(y, c) .

A tangent vector v belongs to Ty (Y) c Ty (X) if and only if there is a differ entiable path a : I -+ Ywith a( O) = x and (cp, o a) ' ( O) = c . This is equivalent to the statement that c = ( c1 , . . . , cd , 0, . . . , 0) . Therefore �P;- 1 (U, X {c E en :

cd+ l

= . . . =

Cn

=

0} )

= T(Y ) I u , ,

and T(Y) is a subbundle of j * T(X) . If G, := then

(oztjoz; I v, p, = 1 , .. . , n) are the transition functions for T(X ) , GU <

. ( ' ) ( gu, (z', 0)

O) Z

-

_

n

gz" ( z' , 0)

g,K = ((8zt/oz;)IY I p, = 1 , .. . , d) , are the transition functions for and gZ" ( z' , O) = ((oztjoz;)( z' , O ) I v, p, = d + 1 , .. . , n) . Let V be a vector bundle of rank If W c V is a subbundle, then we define the quotient bundle VjW by (VjW)x := Vx/Wx. We have to show that there are trivializations for VjW. If E c er is a subspace such that there are trivializations If? : V l u -+ u X e r with Wlu = �J?- 1 (U X E ) , then we can choose a subspace F c e with E ffi F = e and define If? : ( VjW) I u -+ u X F where T(Y),

v,

r.

by

lf?( v mod W ) : = prF (IJ? (v) ) ,

where pr, : E ffi F -+ F is the canonical projection. Using trivializations as above one obtains transition functions

1 82

IV. Complex Manifolds

for V such that g,,. are transition functions for W , and h, are transition functions for VjW. Example

The quotient bundle Nx (Y) := j* T(X) jT(Y) is called the normal bundle of Y in X . Exercises

1. Let G be a complex Lie group that acts analytically on a complex man ifold X . Then for every x E X the stab ilizer G, := { g E G : gx = x} is a closed Lie subgroup of G, i.e. , a subgroup and a (closed) complex submanifold. Prove that there is a unique complex structure on G fGx such that the canonical projection 1r : G -+ G /Gx is a holomorphic submersion. 2 . Let 1r : V -+ X be a complex vector bundle of rank r . For x E X let !JlJx be the set of bases of Vx. Prove that the disjoint union of the !JlJx , x E X , carries the structure of a fiber bundle over X with structure group and typical fiber equal to GLr (C). 3. Let cp : V -+ W be a vector bundle homomorphism over X . Suppose that rk(cpx ) is independent of x E X . Prove that

Ker ( cp) :=

U

x EX

Ker(cpx )

and

Im(cp) :=

U

x EX

Im(cpx )

are subbundles of V , respectively W. Show that Im(cp) "' V/ Ker(cp). 4. Let X = C be the Riemann sphere. Determine the transition functions for T(X) for the canonical bundle Kx and for the normal bundle of { oo} in X . 5 . Let f : X -+ Y be a holomorphic map. Prove that there is a uniquely determined vector bundle homomorphism f' : T(X) -+ f*T(Y) with (f' )x = (f. )x for X E X. 6. Let p : V -+ X be a holomorphic vector bundle. Show that there are vector bundle homomorphisms h : p* V -+ T(V) and k : T(V) -+ p*T(X) over V with Im(h) = Ker(k).

3.

Cohomology

C oh omology Gr oup s . Let X be an n-dimensional complex manifold and 1r : V -+ X a complex vector bundle of rank T. Assume that there is an open covering f/ = { U, : L E I } of X . We consider sections in V on the sets U, and on intersections U,,. := U, n U,. , respectively U,,.>- := U, n U,. n U>- .

3.

Cohomology

1 83

A 0-dimensional cochain with values in V (with respect to � ) is a function s that assigns to every L E I a section s, E r(U" V ) . The set of 0-dimensional cochains is denoted by C0 ( � , V ) . Defin ition .

A !-dimensional cochain (with values in V )is a function � that assigns to every pair 1 L , K) E I x I a section �"' E r(Uu, , V ) .The set of !-dimensional cochains is denoted by C 1 (0/o, V ) . Finally, a 2-dimensional cochain (with values in V ) is a function A that assigns to every triple 1 L, K, v) E I x I x I a section A,, E r(U uw , V ) , and the set of all these 2 -dimensional cochains is denoted by C2 (� ' v ) . Assume that a 0-dimensional cochain s is given. One may ask whether the sections s, E r ( U, , V ) can be glued together to a global section s E r(X, V ) . For that it is necessary and sufficient that s, = s" on U," . This can be expressed in another way: If we assign to every 0-cochain s a 1-cochain 6s by ( Os ) , " := s" - s,, then s defines a global section if and only if 6s = 0. Defin ition .

The coboundary operators

are defined by · .

(on u,l< ) '

A cochain s E C0 (� , V ) (respectively � E C 1 ( � , V ) ) is called a cocycle if 6s = 0 (respectively o� = 0). The sets of cocycles are denoted by Z0 (� , V) (respectively Z 1 (� , V) ) . Remark s

The sets of cochains and the sets of cocycles are all complex vector spaces. 2. We can identify Z0 (�, V ) with r( X , V ) . 3. A 1-cochain � is a cocycle if and only if the following compatibility con dition holds: �w = �'" + ��
4. Sometimes we need cocycles of degree 2. We call an element A E C2 (� , V ) a cocycle if the following compatibility condition holds:

The set of all these cocycles is denoted by Z2 (�, V ) .

184

IV. Complex Manifolds

A co boundary is an element of the image of the co boundary operator. The sets of coboundaries are again vector spaces, denoted by B 1 (Uf/ , V) = OC0 (Uf/ , V) and B2 (Uf/ , V) : = OC 1 ( Uf/ , V). For completeness we define B0 (Uf/, V) : = 0 . 3 . 1 Pr o p o sition .

PROOF:

Bi (Uf/, V) c Zi (Uf/ , V ) to r i = 0, 1 , 2 .

The case i

=

0 is trivial. For �

=

O.s E B 1 (Of/ , V) we have

( TJvl" - 1],1" + TJw ) - ( TJr<J" 1],1" + 1],1< ) + ( TJr< v - TJw + TJ,r< ) T}vl" - TJr<J" + TJr< v = Ar< VJ" -

•

•

Hi (Uf/ , V) : = zi (Uf/ , V ) /B' (Uf/ , V) is called the ith coho mology group of V with respect to 0/o.

Defin ition .

We have H0 (Uf/ , V)

=

r(X, V ) , independently of the covering, and we have

The canonical map from Z1 (Of/ , V) to H 1 (Of/, V) will be denoted by q. We do not want to elaborate on H2 , because we need cases.

Refinement s . Let 'f/ = {V, there is a refinement map T : N To : C0 (Uf/ , V) -+ C0('f/, V) and T1 (To .S )n := ( 8T(n) ) l vn

it

only in very special

: n E N} be a refinement of Of/ . Then -+ I with Vn C U7 (n) . It induces maps : C1 (Uf/ , V) -+ C 1 ( 'f/ , V) by

and

h� ) nm := (�T(n)T(m) ) I Vnm

Then O(To.s) = T1 (6.s), and if 6� = 0, then also O(T1 � ) = 0. Therefore, T induces a map T* : H1(Uf/, V ) -+ H 1 ('f/, V) by T* (q(� ) )

:=

q(T1 �).

By the remarks above it is clear that T* is well defined, and it is a vector space homomorphism.

3.

The map 7* is independent

3 . 2 P r op osition .

is injective.

of

Cohomology

185

the refinement map 7 and

PROOF: Let a : N -+ I be another refinement map and � a cocycle with respect to f/ . We have to show that 71 � - a � is a coboundary: 1

�T(n)T(m) I Vnm - �a(n)a(m) I Vnm (�T(n)a(n) + �a (n)T(m) ) - (�a (n)T(m) + �T(m)a(m)) �T(n)a (n) lvnm - �T(m)a(m) l vnm •

We define TJ E C° C'f/, V) by

Then 71� - a 1 � = OTJ. Now we consider a cocycle � with respect to f/ such that 71 � = O s for some s E C0 (''f/ , V ) . We want to see that � itself is a coboundary. On \;

n U, we have

Since �T(n)T(m) I Vnm = (s m - sn ) l vnm ' it follows that :=

Therefore, we can define h, on U, by h, I u, nvn �
h, - hi< = (�
=

0 ( - h ) . So 7* is injective.

•

We have seen that if Y is a refinement of f/ , then H 1 ( f/ , V ) can be identified with a subspace of H1 (Y, V ) . Therefore, we form the union of the spaces H1 ( f/ , V ) over all coverings f/ and denote this union by H 1 (X, V ) . We call it the absolute, or Cech, cohomology group of X with values in V . Acyclic Coverings. Let

vector bundle over X .

X be a complex manifold and p

:

v

-+

X

a

The covering f/ is called acyclic, or a Leray covering for V , if H1 (U" V) = 0 for every L E I . If Y = {V, : n E N } is another covering, then U, n Y := { U, n Vn : n E N } is a covering of U" and since H 1 ( U, n Y, V) is a subspace of H1 (U" V ) , we have also H1 (U, n Y, V ) 0. =

1 86

IV. Complex Manifolds

3 .3 Theorem. If ut/ = {U, : L E I } is acyclic and Y = {V, : n E N } is a refinement, with refinement map T : N -+ I , then T* : H1 (Of/ , V) -+ H1 ( Y, V) is bijective. PRooF:

We start with an TJ E Z1 (1/, V) and define TJ ( ' ) E Z1 (U, n Y, V) by 1)�,),

: = T}n m l u,nVnm , for all n, m with U, n "

f 0.

Since Of/ is acyclic, there is an element g ( <) E C0 (U, n "f/ , V) with TJ ( < ) = il ' ) . ( ) ) ) ) Then TJnm + g�' = g� on U, n V, m , and therefore gr,::( - g� = g}/: - gr, on U," n Vn m · Now we define � E Z1 ( Uf/ , V) by � " i u,K n Vn : = gk ) - g�" ) , and h E C0 ( Y , V)

' n r )) (on " = v, n ur ( n ) ) · by h, : = g� ( Then on Vnm we have

h � - TJ) nm

�r ( n ) r (m ) - TJnm g� ( n) ) g{r (m )) g �r ( n )) g� (m ) ) h, - hm . _

_

(gF( n ))

_

g �r ( n )) )

_

So TJ = T1 � + Sh, and T* is surjective.

3 . 4 Corollary.

•

Ij 0l/ is an acyclic covering of X , then H 1 (X, V ) = H 1 (Of/ , V).

PROOF: We have H 1 (UZ£ , V) c H1 (X, V). 1f a is an element of H1 (X, V), then there is a covering Y with a E H 1 ( Y , V) . Now we can find a common refinement 1f/ of Of/ and Y. Then H 1 (Y, V) c H 1 (1f/, V) = H1 (Uf/ , V ) , and • therefore a E H 1 (Of/ , V ) .

Gener alization s. The simplest case of a vector bundle over X is the trivial line bundle Ox : = X x C. Therefore, the associated Cech cohomology group H 1 (X, Ox ) plays an important role for the function theory on X . The trivial fiber bundle X x @* is not a vector bundle, but it is not so far from that. If 1r : P -+ X is a general analytic fiber bundle, then a section in P over an open set U c X is a holomorphic map s : U -+ P with 1r o s = id u . If the typical fiber of P is an abelian complex Lie group, then the set r ( U, P ) of all sections in P over U carries in a natural way the structure of an abelian group. In the case of the bundle Ox : = X x C* we have a canonical isomorphism r ( U, Ox ) "' O* (U) : = { f E O(U)

:

f (x ) i= 0 for every x E U}.

3.

Cohomology

187

This is a multiplicative abelian group . If f/ = { U, : L E I } is an open covering of X , then we can form cochains � : ( L , K ) f--t �"' E O* (U," ) . The set C1 (f£ , 0x ) of all these cochains forms a (multiplicative) abelian group, and we can define the subgroups Z 1 (f/ , O:X ) and B 1 (f/ , O:X ) of cocycles and coboundaries : A cochain � is called a cocycle if �w = �'"�"v on U," v , and � is called a coboundary if there are functions s, E O* (U,) such that �'" = s"s;:- 1 on U," .

Since all the groups are abelian, we have the quotient group

which we call the first cohomology group with values in O:X (with respect to the covering 0.?/ ). Just as in the case of vector bundles we can pass to finer coverings and finally form the Cech cohomology group H 1 (X, O:X ) . There is a nice interpretation for the elements of H 1 (X, O:X ). Every cocycle with values in O:X defines a line bundle over X , and this bundle is independent of the covering. Two cocycles e and � II define equivalent line bundles if and only if there are functions h, with ��" = �� h"h;:- 1 , i.e., if and only if the cohomology classes of e and e' are equal. Therefore, H1 (X , O x ) is the set of isomorphy classes of line bundles over X . This group is also called the Picard group of X and is denoted by Pic(X). The group structure is induced by the tensor product. The identity element corresponds to the trivial bundle Ox and the inverse to the dual bundle. In the same way as above we can form cohomology groups of any fiber bundle P whose typical fiber is an abelian group. If the fiber of P is a nonabelian group, things become a little bit more complicated. We can define cocycles with values in P , but they do not form a group. In the set Z 1 ( f/ , P) of cocycles we can introduce an equivalence relation by (I

<,

(II

"-' <,

:

<¢==:?-

(I = h - 1 g'" II h" . 3 h, W1th "''" , •

The set H1(f/ , P ) of all equivalence classes is called the cohomology set with values in P (with respect to f/ ) . Usually it is not a group, but there is a distinguished element, represented by the cocycle � with �'" = 1 for all L , K . Passing to finer coverings and forming the cohomology set H 1 (X, P ) causes no problems. The most important nonabelian case is the cohomology set

whose elements correspond to the isomorphy classes of vector bundles of rank r over X . The distinguished element is represented by the trivial bundle X X
1 88

IV.

Complex Manifolds

abelian group for every open set U c X , and that we can restrict sections over U to open subsets V c U. Having this in mind we can define another sort of cohomology group . Let K c C be a subgroup with respect to the addition, for example K = Z, IR, or C. For an open set U c X we define the (abelian) group K(U) by K(U)

:=

{f : U -+ K

:

f is locally constant } .

Then K(U) is an abelian group . 1f V c U is an open subset and f an element of K(U) , then flv E K(V). If U is connected, then K(U) "' K . We can define groups of cochains C0( %' , K ), C 1 ('32, K ) , and C2 ( %' , K ) just as we did it for vector bundles. If � is an element of C 1 ( %' , K ) , then �"' E K(Uu, ) · Cocycles, coboundaries, and cohomology groups are defined in the usual way. For example, we have

The Singular C oh omology. We want to give a short overview of co homology groups of topological spaces and their relation to Cech cohomology as defined above. For proofs see [Gre67] . For

q

E No the set

A,

:=

{x

=

( xo , . . . , xn ) E JR q + l

n

: L x; i=O

=

l , x; >

0}

is called the q-dimensional standard simplex. The 0-dimensional simplex is a point, � 1 a line segment, � 2 a triangle, and so on. Let X be a topological space. We assume that all spaces here are connected and locally connected. A singular q-simplex in X is a continuous map a : A, -+ X . If X is a complex manifold and if there is an open neighborhood U = U(� q ) and a smooth map a : U -+ X with &l�q = a , then a is called a differentiable q-simplex. A singular q-chain in X is a (formal) linear combination n 1 a1 + . . . + n k ak of singular q-simplices with n i E Z. The set Sq (X) of all singular q-chains in X is the free abelian group generated by the singular q-simplices. For a q-simplex defined by It

a

and i

=

is called the ith face of a.

0, . . . , q the

(q

- I)-simplex

ai

:

� q- 1 -+ X is

3.

Cohomology

1 89

The (q - 1 )-chain aa := '2:.I=o ( - 1 ) i ai is called the boundary of a. The bound ary operator a induces a homomorphism

and it follows easily that a 0 a = 0 . Definition . The group Hq (X ) := { c E Sq (X) : de = O } jaSq + 1 (X) is called the qth singular homology group of X . In general, H0 (X ) is isomorphic to the free abelian group generated by the connected components of X . Since we assume all spaces to be connected, we have H0 (X) "' Z . If X is an n-dimensional complex manifold, then Hq (X ) = 0 for q > 2n. Now for

q

> 0 we define the group of singular q-cochains to be

Then the coboundary operator o : Sq (X) --+ Sq + 1 (X) is defined by

Jj [c] := j[ac] , for f E Sq (X) and c E Sq + 1 (X) . Obviously, we have 6 o 6 = 0. We can define cocycles (elements f of Sq (X ) with o j = 0) and coboundaries (elements of the form og with g E sq-1 (X ) ) . Defin ition . The group Hq (X) : = { ! E Sq (X) : oj = O} joSq - 1 (X ) is called the qth singular cohomology group of X . From above it is clear that H0 (X) "' Z.

A topological space X is called contractible if there is a point x0 E X and a continuous map F : [0 ,1 ] x X --+ X with F(O, x) = x and F(1, x ) = x0 • In that case H1 (X ) = {0} . The space X is called locally conlraclible if every poinl of X has arbitrarily small contractible neighborhoods. Among the connected topological spaces there is a big class of spaces (including the so-called CW complexes) that are locally contractible and have the following properties:

1 . Hq (X) "' Hq (X, Z) for q = 0, 1 , 2, . . ..

2 . Every open covering of X has a refinement tt/ = { U, acyclic in the sense that H1 (U,, Z) = 0 for every L E J .

:

L

E I } that is

We call such spaces good topological spaces. For example, every (connected) complex manifold is a good topological space, and also every irreducible an alytic set. Finitely generated abelian groups are classified as follows:

190

IV. Complex Manifolds

If G is a finitely generated abelian group, then there are uniquely deter mined numbers r E No and n 1 , . . . , ns E N with ni > 2 and nilni+l for i 1 , . . . , s 1 such that =

-

The number r is called the rank of G, and zr the free part of G. The sum of the ZjniZ is called the torsion part of G . In many cases cohomology can be computed from homology:

If X is a compact good topological space (for example, a compact connected complex manifold), then Hq (X) is finitely generated for every q, and for q > 1 there is an isomorphism

3.5 Theorem.

H q (X) '"" (free part of Hq (X)) EB (torsion part of Hq _ 1 (X) ) .

The rank of H 1 (X) is called the first b1 (X).

Betti number of X , and is denoted by

Now we give an application of Cech cohomology methods to singular coho mology. Let X and Y be good topological spaces. 3.6 Theorem (Kiinneth formula) .

We choose open coverings 02/ = {U, : L E I} of X and )/ = {Vn n E N} of Y such that all the U,, Vv and all their pairwise intersections are connected. We write Wm := U, X Vn and Wm,l<m : = Wm n Wl<m = u,l< X Vnm · A cocycle '1/! E Z 1 (%' X 1/ , Z) is given by constant maps '1/Jm , l<m : Wm,l<m --+ z.

P ROOF:

We identify any cocycle � E Z 1 (%' , Z) with a cocycle [ E Z 1 (%' x 1/, Z) by [n ,�<m = �'" and also any rJ E Z 1 (1/, Z) with an fj E Z 1 (%' x 1/, Z). This ' induces natural injections

with Im(j1 ) n Im(j 2 ) =

{0}, and therefore an injective map

j : H 1 (%' , Z)

x

H 1 (1/, Z) --+ H 1 (%'

x

1/, Z)

by j ( a , b) := J 1 (a) + jz (b). Given a cocycle '1/J E Z 1 (%' x 1/, Z), for any L E I we define '1/J, E Z 1 (1/, Z) by ('1/J ,) n m : = '1/J m,,m · The cohomology class of '1/J, in H 1 (1/, Z) is independent of the i n dex We see this as follows. L

3.

Cohomology

191

(a) If Uu, f. 0 , then by the co cycle property we have 1/JLn,K,n + 1/JKn,Km + 1/JK,m,Lm - 1/'K,n ,Km 1/JLn,Kn - 1/Jim,Km ·

Setting ('Pu,)n := '1/!m .>< = 0 , one can find a chain of sets U>q , . . . , U>-.N with UL>-.1 f. 0 , U>-.i>-.,+1 f. 0 , and U>-. N " f. 0 (since X is connected) , and from (a) it again follows that the cohomology classes of '1/JL and '1/J" are equal. Let '1/Jo be a representative of the common cohomology class of the '1/JL . The assignment '1/J H '1/Jo induces a map p : H1(6!/ x Y', Z) --+ H1 ( "f/ , Z). We will prove that for every class c E H1 ( 6!1 x )/, Z) there is a class a E H1 ( 6!1 , Z) with j1(a) = c - j2 (p(c)) , and consequently j (a, p(c)) = c. L

For each there is an 'rJL E C0 ( "f/ , Z) with '1/Jo = '1/!L - 61/ ( TJL ) , and we define 'f) E C0 (6!/ X Y', Z) by TJm : = (TJL )n . Then

1

-

: = '1/J - '1/Jo - OrJ E Z1(6!/

x

"f/, Z) and

IL = '1/!L - ('1/!L - 01/ (TJL)) - 01/ (TJL ) =

Now we construct a (} E Z1 ( 6!1 , Z) with For

n

E N define Qn E Z1 (6!1 , Z) by

Since lm,Lm

=

0 for all

n, m,

and

L,

Q=

0 for every

L.

T

in the case Vnm f. 0 we have

and therefore (}m = (}n . If Vnm = 0, we can argue in the same way as above in (b) , because Y is connected. So we have a (} E Z1 ( 6!1 , Z) with Qn = (} for every (}LI< = lm,><m

Therefore,

Q=

1

-

=

lm,><m

n

and

(for arbitrary

)

n, m .

and '1/! = Q + '1/!o + OrJ. This shows that j is surjective.

•

Exercises :

1 . Let 6!1 = { U1 , Uz , U3 } be the covering of X : = { z E C 1 < l z l < 2} given by U1 : = { x + i y : y < x} , Uz : = {x + i y : y < -x } , and u3 : = {X + iy : y > 0}. Calculate H1 ( 6!1 ' Z). 2. Prove that dime H1 ((C2 - { 0}, 0) = oo , where 0 denotes the trivial line bundle over C2 . Hint: Use Laurent series.

192

IV. Complex Manifolds

3 . Show that H 1 (e, O) 0. 4. Show that %' = {Ui : i 1 , . . ' n} with ui := {z : Zi f. 0} is an acyclic covering of en - { 0} for 0. 5. Let X be an n-dimensional complex manifold such that Hq (X) = Z for q even, 0 < q < 2n, and Hq (X) 0 otherwise. Calculate the singular cohomology of X. =

=

.

=

4.

Meromorphic Functions and Divisors

Let X be an n-dimensional complex manifold and x E X a point. Two holomorphic functions f, g defined near x are called equivalent at x if there exists a neighborhood U U(x) with f l u = g\u- The equivalence class of f at x is called the germ of f at x. We denote the germ by fx and the set of all germs by Ox· Having fixed a complex coordinate system r.p U -+ B C en at x, we may identify the set Ox with the ring Hn of convergent power series by The Ring of Germs.

=

:

fx H

Taylor series of f o r.p - 1 at r.p ( x).

So Ox has the structure of a local e-algebra. 2 An element fx E Ox is a unit if and only if f(x) f. 0. Of course, Ox is also noetherian and a unique factorization domain.

Let f, g be holomorphic functions near x0 E X . If the germs fxo , 9xo are relatively prime in Ox0 , then there is an open neigborhood U U(xo ) C X such that fx, 9x are relatively prime in Ox for x E U. PROOF: We can work in en and assume that x0 0 and that fo and go are Weierstrass polynomials in z1 . Since fo and g0 are relatively prime in Hn ,

4.1 Proposition. =

=

they are also relatively prime in H� _ 1 [z! ] . If Q is the quotient field of Hn _ 1 , it follows from Gauss ' s lemma that fo and g0 are relatively prime in Q[ z1 ] .

We can find a linear combination

where a, b E Hn _ 1 [ z1 ] , and h E (Hn _ I )* is the greatest common divisor of fo and 90· If u U' X U" c e X en - 1 is a sufficiently small neighborhood of the origin, the power series a, b converge to pseudopolynomials over U" and h to a holomorphic function on U'' that does not vanish identically. =

2

A commutative «:::-algebra A with unity is called a local «:::-algebra if the set m of nonunits forms an ideal in A and the composition of canonical homomorphisms C '-t A __,. A/m is surjective.

4.

Meromorphic Functions and Divisors

1 93

Let w = ( w1 , w' ) be a point in U. If i.pw is a common factor of fw and gw , then it divides hw , which has degree 0 as a polynomial in z1 . So r.pw does not depend on z1 and does not vanish identically in zz , . . . , Zn · Therefore, i.pw is z1 -regular of order 0, and by the preparation theorem it is, up to a unit, a Weierstrass polynomial of degree 0; i.e., r.pw is already a unit. •

This shows that fw and gw are relatively prime.

Let X be an n-dimensional complex man ifold. We consider analytic hypersurfaces A C X. Then locally A is given as the zero set of one holomorphic function f. We always assume that f does not vanish identically and therefore A is nowhere dense in X. If locally A = { ( z1 , . . . , zn ) : Zn = 0}, then every holomorphic function g vanishing on A is of the form g(z1 , . . . , zn ) = Zn · g(z1 , . . . , zn ) · We will generalize this result to the arbitrary case. Analyt ic Hypersurfaces .

Every hypersurface A lytic set of dimension n - 1 .

CX

4.2 Proposition.

is a pure-dimensional ana

After choosing appropriate coordinates, we may assume that A is contained in an open set u c en and f is a Weierstrass polynomial w in Zl without multiple factors such that P ROOF:

A = N(w) = {z E U

:

w ( z)

= 0}.

Since N(w) is a branched covering over some domain G C en -I , every irre • ducible component of N (w) has dimension n - 1 . 4.3 Theorem (Nullstellensatz for hypersurfaces) .

Let A C X be an

analytic hypersurface and xo an arbitrary point of A. 1 . There exists an open neighborhood U = U(x0) C X and a holomorphic function f on U such that: (a) U n A = {x E U : f(x) = 0}. (b) If h is a holomorphic function on a neighborhood V = V(x0) C X with hi = 0, then there is a neighborhood W = W(x0) C U n V and a holomorphic function q on W such that hl w = q · (fl w ) . � 2. If f is any holomorphic function defining A in U and h is again a holo morphic function on a neighborhood V = V(x0) vanishing on A, then there exists a k E N and a holomorphic function q on a neighborhood W = W(x0) c U n V such that hk l w = q · U l w ). A

Again we work in en and assume that x0 0. Let f be an ar bitrary defining function for A near the origin. After choosing appropriate

PROOF:

=

194

IV. Complex Manifolds

coordinates we find a unit e and a Weierstrass polynomial w in z1 such that f = e . w. We choose a neighborhood U = U' x U" of the origin such that:

1 . A n U = {(z1 , z') E U

0�.

w(z1 , z' ) = 2 . There is a prime factorization w = w1 1 :

•

•

•

w;1 on U.

We define w : = w1 · . . wz . This is a pseudopolynomial without multiple factors that also defines A in U.

A, which we may assume to be defined on U, then by the division formula there is a holomorphic function q near the origin

1f h is a function vanishing on

and a pseudopolynomial r with deg(r)

<

deg(w) such that near 0 ,

h =q · w+r. Since w has no multiple factors, the greatest common divisor of w and ow j 8z1 is a not identically vanishing holomorphic function g of zz , . . . , zn , and we can find pseudopolynomials q 1 , qz with

We may assume that everything is defined on U. Suppose that there is a z� E U" such that w((, z!J ) E C[(] has a multiple zero (0 . Then w((o , z�) = 8wj8z1 ((o , z�) = 0, and therefore g(z�) = 0. Hence, if g(z') f. 0, then w((, z') has exactly s :=deg(w) distinct zeros. Since h\ N (w) = 0, h((, z') has at least these s distinct zeros. Using this fact and the division formula, it follows that r(z') = 0 for z' E U" N(g). Therefore, by the identity theorem r = 0 and h = q · w. Taking f = w yields the first part of the theorem. -

Let k := max(k 1 , . . . , kz ) . Then

This proves the second part of the theorem.

•

Every local holoniorphic function f that satisfies the conditions of the first part of the Nullstellensatz will be called a minimal defining function for A. Now let A be an irreducible analytic hypersurface in X , and h a holomorphic function on some open subset U c X with h\(AnU) = 0. For xo E U n A there exists a neighborhood V = V(x0 ) c U and a minimal defining function f for A on V . Then ordA,x0 (h) : = max{m E N

:

:l q with h = fm . q near xo }.

It follows from the Nullstellensatz that ordA,x0 ( h ) > 1 , and from the unique factorization into primes that it is finite. Furthermore, it is independent of j, because if h , fz are two minimal defining functions, then we have equations

4.

h

=

q1 q2)

qz fi . It follows that h So q1 and qz must be units.

q1 !z and

= 0.

Meromorphic Functions and Divisors

fz =

=

195

q1qzh and therefore h ( l

-

For every point xo E A there is a neighborhood U = U( xo) such that ordA,x (h) > ordA,x0 (h) for x E U n A.

4 .4

Pr op osition . If A is irreducible, h holonwrphic in a neighborlwod cl A , and h i A = 0, then the number ordA,x (h) is independent ofx E A. PROOF: Let x0 E A be an arbitrary point. In a neighborhood U of x0 there exist a decomposition A n U = A 1 U . . . U A1 into irreducible components and minimal defining functions f>.. for A.\. Since h vanishes on every A.\, there exist k1 , . . . , kz E N and a holomorphic function q on a neighborhood V = V ( x0 ) C U such that =

h

j�l . . . j1kt q, and (f>.. ) x0 f qx0 for A •

= 1 , . . . , l.

Since (!I )x 0 , (fz )x0 are irreducible, it follows that ( f.\ )xo and qxo are rel atively prime for A = 1 , . . . , l . But then ( f>.. ) x and y, remain relatively prime for x sufficiently close to x0 , say in a neighborhood W ( xo ) c V. •

Let

•

•

,

( ) : ordA,x (h) . It is necessary to consider two cases:

n x

=

( a) 1f A is irreducible at x0 , then l = 1 , and it is clear that n ( x ) = n ( x0 ) for x E W n A. Consequently, x H n ( x ) is a locally constant integer-valued function on the set A of regular points of A. Since A is globally irreducible, A is connected and n ( x ) globally constant on A. Let n * E N be the value of this function. •

(b) If 1 > 1 , then f == h . . . fz is a minimal defining function for A at x0 . With m := min(k 1 , . . . , kz ) we have h

=

f1k1 . . . Jlkz . q

=

Jm . (},

where (} is a holomorphic function near x0 . Therefore, n ( x0 ) >

m.

We assume that m = k.\. In every small neighborhood of x0 there are regular points x E A.\ that do not belong to A, for p, f- A . Then n ( x ) = n *, and f>.. is a minimal defining function for A at x. Since h = J�>- . q , with a holomorphic function q, and (h )x f qx it follows that n ( x ) = k.\. So m < n (xo ) < n ( x ) = n * = k.\ = m, and therefore n ( xo ) = n *. ,

•

Now we define ordA (h)

:

=

One easily sees that

{ �onstant value of ordA,x (h)

if h i A = 0, otherwise.

1 96

IV.

Complex Manifolds

Mer omorphic Function s. Let X be an n-dimensional complex mani fold. We consider holomorphic functions that are defined outside of an ana lytic hypersurface. In the !-dimensional case these are holomorphic functions with isolated singularities. Definition . Let A c X be an analytic hypersurface. A complex-valued function m on X - A is called a meromo rphic function on X if for any point x E X there are holomorphic functions g, h on an open neighbor hood U = U(x) c X such that N (h) c A n U and m = g j h on U - A. Obviously, m is holomorphic on X - A. In particular, every holomorphic function f on X is also meromorphic on X . Different meromorphic functions may be given outside of different analytic hypersurfaces. If m>-. : X - A>-. --+ C are meromorphic functions on X , then m1 ± m2 and m1 . m2 are meromorphic functions on X , given as holomorphic functions on X - (A1 U Az). m

:

X - A --+ C is a meromorphic function, for p E A we have two possibilities: If

(a) There is a neighborhood U = U(p) c X such that m is bounded on U - A . Then there is a holomorphic function m on u with ml u-A = m l u - A , and p is called a removable singularity for m. (b) For any neighborhood V = V(p) c X and any n E N there is a point x E V - A with lm(x)l > n. If m = g/h near p , then h must vanish at p , because otherwise we would be in situation (a) . Now there are again two possibilities: (i) If g(p) f- 0, then limx -+ p l m(x) l = + oo, and we have a pole at p. (ii) The other possibility is g(p) = 0. This cannot occur in the case n = 1, since it may be assumed that the germs gp and hp are relatively prime, but it is possible for n > 1 . The behavior of m is extremely irregular in that case: We take any c E C. Then gP - c · hp and h, are relatively prime, and therefore there exists a sequence (xv) of points in N(g - ch) - N( h) with limv -+oo Xv = p . This means that m(xv) = c for every v. We call p a point cl indeterminacy in this case. In the case n = 1 a meromorphic function is a function that is holomorphic except for a discrete set of poles. For n > 1 we have the polar set

P( m)

:=

{p E X

: m

is unbounded in any neighborhood of p } .

The polar set consists of poles and points of indeterminacy. We show that P( m) is an analytic hypersurface. Let p E X be an arbitrary point and U = U(p) c X a connected neighbor hood, where m is the quotient of g and h and N (h) c A . We may assume that p E A and gP, h, are relatively prime. Then

4.

m bounded near p

So P(m) n U

=

{x E U

{::=} {::=} {::=} {::=} :

h (x)

Meromorphic Functions and Divisors

3 rp (holomorphic near p ) with g hp l gp hp is a unit h(p ) # 0. =

=

rp

1 97

.h

o}.

If Z c X is an irreducible hypersurface, then we define ordz ( m) as follows: If m g/h near x , then ordz,x ( m) == ordz,x (g) - ordz,x (h). If we choose gx , hx relatively prime, this definition is independent of g and h. Now it follows exactly as above that ordz,x ( m) is constant on Z. =

4.5 Identity theorem for mer om orphic fun ctions. Let X be connected, m : X - A --+ C a meromorphic fun ction, and U C X a nonempty open set such that ml u - A 0. Th en P(m) 0 and m 0. =

=

=

PROOF: The set X - P(m) is connected, m is holomorphic there, and U (AU P(m) ) is a nonempty open subset of X - P(m). By the identity theorem for holomorphic functions it follows that m 0 on X - P(m). But then m is globally bounded and P(m) 0 . • =

=

The set Al(X) of meromorphic functions on X has the structure of a ring with the function m 0 as zero element. We set =

Al(X) - {0}

{ m E At (X) : m vanishes nowhere identically } If m E Al(X ) * has a local representation m gjh , the zero set N(g) is independent of this representation. Therefore, we can define the global zero set N (m ) , which is an analytic hypersurface in X . Outside of P(m) UN(m) , m is holomorphic and without zeros. Therefore, 11m is also holomorphic there and has local representations l jm hjg. So l jm is also meromorphic, and consequently Al(X) is a field. For this it is essential that X is connected! =

=

4 . 6 Levi' s exten sion theorem. Let A C X be an analytic set that has at least codimension 2 , and let m be a meromorphic fun ction on X - A. Th en there exists a meromorphicfun ction m on X with mix-A m. =

PROOF: Since the statement is true for holomorphic functions, we may assume that P( m) # 0 . So it is an analytic hypersurface in X - A. By the theorem of Remmert-Stein, Q : P(m) is an analytic set in X . By Riemann's second extension theorem the holomorphic function m on ( X - Q) - A has a holomorphic extension m to X - Q. =

Let p E A n Q be a point. We have to show that m is locally meromorphic at p. We choose an open neighborhood U U(p) c X and a function g E O (U) such that : =

198

IV.

Complex Manifolds

1 . Q n U c N (g) . 2 . N(g) = N1 U . . . U Nk is a decomposition into irreducible components.

Since dim ( A) < dim ( Ni ) , there are points a , E Ni - A and neighborhoods Vi = Vi ( a i) c U - Asuch that m = P;/qi on v; - Q and N(qi) c Qnv; c N(g); i.e., g\N(qi ) = 0. From the Nullstellensatz it follows that there is a number si E N and a holomorphic function ri such that g8 i = r, . qi . Then m = Pirig - Si near a,. This means that there exists an s E N such that g5 m is holomorphic near a 1 , . . . , ak •

Thus N : = (U - A) n P(g8m) is empty or an analytic hypersurface that is contained in N (g) -A. In the latter case it is a union of irreducible components of N (g) - A, and this is impossible, since every such component contains a point a,. So N must be empty, and g5m is holomorphic in U - A. By Riemann's second extension theorem there is a holomorphic extension h of • g5m on U. Then g - s h is meromorphic on U with (g- sh) \ u-A = m.

Divisors . Let X be a connected complex manifold, m E .4(X) * , and Z c X an irreducible analytic hypersurface. If Z c P( m), then ordz ( m) is a negative integer, and if Z c N (m) , then ordz (m) E N. In all other cases we have ordz (m) = 0. If P(m) = U, EI P, and N (m) = U.\EL N.\ are the decompositions of the polar set and the zero set into irreducible components, then the formal sums (m),

:=

2 )- ordpJm)) . P,

and

LEI

(m ) o

:=

L ordN"' (m)

.\EL

. N:>,

are called respectively the divisor cl poles and the divisor if zeros of m. Finally, div(m ) : = (m ) 0 - ( m ) , is called the divisor of m. From the remarks above it is clear that

div ( m) =

L

z cx

ordz (m)

·

Z,

where the sum is over all irreducible hypersurfaces

Z in X

Definition . Let (Z,), EI be a locally finite system of irreducible ana lytic hypersurfaces Z, E X . If for every L E I a number n, E Z is given, then the formal linear combination

is called a divisor on X . The divisor is called positive or

effective if n,

> 0 for every L E I.

4.

Meromorphic Functions and Divisors

1 99

Divisors can be added or multiplied with integer constants in an obvious way. Therefore, the set 5?(X) of all divisors on X has the structure of an abelian group. As we just have seen, there is a map div : At'(X )* --+ 5?(X). Since we have div(m 1 m2 ) = div (m 1 ) + div(m 2 ) , div is a group homomorphism. Sometimes it is useful to generalize the notion of a divisor a little bit. Let (A) , E r be a locally finite system of (arbitrary) analytic hypersurfaces in X , and ( n J ,EI a system of integers. Then for every L E I we have a decomposition

into irreducible components. The system {At : L E I, -\, E L,} is agam locally finite, and we define 2:,EI n, . A, : = 2: ,EI 2:: .\ , E L, n, . A � , . With this notation it is possible to restrict divisors to open subsets: If U c X is open and D = 2:,EJ n, . Z, a divisor on X , then

D u := 4 . 7 Pr oposition .

If A c X is an analytic hypersurface and f a m inimal definingfunctionfor A in an open set U with U n A f- 0, then

The proof is more or less straightforward. Now let an arbitrary divisor D on X be given. Then for any point p E X there is an open neighborhood U = U (p) c X , a finite system { Zi : i = 1 .. . , N } of irreducible hypersurfaces Zi c U, and a system of numbers ni E Z such that ,

N

D iu = I >i Zi . ·

i=l In addition, there is a neighborhood V = V(p) c U such that there exist minimal defining holomorphic functions fi for Zi in V. Then

di v

N

N

=l

i =l

(iIJ it'') = L ni . zi n

v

= Dlv

In this way every divisor is locally the divisor of a meromorphic function.

200

IV.

Complex Manifolds

Associated Line B undle s . Let X be a connected n-dimensional com plex manifold. If Z c X is an analytic hypersurface, then there is an open covering %' = { U, : L E I } of X with the following property: If U, n Z -=/= 0 , then there is a minimal defining function f, E O(U,) for Z. Setting f, == 1 if U, n Z = 0 , in Uu, we get the two relations

with suitable holomorphic functions

gu,

and g,"'" Then

Since f, does not vanish identically, it follows that g,"g'" 1 on U," . This shows that gu, E 0* (Uu-,) and g"' = g,"1 . Furthermore, on U,">-. we have the compatibility condition =

The system of the nowhere vanishing functions g," = f,j f" defines a holo morphic line bundle on X, which we denote by [Z}.It is easy to show that this definition does not depend on the covering and the functions k 4.8

Proposition.

1. Th ere is a section sz E f(X, [ Z] }vith Z = {x 2. [Z]is trivial over X Z .

EX

: sz (x) = 0} .

-

PROOF: The system of holomorphic functions f, defines a global section sz with {x E U, : sz (x) = 0) = {x E U, : f, (x) = 0) = U, n z . Then it is clear • that [ 41 - z is trivial.

We can generalize the concept of associated line bundles to the case of divi sors. If D is a divisor on X , then there is an open covering %' = { U, : L E I } of X , and meromorphic functions m, on U, with Di u, = div ( m, ) . It follows that the functions are nowhere vanishing holomorphic functions on U," . They define a line bun dle, which we denote by [D]Jf D = k . Z, then [D) [Z] k . If D = D1 + D2, then [ D ]= [D1] Q9 [D2] . Thu s the map 6 : 51( X) --+ Pic( X ) , is a homomorphism of groups. 4.9

Theorem.

Th e sequence v¢!(X ) *

o(D) d

==

isomorphy class of [D] ,

group homomorphisms

� 51(X) � Pic(X)

4.

Meromorphic Functions and Divisors

201

is exact. PROOF: ( l ) Let m f- 0 be a meromorphic function. Then [div(m)] is given by only one transition function mjm = 1 . Therefore, o o div(m) = 1 .

( 2 ) Let D be a divisor on X with J(D) = 1 . We assume that D i u, = div(m,) and [D] is represented by g,"' = m,jm"' . Since [ D ]ls trivial, there are nowhere vanishing holomorphic functions h, with h, . g,"' = 1 . h, on U,"' . Then h, . m, = h,

·

m,

Therefore, a meromorphic function m on X can be defined by ml u, : = h, . m, . Obviously, div(m) l u, = div(m,) = D i u, , and therefore div (m) = D . •

Meromorphic Sections . Let X be a connected n-dimensional complex manifold. Any analytic hypersurface Z c X leads to a line bundle [ Z jq, gether with a global holomorphic section sz that vanishes exactly on Z. The construction of sz fails in the case of an arbitrary divisor D and its associ ated line bundle [ D] Therefore, we introduce the notion of a meromorphic section. Definition. Let 1r : L --+ X be an analytic line bundle and A c X an analytic hypersurface. A holomorphic section s E f(X A L ) is called a meromorphic section over X in L if for every point x E X there is an open neighborhood U = U(x) c X , a function h E O(U), and a holomorphic section t E f(U, L ) such that: 1. h · s = t over U - A . 2. N(h) c A n U. -

,

If we have a system of trivializations rp, : 1r - 1 (U,) --+ U, x C and transition functions g,"' , then we have the following description of s .

For x E U, - A we define s,(x) by rp, o s(x) = (x, s, (x) ). Then s, = g,"' · s"' on U,"' - A. If we choose the U, small enough, there are holomorphic functions h, t, on U, such that h, . s, = t, over U, - A and N ( h, ) c A n U, . This means that s, is a meromorphic function on U, . We could have as well said that a meromorphic section is a system (s,) of meromorphic functions with s, = g"" . s"' . If Z is an irreducible hypersurface, then ordz(s,) = ordz (s"') , and we denote this number by ordz(s). The sum

div(s)

:= ZC X irreducible hypersurface

ordz (s) . Z

is called the divisor of the meromorphic section s.

202

IV. Complex Manifolds

Now we have the solution for our problem: Let D be the divisor on X given by D \ u div(m, ) . Then [D] is described by the transition functions 9u< = m,m"' 1 , and the system of the m, defines a global meromorphic section sv of [D]with div(sv) = D. ,

=

So far our definitions seem to be purely tautological. But for example, they allow us to determine the space of holomorphic sections of [D ]in terms of meromorphic functions on X . For that we need the following notation: If D�, D2 are two divisors on X , then D 1 > Dz if and only if D1 - Dz is a positive divisor. 4 . 1 0 Theo r em. Let D = 2: z n z . Z be a divisor on X . Th en there is a natural is omorphism {m E v¢!(X ) * : div(m) > - D) ----+ r(X, [D]) . ""

PROOF: Let sv be the global meromorphic section of [D]with div( sv) D . Then for any meromorphic function m on X also t : = m · s v is a meromorphic section of [ D}. =

If

m

is a meromorphic function with div(m) > - D, then div(t) = div(m) + div(sv) = div(m) + D > -D + D = 0.

This means that t is a holomorphic section. The map m H m · s v is obviously injective. Let t E r(X, [D] ) be given. If D \ u

,

=

div(mJ , then

Hence t,m-; 1 = t "' m"' 1 on U,"' ' and there exists a meromorphic function m on X with m\u, = t,m-; 1 . Therefore, div(m) \ u, = div(tJ - div(m,) > - D \ u, • So the map is an isomorphism. ·

Example

Let X

=

C be the Riemannian sphere and D

r(x, [D] )

=

=

n · 00 · Then

{ m E v¢!(X)* : ord00 (m) > - n and ordp(m) > 0 otherwise } .

The holomorphic sections in [D] are just the meromorphic functions on X that have a pole of order at most n at oo . Exer ci ses

1 . Let x be a point in a complex manifold X and let fx , 9x be nonunits in Ox . Prove that fx and 9x are relatively prime if and only if dim(N (f, g)) < n -

2.

5.

Quotients and Submanifolds

203

2 . Let G c en be a domain, A c G an analytic hypersurface, U c G an open subset with U n A -=!= 0 , and f : U --+ C a minimal defining function for A. Prove that Sing(A) n U = {z E U : f(z) = 0 and \7 f(z) = 0}. 3. Consider the meromorphic function m(z 1 , z2) := 2 212 1 on C2 . Show that the closure X of the graph { (z1 , zz , w) : z1 -=/= 0 and w = m(z 1 , z2 ) } in C 2 x C is an analytic hypersurface. Determine a minimal defining function for X at (0, 1 , oo ) . 4. Classify the singularities of m(z1 , zz) :=sin(z1 )1 sin(z1 z2 ) . 5. Let L --+ X be a holomorphic line bundle. Prove that L = [D ]for some divisor D on M if and only if L has a global meromorphic section s -=!= 0. 6. Let X be a compact Riemann surface. Show that for every nonconstant meromorphic function f on X the numbers of zeros and poles are equal (counted with multiplicity) .

5.

Qu otien t s a n d Sub manifold s

T op ological Qu otients. Let X be an n-dimensional complex manifold and '"" an equivalence relation on X . If x,y E X are equivalent, we write x '"" y or R( x, y ) . For x E X let X (x) := { y E X : y rv x} = { y E X : R(y , x ) }

be the equivalence class of x in X . These classes give a decomposition of X into pairwise disjoint sets. The set X1R of all equivalence classes is called the topological quotient of X modulo R. Let 1r : X --+ XI R be the canonical projection given by 1r : x H X (x) . Then XI R will be endowed with the finest topology such that 1r is continuous. This means that U c XI R is open if and only if 1r� 1 (U) c X is open. We call this topology the quotient topology.

A set A c X is called saturated with respect to the relation R if

5 . 1 P r op o si tio n . 1.

2. 3.

A saturated -{==} A = UxEA X ( x ) . If U C XIR is open, the-n <+= l (fff') is open and saturated. If W C X is open and saturated, then 1r(W) C XIR is open.

The proof is trivial.

204

IV. Complex Manifolds

5 .2 Pr oposition. Let Z be an arbitrary topological space. A map f X/ R --+ Z is continuous if and only if f o 1r : X --+ Z is contin uous. This statement is also trivial, since ( f o 1r) - 1 ( U ) = Jr - 1 ( f - 1 ( U )) .

Analytic Decomposition s. If X is an n-dimensional complex mani fold and R an equivalence relation on X , one can ask whether Xj R carries the structure of a complex manifold such that 1r is a holomorphic map. As sume that such a structure exists. Then Xj R must be a Hausdorff space. If rp : U --+ e k is a complex coordinate system for X/ R, then U = = 1r- 1 (U) is an open saturated set in X , and f = = rp o 1r : U --+ ek a holomorphic map with r- 1 (f(x)) = 7r - 1 (1r(x)) = X(x). So the fibers o ff are equivalence classes, and the equivalence classes must be analytic sets. If additionally 1r is a submersion, then rkx (f) = k for every x E U, and the fibers are ( n - k) dimensional manifolds. We now show that these conditions are also sufficient for the existence of a complex structure. �

Let X be an n-dimensional complex manifold and 2 = {Z, : L E I } a decom position of X into d-dimensional analytic sets. For x E X let t(x) E I be the uniquely determined index with x E Z,(x) . Then there is an equivalence rela tion R on X such that the equivalence class X (x) is exactly the analytic set Zt(x) . We consider the topological quotient X I R and the canonical projection 1r : X --+ XjR and assume that the following conditions are fulfilled: 1 . Xj R is a Hausdorff space. 2. For any x0 E X there exists a saturated open neighborhood U of X(xo) in X and a holomorphic map f : Q --+ en- d such that (a) r-1 (f(x)) = X(x) for all X E U. (b) rkx (f ) = n - d for x E U. �

�

5 . 3 Theorem. Under the conditions above, Xj R carries a unique structure of an (n - d)-dimensional complex manifold such that 7f : X --+ XjR is a

holomorphic submersion.

�

PROOF: Let x0 E X be �given. Jhen there is an open n�ghborhood U of X(x0) in X with Jr-1 (1r (U) ) = U, and a submersion f : U --+ en - d whose fibers are equivalence classes X(x). If z0 : = f(x0 ), then there is an open � n neighborhood W = W(z0) c e - d and a holomorphic section s : W --+ U (with s(zo) = xo and f o s = idw ) . For z E W we have f- 1 (z) = X (s (z)), and therefore 7r - 1 (1r(s (W)))

=

U

zEW

X(s (z))

=

U

ZEW

r - 1 (z)

=

f-1 (W)

This is an open set, so 1r(s(W)) C XjR is open as well. We define a complex coordinate system rp : 1r( s(W)) --+ en - d by

5.

cp(7r(s(z)) )

Quotients and Submanifolds

205

:=z.

Then cp (7r (x) ) = f(x). This shows that if is well defined and continuous. It is also bijective, with cp - 1 (z) = 7r(s(z) ) , and therefore a homeomorphism. Now let '1/J be another coordinate system given by '1/J(K(t(z) )) a local section for some suitable submersion g . Then

:=z,

where t is

The coordinate transformations are holomorphic.

•

Pr operly Discontinu ou sly Acting Gr oup s. Let G be a complex Lie group acting analytically on an n-dimensional complex manifold X . Then R (x, y )

: -{=}

:l g E G with y = gx

defines an equivalence relation on X . The equivalence class X ( x) = { y E X :J g E G with y = g x} is called the orbit of x under the group action and is also denoted by Gx. The topological quotient Xj R is called the orbit space and is also denoted by XjG. We consider a very special case. Definition. The group G acts properly discontinuously if for all x , y E X there are open neighborhoods U = U(x) and V = V (y) such that

{g E G : gU n V # 0} is empty or a finite set. Here the orbits Gx are discrete subsets of X and are therefore 0-dimensional analytic subsets. If the action is free, we want to show that all conditions are fulfilled for X I G to be a complex manifold and 1l" : X --+ XjG a holomorphic submersion (which means in this case that 1l" is an unbranched covering). 5 . 4 L emma . Let G act freely and properly discontinuously o n X and let xo , y0 E X be given.

1 . If there is a go E G with Yo = gox0 , then there are neighborhoods U = U(xo) and V = V(yo ) such that gU n V = 0 fo r g # g0 . In the case Yo xo and go e one can choose V U. 2. If gxo # Yo fo r every g E G, then there are neighborhoods U = U(x0 ) and V = V(yo ) such that gU n V = 0 fo r every g E G. =

PROOF:

=

=

At first we cho ose neighborhoods Uo (x0 ) and V0 (y0) such that M

:=

{g E G : gUo n Vo # 0}

206

IV. Complex Manifolds

is finite or empty. There is nothing to prove if M = {g0 } in the first case or M = 0 in the second case. Therefore we assume that there are elements gi , · . . , gN , N > 1 , with M = {go , gi , . . . , gN } in the first case and M = {g i , . . . , gN } in the second case. Then we define Y>.. := g>.. xo , for A = 1 , .. . , N . Since G acts freely, Y>.. -=/= yo for A = 1 , , N . _

.

We choose neighborhoods W>.. = W>.. (Y>.. ) and V = V(yo) c Vo such that W>.. n V = 0, and we choose a neighborhood U = U(xo) c U0 such that g>.. U c W>.. , for A = 1 , .. . ,N. Then gU n V = 0 for g -=/= go in the first case, H and g E G in the second case. 5.5

Let G act freely and properly dicontinuously on X . Th en X/G has the unique structure if an n-dimensional complex manifold, so that 1r : X --+ X/G is an unbranched holomorphic covering. PROOF: Let U C X be an open set. Then 7f - I (1r(U) ) = U E gU is an Theorem.

g G

open set, and therefore 1r(U) is also open. For xo E X we can choose an open neigborhood U = U(x0) such that gU n U = 0 for g -=/= e. Then 1r : U --+ 1r(U) is bijective.

( 1 ) We have to show that X/G is a Hausdorff space. Let X I , x2 E X be given, with 1r(x l ) -=!= 1r(x2 )- Then gxi -=/= x2 for every g E G. There are open neighborhoods U = U(x l ) and V = V(x2 ) with gU n V -=/= 0 for every g E G. Then 1r(U) and 1r(V) are disjoint open neighborhoods of 1r(x l ) and 1r (x2 ) . (2) We verify the other conditions. Let x0 E X be given and choose a small open neigborhood U = U(x0 ) c X such that 1r : U --+ 1r(U) is a homeomor phism and such that there exists a complex coordinate system rp : u --+ en . Then f : fJ :=7r - I (7r(U)) --+ e n can be defined by f(gx) : = rp(x) , for a: E U and g E G. It is clear from above that f is well defined. The fibers of f are the G-orbits, and on gU we have f(y) = .rp(g - I (y) ) . This shows that f is holomorphic, and rky (f) = n for every y E U.

1f U is small enough, then 1r - I (1r(U)) = U gEG gU, with pairwise disjoint sets gU that are topologically equivalent to 1r (U) . So 1r is an unbranched covering. •

c omplex Tori. Let {WI , . . . ,W2n } be a real basis of e n . Then the discrete group r := Zwi + . . + Zw2n acts freely on e n by translation. The set .

Aw is the orbit of w .

:=

f + w = { w + w : w E r}

The group r acts properly discontinuously On e n : Let zo , Wo E e n be given. If w0 w0 + z0 for some wo E r , choose =

5. E

<

-::

Quotients and Submanifolds

207

. inf { II w II : w E r - J 01 1.

Then (w+ Bc (zo)) n Bc (wo) = 0, unless w = Wo. If wo - Zo tf. r and

E

<

-l

. dist(wo , f t Zo) ,

2

then (w+ B6 (zo)) n B6 (w0) = 0 for every w . The n-dimensional complex manifold rn = TF == e n ;r is called a complex torus, and r is called the lattice of the torus. The set P == { z = t i WI + · · · + t2 n w2n : 0 < ti < 1) contains a complete system of representatives for the equivalence classes. Therefore, Tn = 1r( P) is a compact space. The map

h WI + . . . + t2 n W2 n rl ( e 27r it 1 , induces a homeomorphism T" --+ S I X . . . X s i .

•

5. 6

Pr op osition .

H I (S I , Z) = Z.

•

•

27r it2n ) e ,

2 n times

27r it : PR OO F: Let U. I ·. = { e2 1r it . _ l4 < t < 4l} U2 .· 3n e 0 < t < 2 r t and U3 := { e 1 ' : � < t < 1 ) . Then %' == { UI , U2 , 3} is an acyclic o)Je'rl covering for S I with UI 23 = 0. Therefore, every triple � = ( a,b, c ) E Z3 is a cocycle in z i ( %' , Z) . It is a coboundary if and only if there is a triple (u , v, w) E Z3 with ·

a = v - u'

b = w - u'

J

'

and

c=

w-v

This is the case if and only if a + c = b. Since every cocycle has the form (a , b, c) = (0, b - a - c, 0 ) + (a, a + c, c) = ( b - a c ! . (0, 1, 0) + 8 (0, a , a + c ) , it follows that H I (%', Z) is generated by the class of (0, 1, 0). • _

From the Kunneth formula it follows that HI (Tn , .Z) the first Betti number of T" is equal to 2n.

'""

.z2n and therefore '

H opf Manifold s. Let (} > 1 be a fixed real number and n > 1. Then the (multiplicative) group r := {Qk : k E Z} acts freely on e n - { 0 } by z rl (}k . z. The action is properly discontinuous. To see this, we define the sets Ur := {z E e n : r < llzii < Qr}, for r > O.

Then the sets (}k Ur are pairwise disjoint. If two points are given, one can find an r > 0 and a k E Z such that z2 E V : = QkUr . The case k = 0 is allowed. Now (}8 U n V

Z I , Z2 E en - { 0 } z i E U = = Ur and = 0 , unless s = k.

208

IV. Complex Manifolds

So H = Hr : = (e n - {0})/f is an n-dimensional complex manifold, and the canonical projection 1r : en - {0} --+ H is an unbranched holomorphic covering. H is called a Hopf manifold. The map

zH

exp

( (

)

In z II z 2n .l II , In (! W

)

induces a diffeomorphism H --+ S 1 X S2n - l . Here S2n - l is the (2n - 1 ) dimensional sphere in JR2 n = e n with 2n - 1 > 3.

k 5 . 7 Proposition . For k > 2, H1 ( S , Z) PROOF:

{U1 , U2 }

We have Sk with

=

{x

= 0.

= (xi , . . . , x k+l )

{ XE

Sk k { XE S

: :

llxll - 1 ) . Then 'P£ -

- & < X k+ l <1!11 Jl -1 < X k+l < -t

is an open covering of Sk with contractible sets, and ll'ii

=

{x

E sk

:

- &

< Xk +l < -t

= Z 1 ( 'P£ , Z) = Z and C0 ( 'P£ , Z) = Z2 . The !!!!! iii Z) is given by J(a , b) :=b - a. Then • = 0. It follows that H 1 ( H, Z ) = -- x S2 k - l , Z) = Z, and br ( H) = 1 .

is connected. Therefore, C1 ( 'P£ , Z) coboundary map 5 : C0 (f£ , 3111 --+ obviously, B 1 (f£ , Z) = Z and 1111

•••

The C omplex Pr oj ective Sp ace. In X : = e n +l the equivalence relation

R(z , w )

: -{=}

3

AE

C*

with w

- {0} we consider

= AZ.

The equivalence class L, of z is the set L, = ez - { 0 } , the complex line through z and 0 without the origin. So we have a decomposition of X into ! -dimensional analytic sets. We can also look at these sets as the orbits of the canonical action of e• on X by scalar multiplication. Definition . The topological quotient ]pm : = X/ R = is called the n-dimensional complex projective space.

(e n+l - {0}) /e*

Let n : X = e n +l - {0} --+ Jpm be the canonical projection, with n(z) : = L, and let two points z = (zo , . . . , zn ) , w = (wo, . . . , wn ) be given. We have

n(z) = n(w)

3 A E e• with wi = AZi for i = 0 , . . . , n zi wi = for all i , j where the fractions are defined.

wJ·

zJ·

5.

Quotients and Submanifolds

209

So 1r (z) does not determine the entries zj , but the ratios zi : Zj · Therefore, we denote the point x = 1r(z0 , . . . , zn ) by (zo : . . . : zn ) and call zo , . . . , Zn the homog eneous coordinates of x. If z0 , . . . , zn are homogeneous coordinates of x , then so are Azo , . . . , A Z n for every A E @*.

If W c X is an open set, then 1r-1 (1r(W)) = U .\ E IC * A · W is a saturated open set in X , and therefore 1r(W) is open in P". For example, this is true for

fji := {z = (zo , . . . , zn) R C'..,.. The sets Ui

:=

-

{ 0 � : zi f. O )

e X,

i = O, . . . , n .

1r( Ui) form an open covering of Jpm

We show that ]pm is a Hausdorff space: Let z , w E X be given, with L, f. 4 Then w z z * ·. = and w := II z II = 1 }. Therefore, are distinct points of S2n+l = {X E lR2n +2 = e n + 1 : we can find an E > 0 such that B6 (z * ) n B6 (w* ) = 0. Then U == 1r ( B6 (z * )) and V := 1r ( B6 (w* )) are disjoint open neighborhoods of 1r(z) , respectively 7r(w). *

l j;lj

l l xl

Now let a point z0 = ( z6o) , . . . , z�o) ) E X be given. Then there exists an index i with zt) f. 0, and Zo lies in Ui . We define fi : ui --+ e n by �

fi ( zo , · · · , Zn )

: ==

�

Zo Zi- 1 Zi+ 1 (,··., ,

Zn \

,... , - I.

Then

If a point u = ( u0 , . . . , u, ) E U; is given, we define a holomorphic section s : e n --+ � by s(z1, . . . , zn) := ( Ui Zl , . . . , uizi, ui , Ui Zi + 1 , . . . , uizn ) · Then

(

)

o Un s U ' . . . ' Ui - 1 ' Ui + 1 ' . . . ' . = ( Uo ' . . . ' Un ) ' "u,' Ut Ui Ui

and

fi s(z� , . . . , Zn ) = (z1 , . . . , zn ) · Therefore, fi is a submersion, and rkz(fi) = n for every z. o

Altogether this shows that Jpm is an n-dimensional complex manifold and 1r : e n +I - { 0} --+ Jpm a holomorphic submersion. Since every equivalence class Lz has a representative in the sphere s2 n+l , Jpm = 1r( S2 n +l ) is compact.

210

IV.

Complex Manifolds

Local coordinates are given by the maps lf!i : Ui

means that

'Pi ( zo : . . . : Zn ) =

---+

e n with 'Pi o 1r = fi ·

This

\ -v ' . . . ' o ' . . . .. ) . z, z,., -

.

7.r:

' .

The set is biholomorphically equivalent to en . We call it an affine part of Jpm. If we remove Uo from lPm , we get the so-called (projective) hyperplane at infinity

H0

{ ( Zo : · · · : Zn ) E lP'n : zo - {(0 : t 1 : . . . : tn) : (h , .

= .

0}

. , tn ) E e n -

(0) ) .

It has the structure of an (n - I )-dimensional complex projective space. If we continue this process we get

lP'n lP'n - 1

en en -1

u

U

lP'n - 21 ' lP'n - '

It remains to study lP' 1 = {(zo : zi ) : (zo , zi ) E e 2 - ( 0 ) ) . But this is the union of e = (( 1 : t ) : t E e} and oo : = (0 : 1 ) , with t = zi/z0 . In a neighborhood of oo we have the complex coordinate zo / z1 = 1 jt. So we see that lP' 1 = e = e U { oo} is the well-known Riemann sphere. The hyperplane H0 is a regular analytic hypersurface, given by

H0 n ui

=

� ( zo : . . . : Zn ) E ui : �0

=

0 �.

Therefore, U0 is dense in P". It should be remarked that there is no reason to distinguish between Uo and the other sets Ui . Everything above could have been done as well with the affine p art ui and the hyperplane Hi := {7r ( z) E lP'n : Zi = 0 } .

Meromorphic Functions. On a compact complex manifold every global holomorphic function is constant. But we know already from the ex ample of the Riemann sphere that there may exist nonconstant meromorphic functions. In this regard we consider the compact manifolds defined above, beginning with the complex projective space lP'n .

A nonconstant polynomial p(t) = l::�v i =O avt v is a holomorphic function on the set Uo = {(1 : t 1 : . . . : tn ) I t = (t 1 , . . . , tn ) E e n }. In fact, it defines a meromorphic function on lP'n with polar set H0. We see this as follows: 3

The hat signalizes that the i th term is to be left out.

5.

Quotients and Submanifo1ds

211

The functions tM = zM /zo, p, > 1 , are holomorphic coordinates on U0 , and w.\ := z.\/ Zi , A f. i likewise on Ui . Therefore, on ui - Ho = ui n Uo we have

( :o ) .. .

I.e. , p = gj h on ui - Ho , where g(w)

h(w) := w�

L av ( :� )

:=

"

'

...

( :n

v 2:: �vi=O av w� -l l w�' . . . w�n an d

are holomorphic functions on ui with

N(h) = (W E Ui : Wo = 0} = Ui D Ho . So there are numerous global meromorphic functions on projective space. Now let T = e n /f be an n-dimensional complex torus, and rr : e n ---+ T the canonical covering. If m is a meromorphic function on T, then m o rr is a meromorphic function on en ' which is periodic with respect to the generators w 1 , . . . ,w2 n of the lattice I?. In the case n = 1 such meromorphic functions always exist; they are the !?-elliptic functions. We shall later see that for n > 2 the existence of !?-periodic functions depends on the lattice I?. In fact there are complex tori with no nonconstant meromorphic functions. Finally, consider the Hopf manifold H = ( e n - { 0 } ) /f with r = {Qk : k E Z} and n > 1. Let m be a meromorphic function on H. Since the canonical projection rr : e n - ( 0 ) ---+ H is a covering, m := m o rr is meromorphic on en - ( 0 ) . Since n > 1 , it follows from Levi ' s extension theorem that m can be extended to a meromorphic function on e n . On any line L through the origin in en , m must have isolated poles or be identically oo . But since m comes from H, poles on L must have a cluster point at the origin, which is impossible unless m is constant on L. The same argument works for any other value of 6 . A meromorphic function on e n - (0) that is constant on every line through the origin comes from a meromorphic function on the projective space Jpm - I _ This means that if h : H ---+ ]pm- l is the canonical map, then a bijection J!t (lP'n- l ) ---+ J!t (H) is defined by m H m o h . On the n-dimensional Hopf manifold there are not "more" meromorphic functions than on ( n - I)-dimensional projective space.

Gras smannian M anifolds . The set of 1-dimensional complex subvec tor spaces of en +l can be identified with the n-dimensional projective space, and we have given it a complex structure. Now we do the same for the set Gk n of k-dimensional sub spaces of en . The idea is the following: If V0 c e n is a fixed element of Gk ,n, then we choose an ( n - k)-dimensional subspace ,

212

IV. Complex Manifolds

Wo c en such that Vo EB Wo = e n . We are looking for a topology on Gk , n such that the set of all k-dimensional subspaces V with V EB Wo = e n is a neighborhood of Vo in Gk , n·

But how to get complex coordinates? In the case G 1 , n + 1 = ]pm we consider, for example, V0 = Ceo with e0 = ( 1 , 0, . . . , 0 ) and Wo = { ( zo , . . . , zn ) : zo = 0). Then Vo EB Wo = c n + I ' and a vector z = ( zo , . . . ' Zn ) cj 0 generates a 1dimensional space v with v EB Wo = en + I if and only if Zo cj 0. Multiplication by a nonzero complex scalar does not change the space V . Therefore, V is uniquely determined by

) z0- 1 . z = z0- 1 . ( z0, z �

The map

f

:V

c-t

=

1

( ,.z0- 1 . z ) �

With Z = ( Z 1 , · . . , Zn ) .

z0 1 . z E C" gives the familiar local coordinates.

When we try to transfer this procedure to higher k, we use another viewpoint. Every V with V EB Wo = C n +l has the form Graph( tp v ) of a linear map 'PV : c --+ e n given by f ( V) = 'PV ( l ) .lf Vo c e n is a k-dimensional subspace and V0 EB W0 = en , then every other k-dimensional subspace V c e n with V EB Wo = en has the form Graph(cpv) for 'PV E Homc ( Vo , Wo ) · Fixing bases of Vo and Wo , the matrix of 'PV with respect to these bases gives local coordinates in Mk, n - k (C) '"" C k (n - k) . Now we will do this job in detail. An ordered k-tuple of linearly independent vectors a1 , . . . , ak E e n can be combined in a matrix

with rk(A) = k. The set

St (k, n) := {A E Mk , n (C)

:

rk(A) = k}

is called the complex Stiefel manifold of type (k, n ) . Since its complement in Mk, n (C) '"" C k n is an analytic set given by the vanishing of all (k x k) minors of A, St (k, n ) is an open set in Mk , n (C) and therefore a complex manifold. The group GL k (C) acts on St(k, n ) by multiplication from the left, and every orbit of this group action represents exactly one k-dimensional subspace of e n . The topological quotient

Gk , n = St(k, n ) / GL k (C) is called the c omplex Grassmannzan of type ( k,n ) . If, for example, Wo = rw = ( w l , · . . , w n ) : w 1 = · · · = w k = o } , then a matrix A E St ( k, n ) represents a basis of a k-dimension al space V with V EB Wo = e n if and only if A = (Ao [A) with Ao E GLk (C) and A E Mk , n - k (C) .

5.

Quotients and Submanifolds

213

In this case v has the form Grap h ( lf!V) for a linear map lpv : e k -t e n - k . Of course, V is uniquely represented by the matrix A0 1 . A = ( E k IA0 1 . A) , and A0 1 · A is the matrix of rpv with respect to the standard bases. Now we consider the set of multi-indices

yrk , n := { I = (i l , · · · , ik ) E Nk : l < i 1 < . . . < i k < n } . For any A E St(k, n ) there exists an I = (i 1 , . . . , i k ) E yrk , n such that

Ar Then there

IS

(Ar iAr) .

:=

a permutation matrix P r E GL n (C) such that A . P r -

For fixed I we define

Vr := {A E St(k, n ) : det Ar f. 0} . .---....._..-

�

We remark that ( G . A) r = G . Ar and ( G . A) , = G . Ar for G E GL k (q . Therefore, Vr is invariant under the action of GL k (q . 5. 8 Lemma.

Let 7rk , n : St( k , n )

-t

Gk , n be the canonical projection. Th en

Let A E 7r;; � (7rk, n (Vr)) be given. Then there is an A* E Vr with 7rk, n (A) = 7rk , n (A* ) . This means that there is a matrix G E GL k (q with A = G . A * . Since Vr is invariant under the action of GL k (C) , A lies in Vr . The converse inclusion is trivial. • PROOF:

So Vr is a saturated open subset of St( k, n ) , and Ur := Kk , n (Vr) is open in Gk, n · We leave it to the reader to show that Gk, n is a Hausdorff space.

If Er C e n is generated by ei, , . . . , eik , and Fr C en by the remaining ej , then Er EB Fr = e n , and every k-dimensional subspace V c e n with V EB Fr = e n is represented by a matrix A E Vr. The uniquely determined matrix A[ 1 . Ar E Mk, n - k (C) describes the linear map rpv : Er -t Fr . Therefore, we define the holomorphic map fr : Vr -t Jl.fk, n - k (C) � e k( n - k )

by

fr (A) := A r 1 . Ar.

It is clear that fr is holomorphic, and since

fr (G · A) = (G . Ar) - 1 . (G . Ar)

=

fr (A) ,

2 14

IV.

Complex Manifolds

f1 respects the fibers of 1fk,n · It remains to show that fr has maximal rank. For that we construct sections:

Let A E Vr be given. Then we define s : Mk,n - k (q --+ Vr by s (B)

:=

Ar . (Ek !B) · P / .

This is a holomorphic map with s(fr (A) ) = (Ar!Ar) . P/ 1 = A and fr o s (B)

=

f1 (A1 . (Ek ! B) . P / )

=

Ar 1 . (Ar . B ) = B .

So fr is a submersion and Gk,n a complex manifold of dimension k ( n - k). Comp lex coordinates are given by rpr : Ur --+ Mk,n - k (q with rpr (1fk,n(A))

:=

A[ 1 . Ar.

-t

Let Sk,n = = {A E St (k, n ) : A · A = Ek } be the set of orthonormal systems of k vectors in e n . Then Sk ,n is a compact set, and 1fk,n : Sk , n --+ Gk,n is surjective. So Gk , n is compact.

Submanifolds and Nor m al B u n dles. Let X be an n-dimensional complex manifold. A holomorphic map f : Y --+ X is called an embedding if there is a submanifold Z c X such that f induces a biholomorphic map from Y onto Z. Every embedding is an immersion, but in general not every (injective) immersion is an embedding. The following proposition has been already mentioned in Section 1 . 5.9

Pr op osit ion . Let f : Y --+ X be a holomo rphic map between com plex manifolds. If Y is compact and f an injective immers ion, then f is a n embedding. PROOF: Every immersion defines a local embedding. To see this, we may consider a holomorphic map f from a neighborhood V = V (O) c em into a neighborhood u = U(O) c em X e n - m with f(O) = (0, 0) and rk Jr (O) = m. We write f = (f1 , f2 ) and assume that already rk Jr1 (0) = m. Then for the map F : v X e n -m --+ en with

we have det JF (O, 0) = det Jr1 (0) f. 0. So there exist neighborhoods V* V* (O) c V and W = W(O, 0) c U such that F : V* x V* --+ W is biholomor phic. Since F(z, 0) = f(z), the image f(V*) = F(V * x { 0 } ) is a submanifold of W . We have proved that for every point xo E Y there are neighborhoods V = V(xo) c Y and W = W(f( xo)) C X such that J (V) is a closed submanifold of W. We have to show that there is a small open neighborhood U = U(f(x0)) c W such that f(Y) n U = f(V) n U. Suppose that there is

5. Quotients and Submanifolds

215

a sequence Xn E Y - V with f(x n ) --+ f (x0 ) . Since Y is compact, we can assume that (xn ) converges to an element x* E Y - V .Since f is injective, • f(x* ) f. f(x0 ) . But f (x n ) must converge to f(x* ) . This is impossible. If Y is a regular hypersurface in a complex manifold X , then there are two line bundles associated to Y ,namely the normal bundle Nx (Y) , which is defined on Y, and the bundle [ Y]on X . We will show that these bundles coincide on Y.

Choose an open covering %' = (Uc) cEI of Y in X such that there are complex coordinates z� , . . . , z� for X in U" with the following properties: 1 . Y n Uc = { X E U, : Z� ( X) = 0}. 2 . z L . . . , z �, _ 1 are complex coordinates for Y.

We have already seen in Section 2 that the normal bundle Nx (Y) is given with respect to u" n y by the transition functions

On the other hand, the line bundle [ Y)s defined by the transition functions f""' : = z� / z�. But z� ( z� , . . . , z� ) = f""' ( z� , . . . , z� ) · z� implies

az " (z�, . . . ' z� - 1 , 0) = !cl< (z�, . . . ' z� 1 , 0) . a"'�� n This yields the so-called first adjunction formula. 5 . 10

Fir st adj un ction formula. [Y] h,· = Nx (Y) fo r every regular hypersurface Y C X.

The normal bundle Nx (Y) is naturally related to the canonical bundles Kx and Ky . If g""' ' respectively G, are the transition functions for T(Y) , re spectively T(X) , then

Therefore, det G, o j = det g""' . h, and det g" "'1 = (det G" "'1 o j ) is equivalent to the second adjunction formula: 1

5 . 11

. h, 1

which

S econd adjun ction formula. Ky = j* Kx ® Nx (Y) fa r every regular hypersurface Y c X .

This formula can be generalized to higher codimensions; see, for example,

[GriHa78] .

216

IV. Complex Manifolds

Pr oj ective Algebr aic M anifolds. canonical projection and x E Jpm , we define

If

- {0}

1r : c n + l

---+

Jpm is the

This is a complex line through the origin in cn +l , and we have C(1r(z)) = Cz for z E cn +l {0}.

-

A set X c c n + l is called a conical set or a cone if it is the union of a family of complex lines through the origin. This means that �

zEX

�

AZ E X for A E C.

===?

If X is an arbitrary subset of ]pm , then

X :=

U

xEX

U

f(x) = 7f - 1 (X)

(0)

is a conical set.

Let X c c n + l be a conical set, f a holomorphic function near the origin, and f = 2:� 0 Pv its expansion into homogeneous polynomi als. If there is an E > 0 such that f i Bc ( D) nX - 0, then Pv l x = 0 fa r ev ery

5. 1 2 Lemma.

l/ .

PROOF:

Let

z

f. 0 be an arbitrary point of B6 (0) n X. Then 00

u=o

vanishes identically for I A I < I . So Pv ( z) = 0 for every v , and since X 1s • conical, P v I x 0 for every v . Now let F'r , . . . ,Fk be homogeneous polynomials in the variables zo , Then the analytic set �

X

:=

{z

�

E

en +

is a cone. If we set X ' := X set

X In U;

by

1

:

F1 (z)

= . . . = Fk (z) = 0 }

- {0 ) , then the image X

:= 1r (X ' ) c ]pm is the

= { (ZQ : . . . : Zn ) : F ( zo , . . . , Zn ) = = Fk (ZQ ,

= {(zo

1

.. . . . : z,)

=

. . . , z,.

·

•

·

· ··

,

Zn )

= 0}

•

z; f. 0}, we can define holomorphic functions hv zo Zn . , . . . Fv = ' ) J; ,v (zo : . . . : Zn : z, z, {x E U; : fi , 1 (x) = . . . = J; , k (x) = 0}, and consequently X

Then X n U; is an analytic set.

:

(

)

5. Quotients and Submanifolds

217

Defin it ion . An analytic set X c Jpm that is the zero set of finitely many homogeneous polynomials is called a (p rojective) algebraic set. The subsets X n Ui are called (affine) algebraic.

E

A complex manifold X is called projective algebraic if there is an N N and a holomorphic embedding j : X --+ lP'N such that j (X) is a regular algebraic set.

Every analytic set X in projective space is the zero set of finitely m any homogeneous polynomials F1 , . . . ,F8 such that if x X is a regular point of codimension d, then rkz ( F1 , . . . , F, ) = d fa r every z 1r - 1 (x) . 5 . 1 3 Theorem of Chow.

E E

PROOF: If X c lP'n is a nonempty analytic set, then X' = 1r - 1 (X) is also analytic. Since dimz (X') > J for all z c n +l _ { 0}' by the extension theorem of Remmert-Stein its closure X = X' u { 0} is analytic in cn + I .

E

By Theorem 6.5 in Chapter III we can find an open neighborhood U = U(O) c C n + I and finitely many holomorphic functions ft , . . . ,fm on U with N(ft , . . . , f ) = U n and rkz ( !I , . . . , fm) = d at any regular point z of m dimension n+1 -d in U n . Now we expand fi into homogeneous polynomials Pi ,v· Then Pi ,v l x = 0 for all i , v.

X

X

Let h c (')0 Hn + I be the ideal that is generated by all i = 1 , . . . , m . Since 11 C I, C . . . C 00 '""

Pi, v,

v < k,

is an ascending chain of ideals in a noetherian ring, it must become station ary. Thus there are homogeneous polynomials F1 , . . . , F8 such that every Pi v is a finite linear combination of the Fa . But then every fi is also a linear combination of the F,: ,

a=l It is clear that N ( ft , . . . , fm ) = N (F1 , . . . , F8) near the origin. But since X is a cone, even X = N ( F1 , . . . , F, ) . Setting �

A (z) = ( ai, a(z) we have

i = 1, . . . , m CY = 1 , . . . , s

)'

A (z) . J(F, , . . . , Fs ) (z) for z E X near 0. Therefore, d = rk, (!I , . . . ,Jm ) < rk, (F1 , . . . , Fs) . If X is regular of codimen sion d at x, then X is also regular of codimension d at every z E 1r- 1 (x) , J(f, , ... ,Jm) (z) =

218

IV. Complex Manifolds

because 1r is a submersion. It follows that rkz (F1 , . . . , Fs) cannot be greater than d at these points. Since the rank is constant along 1r - 1 ( x) , it must be • equal to d at every z E 1r - 1 (x) . Examples

1 . Let L C e n+ I be a complex linear subspace of codimension q . Then there are linear forms y l , . . . , yq On en + l SUCh that L

=

{z E e n+ I

:

tp 1 (z)

=

...

=

tpq (z)

=

0}.

Since the linear forms are homogeneous polynomials of degree 1 ,

IP' ( L)

:=

{ ( zo : . . . : z,) E IP'n

:

tpf.t ( zo , . . . , Zn )

=

0 for f..L

=

L

. . . , q}

is a regular algebraic set. We call !P'( L) a (projective) linea r subspace. It is isomorphic to IP'n - q . 2. We now show that the Grassmannian manifolds are projective �lgebraic. Let 0 < k < n be given, and N : = G) - 1 . We identify 1\ e n with eN+ l and define the Plucker embedding pl : G k . n --+ IP'N as follows: 1f a subspace v c e n has the basis { al , . . . , ak } , then

It is clear that this is a well-defined injective map . To see that it is a holomorphic immersion, we choose another descripton. As above, when we introduced the Grassmannian, we use the set of multi-indices

To any I E Y'k, n there corresponds a permutation matrix P r E GL n (e) such that for A E St(k, n) we get A . P I = (Ar iAr) . We define p : St(k, n) --+ e N + l - { 0} by p(A)

:=

(det Ar I I E Y'k , n ) ·

Then p(G . A) = det G . p(A) , so p induces a map p : G k, n --+ IP'N such that the following diagram commutes: p

-

St(k, n)

cl'k, n

p

5.

Quotients and Submanifolds

L

a 1 /\ · · · /\ ak =

I E..Yfk ,n

219

det AI - ei ,

and therefore p = pl. Let 'P I : UI --+ Mk , n - k (C) be a complex coordinate system for Gk , n · Then rp i 1 (B) = 7rk,n ((Ek iB) - P[ 1 ) and 1 1 p o rp [ ( B ) = Jr o p ((Ek i B ) · P[ ) = 1r (det ((Ek i B) · P[ 1 ) J :

J E

Y'k , n) ·

Obviously, p is a holomorphic map. Every matrix

E

B E Mk ,n - k (q can be written in the form B = (b� 1 , . . . , b�) +

with b, C k for p, = k + 1 , .. . , n . We have det ((Ek iB) . P/) I = 1 , and if J = (I- {iv }) U {p,} for some 11 { 1 , . . . , k } and p, {k + 1 , .. . , n } , then

E

E

and therefore det ((En iB) . P[ 1 ) J = bvw So p o rp / (B) contains the components 1 , b, for all 11, p, and some other components. It follows that pl is an immersion. Since Gk,n is compact, pl is an embedding.

Pr oj ective H yper sur fa ces . The simplest example of an analytic hy persurface in Jpm is the hyperplane H0 { z0 0). =

=

If S E Mn+I (q is asymmetric matrix, then qs (z) := z·S·z t is a homogeneous

polynomial of degree 2. The hypersurface Q s : = { ( zo : . . . : z,)

: qg ( zo , . . . , Zn) = 0}

is also called a hyperquadric. It follows easily from the classification of sym metric matrices that Q 8 "" QT (biholomorphic) if and only if rk(S) = rk(T) . In particular, every quadric of rank n + 1 is biholomorphically equivalent to the standard hyperquadric Qn -1 = { (zo : . . . : z, ) : z5 + · · . + z� = 0). Since Qn - 1 n Uo = (( 1 : t 1 : . . . : tn) : ti t . . . + t � = - 1 }, Qn - 1 has no singularity in Uo . The same works in every Ui , so Qn _ 1 is a projective algebraic manifold. Now consider the Grassmannian G 2 ,4 '-+ lP'5 . Since dim (G 2 4) = 4, it is a , hypersurface. From multilinear algebra one knows that a 2-vector w E f\ 2C 4 is decomposable (i.e., of the form w = a A b ) if and only if w 1\ w = 0. So G2 ,4 = {1r(w) : w12W34 - w13 W24 + W14 W2 3 = 0}

220

IV.

Complex Manifolds

is a hyperquadric that is isomorphic to Q4 . This means that there is a 1 to 1 correspondence between the set of proj ective lines in lP'3 (which is the same as the set G2 ,4 of planes in C4) and the set of points of a 4-dimensional quadric.

Every analytic hypersurface Z C lP'n single homogeneous polynomial. 5 . 14 Theor em .

a�

the zero set of a

�

PROOF: If Z c lP'n is an analytic hypersurface, then Z = 1r -1 (Z) U ( 0 ) c c n+ 1 is also an analytic hypersurface. Therefore, there exists an open neigh borhood U = U(O) c c n + 1 , a point z E U n Z ,and a holomorphic function f : U -+ C with \7 f(z) f. 0 and Z n U c N(f). Making U smaller if necessary, we can find a holomorphic function g dividing f such that U n Z = N(g) . Without loss of generality we may assume that u U' X U" c c X e n and g (z 1 , z') = Z � + ak- 1 (z')z� - 1 + · . . + ao (z') �

�

=

is a Weierstrass polynomial. If a�<,v is the homogeneous part of a, of degree v, then

Pk (z1 , z')

:= z� + qk -1 , 1 (z')z� - 1 + · · · + qo,k (z')

is the homogeneous part of g of degree k. Since Z is conical, it follows that Pk l z 0.

=

Now there is a dense open subset V c U" such that { t E U ' : g(t, z') = 0} consists of exactly k points, for every z' E V. Since N(g) n U c N(p k ) n U and deg(pk ) = k, we have g Pk over V and then everywhere in U, by the • identity theorem. So Z = N(p k ) · �

=

We can choose a polynomial p with minimal degree such that

Z

=

{ (zo : . . . : Zn ) E lP'n : p(zo , . . . , Zn ) = 0} .

Then by the degree of Z we understand the number deg(p). For example, deg(H) = l for any hyperplane, and deg(Q) = 2 for any hyperquadric. Now, let Z c lP'n be an arbitrary hypersurface of degree k, defined by some homogeneous polynomial p of degree k. Then

Z n Ui

= { (zo : . . . : Zn )

E

Ui : zi k .p(zo , . . . , Zn )

=

0},

( ZJ I zi) k . and the line bundle [ Z !b given by the transition functions 9i; In particular, for every hyperplane H we have the same line bundle [H] with transition functions ZJ I zi · =

Defin ition . If H c lP'n is a hyperplane, then the line bundle 0 ( 1 ) [H] is called the hyperplane bundle. The kth tensor power of the hyperplane bundle is denoted by O(k).

5.

Quotients and

Submanifolds

221

If Z c ]pm is a hypersurface of degree k, then [ Z,j: O(k).

A homogeneous polynomial F of degree k induces a global section sF f(lP'n , O(k)) by

(sF ) i ( zo : . . . : zn ) : = zj k F(zo, . . . , zn ) In fact, (sF) i is a holomorphic function on ui ' with

Obviously,

{X

E JP'n

:

SF (X ) =

0}

=

E

for zi f. 0.

{ ( Zo : . . . : Zn ) : F ( Zo ,

.

·

. , Zn ) = 0}.

On the other hand, let s be an arbitrary global holomorphic section of O (k) . Then s can be represented by holomorphic functions S i : ui --+ c with

: Zn ) On Ui n Uj , Z� · Si ( Zo : and we obtain a holomorphic function f : c n + l - ( 0 ) --+ c with ZJ . S j ( Zo : . . . : Zn ) f( z )

=

.

z� si (1r(z))

=

·

·

·

on 7r -1 (Ui) ·

There is a holomorphic continuation F of f on c n+ l with F(>..z) = A • • F ( z ) . This means that F is a homogeneous polynomial of degree k, with sF = s . Consequently, we have the following result.

EN

For k the vector space f(lP'n , O(k) ) is isomorphic to the space of homogeneous polynomials of degree k in the variables z0 , . . . , zn . Analytic hypersurfaces in lP'n are exactly the zero sets of global holomorphic sections cf O(k) . there is a homogeneous polynomial For any Zo c n + l - ( 0 ) and any k F of degree k with F(z0) f. 0. Therefore O(k) is generated by global sections. 5 . 1 5 Pr oposition .

EN

E

The bundle 0(1) can be described in purely geometric terms. For this we note that if -1 Po := (0 : . . . : 0 : 1 ) lP'n 1 , then a projection 1r0 : JP'n + 1 - {Po} --+ lP'n is given by

E

7ro (zo : . : zn : Zn + l ) : = ( zo : . . . : zn ) We obtain local trivializations Yi : Jr() 1 (Ui) --+ ui X c by ·

·

(

·

)

lf!i ( Zo : . . . : Zn : Zn+ 1 ) : = ( Zo : . . . : Zn ) , Zn+ l Zi with lf!i 1 ( (zo : . . . : zn ) , c) = (zo : . . . : Zn : czi) Obviously this gives the 1 + !ran_sition functions Zj zi _If F . is a linear form on e n ! ' then the section sF IS g1ven at x = 1r ( z ) = �zo . . . . . z ) by n SF (x) = 'Pi 1 (x, z;- 1 F(z) ) = (zo : . . . : Zn : F(z) ) . ·

.

,

222

IV. Complex Manifolds

The Euler Sequence. Here we discuss the tangent bundle of the pro jective space in more detail. Let 1r : c n+ l - {0} ---+ ]pm be the canonical projection. Then for any point z E c n + l - {0} we have a linear map lf!z : c n+ l ---+ T7r (z) (lP'n ) defined by

lf!z (w ) [ j]

:=

d dt

0

(f o 1r(z + tw) ) .

The vector w can be interpreted as tangent vector 0:(0) = a. , o ( (djdt)o) with a(t) := z + tw. It follows that

7f, ,z (w ) [ f]

= ( 1r o

a) .,o(d j dt)o [f]

=

(dj dt)o [f o 7f o a] = lf!z (w ) [ f] .

Since 1r is a submersion and lf!z ( z ) = 0, the linear map lf!z is surjective with Ker( lf!z ) = Cz. Ifwe use the local coordinates tv : = zv / zi in Ui = { (zo : z1 : . . . : zn ) l zi f- 0} , then for any z with zi f- 0 it follows that

ZiWv - WiZv

�(

)

0

.

lf!z (w ) = i atv Zi2 'v# Thus a trivialization 'l/Ji : T(lP'n ) l u, ---+ ui X en is given by 'l/Ji ( 'Pz (w ))

:=

(

1r ( z )

,

with a , = w, - ( wiz;,- 1 )zv for 11 f- i.

� (aa , . . .

, iii , .

)

. . , an ) ,

Let 0(1 ) Eil ( n +l) be the direct sum of n + 1 copies of the hyperplane bundle. Recall that every linear form F on cn + l defines a global section SF of 0(1) by ( sF )i = z; 1 F. Then two vector bundle homomorphisms j : OJI'n = lP'n x C ---+ 0( 1 ) Eil (n+ l ) and q : 0 ( 1 ) Eil ( n+ l ) ---+ T(lP'n ) can be defined by

j ( X, ) C

:= C '

( S zo ( X ) , . · , S Zn ( X )) ·

and

q( sF0 (7r (z) ) , . . . , sFn (1r (z) )) : = lf!z (Fo(z) , . . . , Fn ( z ) ) , where Fa , . . . , Fn are linear forms on c n + l .

Using the canonical trivializations, j and q can be given over Ui by

j : ((zo : . . . : zn ) , c) H ((zo : . . . : zn ) , c · (zo z; 1 , . . . , zn zi 1 )) and

� . , a , , . . . , a, )) ,

q : ( ( zo : . . . : Zn ) , (Wo , . . . , w n )) H ( ( zo : . . . : Zn ) , ( ao , . with av = wv - (wiz;,- 1 )zv. This shows that both of these homomorphisms have constant rank. Since there is always an index i such that sz, ( x ) f- O, j is injective. Since 0(1) is globally generated, q is well defined and surjective. ·

5.

Quotients and Submanifolds

223

Defin ition . A sequence 0 ---+ V' � V 4 V" ---+ 0 of vector bundles called exact if the following hold: 1. j and q have constant rank. 2 . j is injective and q is surjective. 3 . Im (j ) = Ker(q).

5 . 1 6 Theorem .

is

We have a n exact sequence of vector bundles

The sequence is called the Euler sequence. PROOF: It is clear that q o j (7!' ( z ) , c ) = 'Pz ( c z) = 0. If, on the other hand, x = 7l'(z) and q(sF0 (x), . . . , sFn (x)) = 0, then (Fo (z) , . . . , Fn ( z )) must be a multiple of z . So there exists a c E C such that (sF0 (x), . . . , sFn (x)) (X ) , . . . , S Zn ( X )) . C ' (S • ·

zo

Ration al Fun ctions. We consider some connections with complex alge braic geometry. A meromorphic function m on ]pm is called rational if m = 0, or if there are homogeneous polynomials F and G of the same degree such that F f- 0 and

) m ( za .. . . . .. zn ) _ - F ( ZQ , , Zn . G(zo, . . . , Zn ) ·

5. 1 7 Theorem .

PROOF:

If

.

.

Every meromorphac functson on ]pm

ai

rational.

m E .4'(1P'n ) , then its divisor has a finite representation

with n, E Z and irreducible hypersurfaces Zi C lP'n . By the theorem of Chow every Zi is the zero set of a homogeneous polynomial F; of degree d, > 1 . Then F = = Il FJ'i is a rational function on cn+ l that is homogeneous of degree d := 2:: i n i di E Z . Since div(m o 7!' ) = div(F) on cn + l - {0 } , the function f := ( m o 1l' ) . F- 1 is

holom_!?rphic there witho_Et zeros. It has a holomorphic extension j to cn + l with f( O) f- 0, because f cannot have isolated zeros.

For c E C* , the equation f( c z ) = c - d . [( z) is valid on an open subset of c n+ l , and then (by the identity theorem) everywhere on c n + 1 . Thus d < 0 ·

224

IV. Complex Manifolds �

�

and c- d = l. From [( c . z ) = f (z) it follows that f ( z ) is a constant wo , and • m o 7l' ( z ) = w 0 . F ( z ) a rational function. submanifold X c Jpm is the restriction m i x of a rational function m on P". We have already seen that every polynomial p on U0 en can be extended to a rational function m on Jpm . Therefore, on every algebraic manifold there is a large number of rational functions.

A rational function on

a

rv

]pm

is a projective algebraic set, then A; := A n U; is called an affine algebraic set. A complex manifold is called an affine algebraic manifold if it is biholomorphically equivalent to an affine algebraic set. A regular function on an affine algebraic manifold j : X '---+ U; is a holomorphic function f : X ---+ e such that there exists a polynomial p on U; e n with f = p o j . It can be shown that a rational function on an affine algebraic manifold is always a quotient of regular functions. If A c

'""

In algebraic geometry there is a more general definition for regular functions, which coincides with our notation in the case of affine algebraic manifolds. On projective algebraic manifolds all regular functions are constant, whereas there are many rational functions. In the affine case the field of rational functions is exactly the quotient field of the ring of regular functions. If Z c ]pm is a hypersurface of degree k , then X : = algebraic manifold. We can see this as follows:

]pm

- Z is an affine

Let I be the set of multi-indices 11 = ( 110 , . . . 11n ) with 110 + · . . +11n = k. Then # ( I ) = (nk k) is the number of monomials z v = z�0 z�n , 11 E I. We set N : = # ( I ) - 1 and define the Veronese m ap vk, n : pn ---+ pN by ,

•

Vk, n ( zo : . . . : Zn )

:=

•

•

( Zv 1 11 E I) .

One can show that vk, n is an embedding, and so its image is Vk, n = vk, n (JPn ) an algebraic submanifold of pN . If p is a homogeneous polynomial of degree k with zero set Z, then there are complex numbers a, 11 E I, such that p = � vEl av z v . It follows that Vk,n (Z) = Vk,n n

{ (wv ) vEl : I:> vwv = 0 } vEl

is the intersection of Vk, n with a hyperplane H c pN . Therefore, pn - Z '"" Vk, n ( Pn - Z) = Vk, n n ( JPN - H ) is affine algebraic. Exercises

0}

1. Show that M : = { ( zo : z1 : zz : z3 ) E 1P3 : zS + · · · + z� = is a projective manifold. Consider the group G := {gm : 0 < m < 4}, where g : JP 3 ---+ JP3 is defined by

5.

Quotients and Submanifolds

Q

g(zo : ZI : zz : z3) : = (Qzo : Q2ZI : Q3z2 : Q4z3 ) ,

=

225

e2 1l' i /5

Prove that X := M/G is a complex manifold (called a Godeaux surface). 2. Let F : C 3 ---+ C 3 be defined by F(t, ZI, zz) := (t, azi

3.

+ tzz , azz) ,

a E JR, 0 < Ia I < 1 .

Set G = {F m : m E Z} and prove that X : = (C x (C 2 - {0}))/G is a corn pact complex manifold. Consider the holomorphic map 1r : X ---+ C that is induced by the projection onto the first component. Then Xt : = 7r - I ( t) is a 2-dimensional complex manifold. Show that Xt is biholomorphically equivalent to XI for every t f- 0, but that there is no biholomorphic map between Xo and XI . Show that Xo is a Hopf manifold. For WI , . . . , Wzn E e n define n := (wf , . . . , win ) E Mn ,2n (C) and prove that

WI , . . . ,wzn linearly independent

-{==}

det

(�) f- 0 .

Let To be the torus defined by the lattice r := Zwi t . . . + Zwzn · Prove that To is biholomorphically equivalent to To' if and only if there are matrices G E GL n (C) and M E GLzn (Z) such that f!' = G . f! · M . 4. The Segre m ap (J'n : lP'I x lP'n ---+ lP'2n + I is defined by

(J'n ( (xo : xr ) , (Yo : . . · : y)) := (XoYo : . . · : xoYn : x l Yo : . . · : xi Yn ) · Show that � n : = (J'n (lP' I x lP'n ) is a complex manifold and (J'n an injective

holomorphic immersion. 5. Show that for n > 2, two irreducible hypersurfaces in nonempty intersection. 6. Decompose A := { (zo : ZI : zz) : z6 + 3z6zi

lP'"

always have a

+ 2z6 + 4z6z� + 5z?z� + 3zi = 0}

into irreducible components. 7. Let d > 0 and n > 0. Prove the following theorem of Bertini. If Hn + I (d) is the vector space of homogeneous polynomials of degree d in z0 , . . . , zn , then there is a dense open subset U c Hn +l (d) such that N (F) = {(zo : . . . : Zn ) E lP'n : F(zo , . . . , Zn )

=

0)

is a projective manifold for every F E U. 8. Use the notation from the introduction to Grassmannian manifolds. For I E Y'k , n set Vr : = {A E St (k, n) : det Ar =/= 0) and Ur : = 1fk, n ( Vr) C Gk n · Then '

give the transition functions of a vector bundle Uk on Gk ,n · there are n independent global holomorphic sections in uk .

Show

that

226

IV. Complex Manifolds

9. Show that H 1 (ll1m , 0) 0 for n > 1 and H 1 (1P'1 , 0 ( k)) = 0 for k > 0. Prove that there is an exact sequence of vector spaces =

for n > 2 and k > 1 . Conclude that H1 (1P'n , O(k)) = 0 for n > 1 and

k > 0.

10. Let X be a compact complex manifold and 1r : V ---+ X a holomorphic vector bundle of rank r > 2 . Show that there is a holomorphic fiber bundle P ---+ X with typical fiber lP'r-1 and a line bundle L ---+ P such that Px lP' (Vx) and LIPx rv OpJ 1 ) . 1 1. Prove that for every automorphism f of lP'n (i.e., every biholomorphic m a p f : ]pm ---+ P") there is a n A E GL n +1 (C) with j(1r(z)) 1r(z · A t) . This is formulated as Aut( lP'n ) P G L n+ l (C) := GL n + 1 (C)/C* . 12. Let r = Zw1 + Zwz be a lattice in C and p the Weierstrass function for this lattice. Show that the map z H ( 1 : p(z) : p' (z)) induces an embedding rp of the torus T C/f into lP'2 . Determine the equation of the image rp (T) in lP'2 . Determine transition functions for rp*O( 1 ) . =

=

=

=

6.

Br anched Riem ann Dom ain s

Br anched Analytic C over ings. Let f : X ---+ Y be a continuous map between Hausdorff spaces. Then the image of compact sets is compact, and the preimage of closed sets is closed. Definition. The map f is called closed if the image of closed subsets of X is closed in Y. It is called proper if the preimage of compact subsets of Y is compact.

6 . 1 Pr op osition.

Iff : X ---+ Y is closed and A C X a closed subset, then f l A : A ---+ Y is closed. 2. Iff is closed, y a point off ( X ) , and U a n open neighborhood of the fiber f - 1 (y) , then there is a n open neighborhood W = W(y) C Y such that f - 1 ( W ) c U. 3. Let X and Y be additionally locally compact. (a) If every point y E Y has a neighborhood W such that f- 1 (W) ts compact, then f is proper. (b) Iff is proper, then f is closed. ( c ) Iff is closed and every fiber f - 1 (y) compact, then f is proper. 1.

PROOF:

(1) is trivial.

6. Branched Riemann Domains

227

(2) Let f be closed and U be an open neighborhood of some nonempty fiber f - 1 (y) . Then W := Y - f ( X - U) is an open subset of Y. If y were not in W , then there would be a point x E f - 1 (y) - U. This is impossible, and therefore W is an open neighborhood of y with f - 1 (W) C U. (3a) Let Y be locally compact and K c Y compact. It follows from the criterion that there are finitely many sets W1 , . . . , Wm such that f- 1 (K) is contained in the compact set f - 1 (WI ) U · · · U f - 1 (Wm) · Since f - 1 (K) is closed, it must itself be compact.

(3b) Let f be proper and A c X be a closed set. For every y E j(A) there is a sequence of points xv E A with lim, f(xv) = Y · The set N : = { y } U {f(xv) : v E is compact (using the local compactness of Y) , and consequently f- 1 (N) is compact. Thus there is a subsequence (xvf.t) converging to some point X E f - 1 (N) n A . It follows that y = f(x) E f(A) .

N}

(3c) Let f be closed and assume that every fiber is compact. For y E Y choose a compact neighborhood U of f- 1 (y) . This is possible, since X is locally compact. By ( 2) there exists an open neighborhood W of y in Y with f - 1 (W) c U. Then f - 1 (W) c U as well. As a closed subset of a compact set it is itself compact, and from (3a) it follows that f is proper.

•

Definition . A continuous map f : X ---+ Y between locally compact Hausdorff spaces is called finite if it is closed and if each fiber has only finitely many elements. Obviously, every finite map is proper. Conversely, if a proper map has only discrete fibers, then it must be finite. Definition . A holomorphic map 1r : X ---+ Y between n-dimensional complex manifolds is called a branched (analytzc) covenng ifthe following hold: 1 . The map 1r is open, finite, and surjective. 2 . There is a closed subset D c Y with the following properties: (a) For every y E Y there is an open neighborhood U = U(y) c Y and a nowhere dense analytic subset A c U with D n U c A. (b) 1r : X - 1r - 1 (D) ---+ Y - D is locally biholomorphic. The set D is called the critical locus. A point x E X is called a branch poznt if 1r is not locally biholomorphic at x. The set B of branch points is called the branch locus . The covering is called unbranched if B is empty. The branch locus B is nowhere dense in X . Now we consider a domain G c en and d pseudopolynomials w; (wi , z ; over G and define

228

IV.

Complex Manifolds A == {(w, z) E red X G

Let

7f :

:

w; (w;, z) = 0 for i = l , _ . , d} .

A ---+ G be the restriction of the projection

6 . 2 Pr oposition.

prz

: red X G ---+ G.

The map 1r is surjective, finite, and open.

PRooF:

( 1 ) The surjectivity follows from the fact that, for fixed z E G, each w; (w; , z) has at least one zero. ( 2) Let M c A be a closed subset and z0 a point of 1r( M) . Then there is a sequence (wv, zv ) in M with z, ---+ z0 . Since the coefficients of the wi are bounded near z0 , the components of the zeros w, are also bounded. Consequently, there is a subsequence (wv M ) of (wv) that converges in red to some w0 . It is clear that (w0 , zo) lies in M . Thu s zo belongs to 1r ( M ) and 1r is closed. ,

Since 1r has finite fibers, it is a finite (and in particular a proper) map. (3) It remains to show that 1r is open. Let (w0 , z0 ) E A be an arbitrary point. If V(w0) c red and W(z0 ) c G are open neighborhoods, we have to find an open neighborhood W'(zo) c 1r(A n ( V x W) ) . �

By Hensel 's lemma there are pseudopolynomials w:; and w:;* such that

Define

A : = {(w , z) E V x W : w; (w 1 , z) = · · · = w'd (wd , z) = 0} . Then A c A n (V x W ) ,and (7riA) -1 (z 0 ) = {(wo, z0 ) } . Since 1riA is a closed map, there is a neighborhood W'(z 0 ) c W such that A n (Cd x W ' ) = (n iA) - 1 (W ') c A n (V x W) . Since also KIA : A ---+ W is surjective, it follows that W' = 7r(A n (cd x W') ) c 1r (A n (V x W)) . • Thus 1r is open. Remark. If X c A is an irreducible component that is everywhere regular of dimension n, then the same proof shows that 1r l x i�also surjective, finite, and open. If D c G is the union discriminant set for A , then 1r : X ---+ G is a branched analytic covering with critical locus D.

Br anched Dom ain s. A continuous map f : X ---+ Y between Hausdorff spaces is called discrete at x E X if the fiber f - 1 (f(x) ) is a discrete subset of X . The map f is called discrete if it is discrete at every point x E X . A holomorphic map f : X ---+ Y between complex manifolds is discrete at x0 if and only if there is an open neighborhood U = U ( xo ) c c X with

f(xo) � f(8 U) .

6. Branched Riemann Domains

229

Now let M be an n-dimensional complex manifold. Definition . A branched domain over M is a pair (X, 1r) with the fol lowing properties: 1 . X is an n-dimensional connected complex manifold. 2 . 1r : X ---+ M is a discrete open holomorphic mapping. Let 1r : X ---+ M be a branched domain and x0 E X an arbitrary point. We can choose a coordinate neighborhood B of 1r (x0 ) in M such that B is biholomorphically equivalent to a ball in en and 7r ( Xo ) = 0 . Since 7r is discrete and open, there is a connected open neighborhood U = U(x0 ) cc X such that 1r(U) c B is open and Jr - 1 (0) n U = {x0 } . We assume that 1r(U) = B . Define j : U ---+ U x B by j(x) : = (x, 1r (x)). Then j is a holomorphic embed ding, and A : = j ( U) = {(x , z ) E U x B : z = Jr(x) }

is a regular analytic set in U x B . We have a factorization 1r = 7r o j , with 7r := pr2 IA · Let w1 , . . . , w, be complex coordinates near x0 in U. It follows from the results of Section 111.6that over a small neighborhood of 0 in B there are pseudopolynomials wi (wi , z ) such that A is an irreducible component of the embedded analytic set �

A = { (w , z ) : wi (wi , z ) = 0 for i = 1 , . . .

, n}.

�

If D c B is the union of the discriminant sets of the wi , then A (and therefore also A ) is unbranched over the complement of D. Therefore, if 1r : X ---+ M is a branched domain, then for every x E X there is a neighborhood U ( x) c X and a neighborhood W(f(x) ) c M with 1r(U) = W such that 1r l u : U ---+ W is a branched analytic covering. If M = en or M = Jpm , then a branched domain over M is also called a

branched R ie mann domain.

Definition . Two branched domains (XI,1rl ) (over M ) and (X2, 1r2 ) (over N ) are called equivalent if there are holomorphic maps rp : M ---+ N and ip : X 1 ---+ X2 such that the following diagram commutes: 7r l

x1 -1 M

�

N

Tor sion P oints . As an example we consider a polydisk P = pn (o, r ) c en ' a positive integer b, and the set

Xo : = { (z, w ) E P

x

e : wb - z 1 = 0 } ,

230

N.

Complex Manifolds

together with the projection 1l'o : (z , w ) H z. Then X0 is a regular analytic hypersurface in P x C and consequently an n-dimensional complex manifold, which is the Cartesian product of a polydisk P' c c n -1 and a neighborhood of the origin in the Riemann surface of w = ifZ. It is unbranched outside {z : z 1 = 0) . Definit ion . Let ( X ,1l' ) be a branched domain over some manifold M . A point x0 E X is called a torsion point (or winding p o int) of order b if there is an open neighborhood U = U(xo) c X such that 1l' : U ---+ 1l'(U) is equivalent to 1l'o : Xo ---+ P. In this case we say that 1!'1 u is a winding covenng. We consider the local case, i.e., a branched analytic covering 1l' : X ---+ Q over some polydisk Q c en around the origin, and we assume that it is branched of order b over 0 and unbranched outside { z E Q : z 1 = 0). Then X ' : = X-1!'-1 ( { z1 = 0}) is still connected, and the covering 1l' : X ---+ Q is equivalent to 1l'o : Xo ---+ P. For a proof see [GrRe58] , §2.5, Hilfssatz 2. One shows that the two coverings outside the branching locus have the same fundamental group. Therefore, they must be equivalent there. Using the theorem of Remmert Stein it follows that X and X0 are equivalent. The following is now immediate. 6 . 3 Pr op osition. Let (X, 1l') be a branched analytic covering over M , with b ranching locus D . If x0 E X is a point such that z0 = 1l'(xo) is a regular point of D , th en the re are neigborhoods W = W(xo) C X and U = U (z0) C M such that 1l' : W ---+ U is a winding covering. The proof follows simply from the fact that one can find coordinates such that locally the situation above is at hand, and then X is locally equivalent to Xo .

concrete Riemann surface is a branched Riemann domain 1l' : X ---+ C or 1l' : X ---+ lP'1 . It is an abstract connected Riemann surface, and in the latter case it may be compact. The infinitely sheeted Riemann surface of w = logz is not a concrete Riemann surface in our sense. Concrete Riem ann Sur faces. A

Here is a method for constructing concrete Riemann surfaces. Let X be a Hausdorff space, M = C or M = lP' 1 , and 1l' : X ---+ M a continuous mapping such that for every x0 E X there is an open neighborhood U = U(x0) C X , a domain V c C (in the case M = C ) , respectively a domain V in an affine part rv C of M (in the case M = lP' 1 ) and a topological map 'l/J : V ---+ U with the following properties: 1 . 1l' 'lj; : V ---+ C is holomorphic. 2. (1!' o 'lj;)' does not vanish identically. o

6. Branched Riemann Domains

23 1

The map '1/J is called a local uniformization. Obviously, (U, 'I/J - 1 ) is a complex coordinate system. We have to show that two such systems ( U1 , '1/!1 1 ) and (U2, '1/!2 1 ) are holomorphically compatible. We consider only the case M = C . The map

'1/!1 1 o 1/!z : 1/J2 1 (U1 n Uz) ---+ '1/J1 1 (U1 n Uz) is a homeomorphism. We denote the set W>-. 1 (U1 n Uz) by v; . Then D 1 := { t E Vt : (7r o '1/J I )' (t) = 0} is discrete in Vt , and therefore Dz := '1/J - 1 (D I ) C V2* is also discrete. Let t0 be a point of V2* - D 2 . Then '1/!

:=

(7r o '1/J I ) ' ( '1/J( t0 )) f- 0. Therefore, there are open neighborhoods U = U('lj!(t0 )) c Vt and W = W(z0 ) c C (for z 0 : = 7r('I/J2(t 0 ))) such that 7r o '1/J 1 : U ---+ W is biholomorphic. Then 7r l ,p, ( U) : '1/! 1 (U) ---+ W is a homeomorphism, and

'1/J = 'l,61 1 o (7r l ,p, (U) ) - 1 o 7r o 'I/Jz

=

(n Oij;l ) - 1 o (7r o '1/Jz) , on

V

:=

'1/J - 1 (U) .

Since V is an open neighborhood of to, we see that '1/J is holomorphic on V2* -D2. As a continuous map it is bounded at the points of D2 , and therefore it must be holomorphic everywhere in V2* . From the above it follows that X is a Riemann surface, and obviously 7r is a holomorphic map. We still want to see that it is a finite branched covering. If x 0 E X , then there is an open neighborhood U = U(xo) C X and a local uniformization '1/J : V ---+ U. We define : U ---+ V x 7r(U) C C 2 by

(x) := ('1/J - 1 (x) , 7r(x)). Then (U) = {(z, w ) E Vx 7r(U) : 7r o 'lj!(z) = w }. Restricting the projection (z, w) H w, we obtain a branched covering p : (U) ---+ 7r(U) that is equiva lent to 7r l u : u ---+ 7r(U). Let be f := 11' 0 1/J and Zo == w -1 (xo ) . If f'(zo) i- 0, then p is unbranched. If f' has an isolated zero of order k at z 0 , then

f(z)

=

7r(x0 ) + (z

_

z0) k · h(z),

where h is a holomorphic function with h(zo) f- 0. It is clear that then p is branched of order k, equivalent to the Riemann surface of ifZ.

Hyperelliptic Riem ann Sur faces .For g > 2 choose 2g + 2 different points z 1 , zz, . . . , zz9+I , z2g+ 2 in Xo := 1P' 1 . We assume that they are all real, in natural order, and not equal to oo .

Now we take two copies X1 , X2 of lP'1 regarded as lying above Xo and cut them along the lines £i between z2,+ 1 and z2i + 2 , i = 0 , . . . , g . We define

X.\ := {f E X>-. : lm(t) < 0 }

and

Xt := {t E X>-.

:

Im(t) > O}

232

IV. Complex Manifolds x2+

-

-

-

-<

x1+

Figure I V . 2. The hyperelliptic surface and glue xt to X ! along the cuts

£i,

and similarly X2 to X{.

We obtain a topological (Hausdorff) space X that lies as a branched cover ing of order 2 over lP'1 (see Figure IV. 2) . It is branched only at the points z1 , . . . , z2 +2 , and at each of these points it looks exactly like the Riemann 9 surface of Fz over the origin. Therefore, it is clear that X , together with the canonical projection 1r : X ---+ P', is a concrete Riemann surface. It is called a hyperelliptic surface. 4 A curve going in X1 from Z 2i +l to zzH 2 and then in X2 back to zzi + I defines a cycle c; in X , i = 0 , . . . ,g. Similarly a curve in Xt which goes from Z2i +2 to Z2 i + 3 and then in Xz back to Z2i +2 gives a cycle C:' in X , i = 0, . . ' g - 1 . Finally we define c� by going in Xt from Z2g + 2 through oo to zt and then in Xz back to Z 2g +2· So H1 (X, Z) is generated by the 2g + 2 cycles Cb , . . . , C� , Cg , . . . , C� . But C' : = Cb + · . . +C� is the boundary of Xz, and C" : = Cg + . . . + C� is the boundary of X{ U X2 . Therefore, H1 (X, Z) = Z 1 (X, Z)/B1 (X, Z) rv Z 29 �and Ho (X, Z) = Z, because X is connected) . It follows that H1 (X, Z) rv Z 9 . The number .

1

1 1 g = g (X) : = � b1 (X) = 2 rk (H (X, Z)) 1 is the Riemann sphere, we replace is called the genus of X . Recalling that lP' Xz by the complex conjugate Xz and glue X1 and Xz inserting tubes into the cuts £i · Thu s we realize X as a sphere with g handles.

Now we give a more analytical description of the hyperelliptic surface. Define X ' := 4 In the case

g =

{(s,t)E C2

1 we would get an

dimensional complex torus.

2g+ 2

2 : s = IT (t i=O

- zi) } ,

elliptic surface that is isomorphic to a !

233

6. Branched Riemann Domains

and 1r1 : X ' ---+ C by 1r ( s , t ) := t. This is a branched covering with branching locus D = {z 1 , . . . , zzg + z}. For F(t) := f179 ;i 2 (t- zi) , the function s = -/F(t) is well defined on X ' .Over { t E C : It I > R } , X ' consists of two disjoint punctured disks. Therefore, we can complete X ' to a compact Riemann sur face X by filling the points at infinity, i.e., the punctures. The map 1r' can be extended continuously to a map 1r : X ---+ 1P' 1 . It follows by the Riemann extension theorem that 1r is holomorphic, and X is a branched covering of order 2 over lP' 1 with branching locus D. Remar k. The Riemann surface X is not the completion of X ' in lP'2 . The projective analytic set X', which has {s 2 = F(t)} as affine part, has a singularity at infinity.

The functions t and s = \.1 P(t) extend to meromorphic functions f and X that have poles of order !, respectively g + l, over oo .

g

on

We now calculate H 1 (X, 0). For this the following is an essential tool.

6 .4 Serre' s duality theor em. If X is a compact R iemann surface and V an analytic vector bundle over X , then H 1 (X, V ) and H 0 (X, Kx Q9 V ' ) are

fin ite-dimensional vector spaces of equal dimension. PROoF:

See [Nar92] , page 47.

In the case V = Ox = X x C we have H 1 (X, 0) rv some remarks about the canonical bundle Kx .

H0 (X, Kx ) .

•

So we need

1 . Let U = (Uc)cEI be an open covering of X such that there are complex coordinates r.p" : U" ---+ C. We denote the complex coordinate on U" by t, . Then the line bundle Kx is given by the transition functions

9cl< =

dtl< . dt"

.

A holomorphic 1 -.form on X is a global section w E f(X, T (X)' ) = H0 (X, Kx ) We can write wlu, = w"dt" . Then w"dt" = w"'dt "' on U""' ' and therefore w, = g""' . w, . 2. On lP' 1 we have two systems of complex coordinates, namely, t0 := z i / z0 on Uo = {(zo : z l ) : Zo i- 0} and h := zo /z l on ul . We denote to by t and -h by s . Then on Uo 1 we have s = -ljt. Using s and t as complex coordinates, the canonical bundle on JID1 is defined by

90 1 =

�� = :l ( ) t� = c�) -!

=

2

.

This is also the transition function of the bundle [D] associated to the divisor D = - 2 . oo. Therefore, o o l rv H H (lP' ' Kll" ) (lP' l ' [D] ) { m E M( lP'1 )* : ord00 ( m) > 2 and ordp (m) > 0 otherwise } .

234

IV. Complex Manifolds

But such a function m is holomorphic and vanishes at oo , and conse quently is identically 0. Thus H 0 ( 1P' I , Kll'' ) = 0. 3 . Let X be a compact Riemann surface, and W I , wz E H 0 (X, Kx) two holomorphic 1-forms. In local coordinates w.\ l u, = wf l dt for ,\ = 1 , 2. Then f l u, := wi2l jwi ' l defines a global meromorphic function f on X such that wz = f . WI · '

"

Now we return to the hyperelliptic surface X . A holomorphic map j : X ---+ X is defined by (s,t )H (-s, t ) . Clearly, j permutes the two leaves, and j 2 = id. This map is called the hyperelliptic involution, and it induces a linear map j* : H0 (X, Kx ) ---+ H0 (X, Kx ) with (j* ) 2 = id. We get a decomposition of H0 (X, Kx ) into the eigenspaces associated with the eigenvalues ± 1 . But if j*w = w , then w = 7!' * ( cp) , where 'P is a holomorphic 1-form on lP'I . Since H0 (1P'I , Kll'' ) = 0, it follows that w = 0. Thus j*w = -w for every w E

H0 (X, Kx ).

The equation s 2 = F(t) implies that 2s ds = F' (t) dt. If s (t) = 0, then t = zk for some k. But F' (zk ) = IT;# (zk - z; ) =/= 0. Therefore, 2ds wo := s = F' (t) dt

is a holomorphic 1-form on X ' . Near oo we have s ( t) � t9 + I and therefore w0 � ( 1 /t) 9 - I d( 1 /t) . This shows that w0 is a holomorphic 1-form on X that has a zero of order g - 1 over oo . The same argument shows that the 1-forms w, : = t11 ( dt / s) are holomorphic on X , for v = 1 .. . , g - 1 . ,

•

Now let w be an arbitrary element of H 0 (X, Kx ) . Then there is a meromor phic function f on X with w = f · w0 . Since w0 has no zero over lP' I - {oo } , f is holomorphic there . And since j *w = -w and j*w0 = -wo , we have j* f = f . This means that f = h o 1l' for some meromorphic function h on lP'I that is holomorphic outside oo . Consequently, h is a polynomial, say of degree d. It has d zeros in C and a pole of order d at oo . It follows that f also has a pole of order d at every point above oo . Since w f . w0 is holomorphic, d < g - 1 . Therefore, w is a linear combination of the 1-forms =

dt s

dt

ts

'

...

'

It follows that dime H0 (X, Kx ) = g. 6. 5

Pr op osition. If X is a hyperelliptic R i emann surface, then the genus g = g (X) is equal to dim H0 (X, Kx ) . We will later see that the same result is true for arbitrary compact Riemann surfaces.

7. Modifications and Toric Closures

235

Exer c ises : X ---+ M be a branched analytic covering with critical locus D c M . Show that if there is a point y0 E D such that D has codimension 2 at y0 , then there is no branch point in 1r- 1 ( y0 ). 2. Let 1r : X ---+ M , and 1r1 : X ' ---+ M , b e branched analytic coverings and D, respectively D ', the critical locus. Prove that if X - 1r- 1 (D) is equivalent to X ' - (1r' ) - 1 (D' ) , then X is equivalent to X ' . 3. Consider the holomorphic map F : C 2 ---+ C 3 defined by 1 . Let

1r

(w, z t , zz) = F (h , tz) = ( h · tz, t i , t � ) . It is the uniformization of a 2-dimensional analytic set A c C3 . Show that A - {0} is a branched domain over C 2 - {0}. What arc the torsion points? Describe A as the zero set of a pseudopolynomial w (w, z ) . Is A also a branched domain in our sense? 4. Let F be a homogeneous polynomial of degree k and

{ (zo : . . . Zn Zn + ) E lP'n +l Z/{ + 1 - F(zo , . . . , zn ) = 0 } . Assume that D = {(zo : . . . : zn ) E lP'n : F(zo , . . . , zn ) = 0 } is regular of codimension 1 and prove that the canonical projection 1r : X ---+ lP'n X

=

:

:

:

t

:

is a branched Riemann domain. Determine the torsion points and their orders. 5. Define rp : 1P' 1 ---+ lP'n by rp(z0 : zt ) := (z0 : Z(t - lz 1 : . . . : zf) and prove that the image C : = rp(lP' 1 ) is a regular complex curve in lP'n . Define

{(p, H) E C Gn , n +l P E H}, where an element H E Gn , n + I is identified with a projective hyperplane in P". Show that the projection 1r = pr, : X ---+ Gn , n + I is an n-sheeted X :=

x

:

branched covering. 6. Let X c lP'n be a d-dimensional connected complex submanifold. Prove that there is an ( n - d - 1 )-dimensional complex projective plane E c lP'n not intersecting X , a d-dimensional complex projective plane F c lP'n with E n F = 0 , and a canonical projection 1r : lP'n - E ---+ F such that 1r l x : X ---+ F is a branched domain.

7.

M odification s and Tor ic C losur es

Pr oper M odifications . Assume that X is an n-dimensional connected complex manifold and A C X a compact submanifold. Is it possible to cut

out A and replace it by another submanifold A' such that X ' = ( X - A ) u A' is again a complex manifold? In general, the answer is no, but sometimes such

complex surgery is possible. Then we call the new manifold X ' a modification of X .

236

IV. Complex Manifolds Definition . Let f : X ---+ Y be a proper surjective holomorphic map between two n-dimensional connected complex manifolds. The map f is called a (proper) modification of Y into X if there are nowhere dense analytic subsets E c X and S c Y such that the following hold: 1 . f(E) c S. 2. f maps X - E biholomorphically onto Y - S. 3 . Every fiber f- 1 (y) , y E S, consists of more than one point. The set S is called the cente r of the modification and E = f- 1 (8) the exceptional set.

Assume that we have a proper modification define X

f

: X ---+ Y with center S and

:= {(y, x) E Y x X : y = f(x)}

Then X is an n-dimensional connected closed submanifold of Y x X. If K c Y is compact, then 1r- 1 (K)� = ( K x X) nX is a closed subset of K x f- 1 ( K) and therefore compact; 1r : X - 1r- 1 (S) ---+ Y - S is biholomorphic, with inverse y H (y, f-1 (y) ) ; and for Yo E S we have 7r- 1 ( Yo) = {(yo , x) : f (x) = Yo} rv f-1 (y0) . This is a set with more than one element. Therefore, we can generalize the notion of a proper modification in the fol lowing way. Let X and Y be two connected complex manifolds. A generalized (proper) modification of Y in X with center S is given by an irreducible analytic subset � X of the Cartesian product Y x X and a nowhere dense analytic subset S C Y such that 1.

2. 3. 4.

.5 .

�

1r :=pr 1 lx : X ---+ Y is proper. 1r- 1 (S) is nowhere dense in X . f - 1r- 1 tS) is a complex manifold. � maps X - 1r-1 (S) biholomorphically onto Y - S . X has more than one point over each point of S.

The center S is sometimes also called the set of indeterminacy. If X is a manifold, then with center S .

7 . 1 Pr op osition.

1r : X ---+ X

1r- 1 (S)

is an ordinary modification of X into

X

�

has codimension 1 in X .

PRooF: We consider only the case where X is regular. A proof of the general case can be found in [GrRe.55] .

Assume that x 0 E 1r-1 (S) is isolated in the fiber of 1r over Yo�:= 7r(x0 ) . Then there is a neighborhood U of xo in Y x X such that U n X is a branched covering over Y. Since yo lies in S , there must be at least one additional

7. Modifications and Toric Closures

237

point x E X in the fiber over y0 . But then in any small neighborhood 1 of y0 there are points y E Y - S with several points in 1r-1 (y) . This is a contradiction. Therefore, dim, Jr-1 (1r(x 0 )) > 0. This is possible if and only if rkx0 (7r) < n :=dim(Y), and this shows that 1r- 1 (S) in local coordinates is given as the zero set of det In: . Therefore, it has codimension 1 . •

Blowing Up . Let U = U(O) be a small convex neighborhood around the origin in Cn + l . We want to replace the origin in U by an n-dimensional complex projective space. If 1r : cn + l - {0} ---+ Jpm is the canonical projection, then every line Cv through the origin determines an element x = 1r(v) in the projective space, and x determines the line £(x) = 1r- 1 (x) U (0) such that Cv = £(1r(v)). Now we insert ]pm in such a way that we reach the point x by approaching the origin along £(x). We define

X := {(w, x) E U X lP'n

:wE

£(x) }.

This is a so-called incidence set (another example is considered in Exercise 6.5) . We first show that it is an (n + I )-dimensional complex manifold. In fact. we have

( w,1r(z)) E X

z f- 0 , and 3 ,\ E C with w = Xz 3 i with zi f- 0 and Wj = '114- · Zj for j f- i z f- 0 and ZiWj - wizj

U x Jpm , with X n (U Uo ) rv {(w, t) E u X en : Wj

Zi

= 0 for all i, j.

So X is an analytic subset of X

= Wo tj for j = l , .. . , n }.

In u X ui (for i > 0) there is a similar representation. It follows that X is a submanifold of codimension n in the (2n + I )-dimensional manifold U x lP'n . The map q : = pr 1 l x : X ---+ U is holomorphic, mapping X - q - 1 (0) biholomorphically onto U - {0} by q : ( w ,x) H w and q- 1 : w H ( w ,1r(w) ) . Obviously, q is a proper map. The preimage q - 1 ( 0) is the exceptional set { ( 0, x) : 0 E £( x)} = { 0} x lP'n . So q : X ---+ U is a proper modification. It is called Hopf's 0'-process or the blowup of U at the origin. If w # 0 is a point of U and (>- n ) a sequence of nonzero complex numbers converging to 0, then q- 1 (>-n w) = (>- nw , 1r(w)) converges to (0, 1r (w)). This is the desired property.

The Tautological Bun dle . We consider the case U = manifold F := { (w, x) E c n+ l x lP'n : w E £(x)}

c n+ I ,

i.e., the

238

IV.

Complex Manifolds

and the second projection

Then p - 1 (x) = £(x) is always the complex line determined by x. The manifold F looks like a line bundle over the projective space. In fact, we have local trivializations 'Pi : p - 1 (Ui) ---+ ui X c defined by 'Pi(w, x) : = (x, wi)· Clearly, 'Pi is holomorphic. In fact, it is biholomorphic with inverse map

So over UiJ ,

Hence F has the transition functions giJ = zi/ ZJ , and consequently, it is the dual bundle of the hyperplane bundle 0 ( 1 ) . It is denoted by 0( - 1 ) and is called the tautologzcal bundle, because the fiber over x E pn is the line £(x), which is more or less the same as x. Sometimes F is also called the Hopf bundle, because it lies in c n + l X pn and the projection onto the first component is Hopfs a-process (see Figure IV.3).

j : F ---+ pn c n+ l Ui X c n+ l by

Let

X

be defined by

j(w, x) := (x, w ) , and Ji : Ui X C ---+

J,(1r(z) , c) := ( 1r(z), : . z ) .

Then we have the following commutative diagram :

F / u; jt

Ui X (: n +l

�

ui c x

-1 Ji

rv

Ui X (: n +l

This shows that F is a subbundle of the trivial vector bun< � p n X c n+ l. In particular, it follows that r(lP'n , 0 ( - 1 )) = 0; i. e . , there is no global holomor phic section in the tautological bundle. There is an interesting geometrical connection between F and F'. Consider the point x0 : = (0 : . . . : 0 : 1 ) E pn+l and the hyperplane

Ho : = {7r(z) E pn+l : Zn + l = 0}. Then the hyperplane bundle F' = 0 ( 1) is given by the Jpm+ l - {xo} ---+ Ho rv lPm, with 7r + ( ZQ : . . . : Zn : Zn+ 1 ) : = ( ZQ : . . . : Zn : 0).

projection

1r+

7.

Modifications and Toric Closures

�

.

Figure I V .3 .

.

239

£(x )

'

Hopf's a-process

Here H0 is the zero section of 0(1). Removing the zero section gives a man ifold isomorphic to @"+' - { 0}. and the fibers of the hyperplane bundle correspond to the lines through the origin. Blowing up the point x0 gives us an additional point to each fiber and so a projective bundle F' over Jpm with fiber lP' 1 . But looking in the other direction, F' - H0 is nothing but the tautological bundle 0( -1) over the exceptional set of the blowing up.

Quadratic Transformatio ns . We consider the case n = 1. Let M be a 2-dimensional connected complex manifold and p E M a point. Let U = U(p) c M be a small neigborhood with complex coordinates z, w such that (z(p), w(p)) = (0, 0) . Let X c U x lP' 1 be the blowup of U at the origin. Then

Qp (M) := ( M - U ) U X = ( M - {p ) ) U lP' 1

is again a 2-dimensional complex manifold, with a nonsingular compact an alytic subset N = 1P'1 and a proper holomorphic map qP : Qp(M) + M such that q; 1 (P ) = N and qP : Qp(M) - N ---+ M - { p } is biholomorphic. We call Qp(M) the quadratic tran5fo rmation of M at p. Let F : M1 ---+ M2 be a biholomorphic map between 2-dimensional complex manifolds and F(p l ) = P2 · Let q1 : Qp1 (M1 ) ---+ M1 and q2 : Qp2 (M2 ) ---+ M2

240

IV. Complex Manifolds

be the quadratic transformations. Then there exists a biholomorphic map F : Qp1 (Ml ) ---+ Qp2 (M2 ) such that q2 o F = F o q 1 . This follows directly from the construction, and it shows that the quadratic transformation is a canonical process. �

In [Ho55] Hopf proved that every proper modification with a single point as center is a finite sequence of quadratic transformations. 5 We consider the quadratic transformation U

Then Q0(U)

= { ( z , w ) E C2

q

:

Q0(U) ---+ U, where

: l z l < r 1 and

lwl

< r2 } .

1P' 1 : ZTJ - w( = 0) = Ua U U1 , with {(TJ , z) E C2 : l zrJI < r2 and l z l < r l }, {((, w) E C2 lw(l < r 1 and lwl < r2 }

= {(z, w )(

: rJ)) E U

x

:

and W

Thus

q

=

ZTJ

1

( = TJ

and

-

on Ua n

U1

is given by

and

((, w) H (w(, w). Consider, for example, the curve C := {(z, w) E U : w 2 = z3 }, which has (TJ, z) H (z, zrJ)

singularity at the origin. Then

q - 1 (C - {o}) n Ua q - 1 (C - {O}) n U1

{(TJ, z) E Ua {((, w ) E U1

a

: z # 0 and z - rJ2 = 0}, : w # 0 and l - w(3 = 0}.

In U1 - Ua we have ( = 0. So q - 1 (C - {0}) lies completely in Ua , and its closure in Q0 (U) is the singularity-free curve C' := {(TJ, z) : z = TJ 2 }. We call it the strict transform of C. The map q : C' ---+ C is called a resolution of the sing ularities of C. One can show that for any curve in a 2-dimensional manifold the resolution of the singularities is obtained by successive quadratic transformations. A proof can be found, for example, in [Lau71] . If a 2-dimensional manifold M contains a compact submanifold Y rv lP' 1 , one can ask whether Y is the exceptional set of a blowup process (then one says that Y can be "blown down" to a point) . Here we give a necessary condition.

We determine the transition functions for the normal bundle of the fiber Fa := q - 1 (0) in Q0 (U) . The divisor Fa is the zero set of the functions fa : Ua ---+ C with fa (rJ, z) : = z and h : U1 ---+ C with fl ((, w) := w . Therefore, the associated line bundle [Fa] on Qa is given by the transition function 5

In algebraic geometry this was already known

century, for example by the italiens

at the beginning of the twentieth G. Castelnuovo and F. Enriques.

7.

Modifications and Toric Closures

( z 9oi ( z, w, (( : TJ )) = -:-: = -:-:TJ

on

241

Uo n U1 .

But ( are the homogeneous coordinates on Fa 1P'1 . S o NQo (Fa ) = [Fo] I Fo is described by the same transition function the tautological bundle 0 ( -1) It follows that a submanifold Y lP'1 in a complex surface M can be excep tional only NM(Y ) 0(-1). We give a topological interpretation of this criterion. We set so := 1 on Uo and which defines a meromorphic section s in 0 ( - 1 ) that has no zero and only one pole at = (0 1).We can use this section to calculate the self intersection number Y . Y. Since more knowledge about topology is needed for the complete calculation, we can give only a sketch. The self-intersection n umber of a compact submanifold Y c M can be de scribed follows. Take a copy Y' of Y (which need be only a topological or piecewise differentiable submanifold of M ) such that it intersects Y tr n ver sally. Then the intersection Y n Y' is a zero-dimensional submanifold, i.e., a finite set. Now count the number of these intersection points respecting orien tation. The result is the self-intersection number. Furthermore, one can show that if Z c NM ( Y ) is the zero section, then Y . Y = Z Z, and the number Z Z can be calculated by intersecting Z with some not identically vanishing (continuous) section Z' . We obtain such a Z' in the following way. Let s be the meromorphic section in N = NM ( Y) mentioned above. Choosing appro priate coordinates near we can there write N = { (z, w) C2 : lzl < 1) and s ( z) = (z , 1 / z) (since s has a pole of order one). For 0 < E < 1 and lzl = E we have s ( z) = ( z,z/c2 ) . Then we define s by for lzl > E, s ( z) s(z) : = lzl < E. (z, 2: ) So s is a piecewise differentiable section, homologous and transversal to the zero section with one zero of order 1. Since gives an orientation opposite to that defined by z, we have the self-intersection Y . Y = - 1 . If y lP' 1 is exceptional, then y . y = -1. It is a deeper result that the converse of this statement is also true (cf. [Gr62]). M onoidal Tr ansfor mations. We consider a domain G c en and holomorphic functions fo , . . . , fk on G such that A := N(J0, . . . , fk) G is a singularity-free analytic subset of codimension k + 1. Then we define k+ f (fo, . . . , fk ) : G ---+ C 1 and ,

rv

as

rv

if

rv

oo

:

as

a s

·

·

oo,

€

for

z

rv

c

:=

242

IV.

Complex Manifolds

{(w, x) E G x lP'k : f(w) E £(x)} {(w, 7r ( z)) E G x lP'k : zdj (w ) - Zj ji(w)

X

= 0 for i , ]= 0, . . .

,k }.

The map p : = pr 1 l x : X --+ G is holomorphic and proper. For wo E G - A we have p -1 (wo) = { wo} x lP'k , whereas p : X -p- 1 ( A) --+ G- A is biholomorphic with p- 1 : w H (w, (fo(w) : . . . : fk (w))) . The set X is called the monoidal transformation of G with center A. The exceptional set E( f) : = p - 1 ( A) is the Cartesian product A x lP' k . In the case k = n - 1 and fi = zi for i = 0, . . . , k this is the a-process. One can show that the monoidal transformation is also a canonical process. Therefore, it can be generalized to an n-dimensional connected complex man ifold M and a submanifold A c M of codimension k + 1 . Then the monoidal transformation of M with aner--A is an n-dimensional manifold M with a hypersurface E c M such that M - E rv M - A. The exceptional set E is a fiber bundle over A with fiber lP'k . The fiber over x E A is the projective space lP'(Nx), where N = NM ( A) is the normal bundle of A in X . The divisor E determines a line bundle L : = [E] on M with L !JP'( Nx) = OIP'(Nx ) (- 1) . We leave the details to the reader (see also Exercise 5.10). -

Mer omor phic M ap s . Let X be an n-dimensional complex manifold and f a meromorphic function on X with polar set P and S c P its set of indeterminacy points. Then f is holomorphic on X - P , and we can extend it to a holomorphic map f : X - S --+ 1P'1 by setting f( x ) : = oo for x E P - S. The set Gt : = { (z,y) E (X - S ) x 1P' 1 : y = f(x) } is called the graph of X X lP' 1 •

f.

Let Gf c X

x 1P'1

be the closure of the graph in

7.2

Prop o sition. Th e graph Gt is an irreducible analytic subset of X x 1P'1 and defines a proper modification of X in 1P'1 with S as set of indeterminacy. PROOF: Let xo E P be an arbitrary point. Then there is a neighborhood U = U(x0 ) c X and holomorphic functions g , h : U --+ C such that f( x ) = g(x)jh(x) for x E U - P . We can assume that 9x and h, are relatively prime for x E U. Then P ll U = { Z E U : h(x) = 0 } and S n U = { x E U : g ( x ) = h(x) = 0} . Hence P is a hypersurface and S has codimension 2 in X . Then dim( S x lP'1 ) < n and Gf c (X x lP' 1) - ( S x lP' 1 ) is an n-dimensional analytic set. B y the theorem of Remmert and Stein its closure G f in X x 1P' 1 is also analytic. In fact, we have The map 1l' : = pr, : Gf --+ X is obviously a proper holomorphic map, with 7l' - 1 (x) {x} x JID1 for 3: E S and =

7. Modifications and Toric Closures

1!'- l (x) =

{ (X,(x, (( 1 : f1 (z) ) ) ) ) Q

243

for x E X - P, for X E p - S.

:

So 1l' : G1 - 1!'- 1 (S) ---+ X - S is biholomorphic, and G1 defines a proper modification of X in lP' 1 with indeterminacy set S. As usual, X is assumed to be connected. Then Reg ( G1) is also connected, and G1 irreducible. •

Now we consider the notion of a meromorphic map. To develop the correct point of view, we start with a very special case. Let

I(Jim

:=

JP' l

X . . . X m

JP'l

times

be the so-called Osgood space. If h , . . . ,fm are meromorphic functions on X without points of indeterminacy, then they define a holomorphic map f = ( !I , . . . , fm ) : X ---+ I(Jim . 1f fi has a set Si as set of indeterminacy, we define S := 81 U . . U Sm . Then f is a holomorphic map from X - S into I(Jim . Again we have its graph: ·

Gr = { (x ; yl , · . . , ym ) E ( X - S ) x I(Jim : Yi = fi(x) for i = l , . . , m}. For x0 E S we define Tr(xo) := {y E I(Jim : 3 xv E X - S with x, ---+ x0 and f(xv) ---+ y } . Then Tr(x0 ) is a nonempty compact subset of I(Jim . Setting Tr(x) := {f(x) } for x E X - S , we get a map Tf from X into the power set P ( I(Jim) .6 We define Gr { (x, y) E X i(Jim : y E Tr(x)}. One sees easily that Gr is the topological closure of Gr in X x I(Jim . We cannot apply the Remmert-Stein theorem to show that Gr is an analytic set, but .

:=

x

locally it is an irreducible component of the analytic set A with

{(x ; (z6 : z i ), . . . , (z[J" : z;:")) E U x i(Jim : z(tg 1 (x) - z�h 1 (x) . . zlJ"gm (x) - zfhm (x) 0 } , where fi = gi/ hi on U - Pi, for i = 1 , . , m. We could carry out the simple =

·

=

=

. .

proof here, but we refer to [Re57] , Satz 33.

Definition. A meromorphic map between complex manifolds X and Y is given by a nowhere dense analytic subset S c X , a holomorphic map f : X - S ---+ Y, and an irreducible analytic set X c X x Y such that

f o pr 1 = pr2

6

�

on X n ((X - S) x Y)

and 1l' : = pr 1 1 x : X ---+ X is proper and surjective. The set S is called the set cf indeterminacy of f. We also write f : X ---' Y. A s usual, the

p o w e r set

P(M) of a set M is the set

of all subsets of M .

IV. Complex Manifolds

244

We define TJ : X --+ P(Y) by nonempty compact set, and

T1 (x) := pr 2 (n- 1 (x)) .

Then

TJ

(x)

is always a

X = {(x, y) E X x Y : y E TJ (x) } . For x E X - S we have n- 1 (x) = {(x, f(x))}, and for x E S we have n - 1 (x) = X n ( { x} x Y) .This is a compact analytic set. If S is minimal, then n : X ---+ X is a proper modification with center S. �

�

So alternatively we could have defined a meromorphic map between X and Y to be a map T : X --+ P(Y) such that:

1. 2.

X := { (x, y) X x Y.

n :=pr 1 lx

E X

x

Y :

y

E

T

(x) }

is an irreducible analytic subset of

�

: X ---+ X is a proper modification.

This definition is due to Remmert. A meromorphic map is called surjective if pr 2 (X) = Y . If f is a surjective meromorphic map between X and Y and g a meromorphic map between Y and Z, then it is possible to define the composition g o f as a meromorphic map between X and Z (cf. [Ku60] ) . But it is not a composition of maps in the set-theoretic sense ! Any meromorphic function f on X defines a meromorphic map between X and 1P' 1 , any m- t uple ( h , . . , fm ) of meromorphic functions a meromorphic map between X and I(Jim . .

7.3 Proposition. S. Then f - 1 : Y indeterminacy. PROOF:

Let f : X -+ Y re a proper modification of Y with center X is a surjective meromorphic map with S as set cl

--"

We use the analytic set y

= f(x)} = { (y, x) E Y X : x E r 1 (y)} . Then f - 1 : Y - S --+ X is holomorphic with f - 1 o pr 1 (y, x) = j- 1 (y) = x = pr 2 (y, x) for (y, x) E Y n ( ( Y- S ) x X ) . Obviously, n := pr 1 1¥ : Y ---+ Y is a proper holomorphic map. It is surjective, since f is surjective. Finally, we • have pr2 (Y) = X . So f- 1 defines a surjective meromorphic map. �

Y : = {(y, x) E Y X

�

X

:

X

�

Toric Clo sures. Let X be an n-dimensional complex manifold. A clo sure of X is an n-dimensional compact complex manifold M such that X is biholomorphically equivalent to an open subset of M . We are interested in closures of X = e n . In the case n = 1 the situation is simple. Assume that we have a compact Riemann surface M that is a closure of X = C. Then we define f : M - {0} ---+

c by

7. f(z) : =

{

Modifications and Toric

Closures

245

for z E C - { 0}, otherwise.

1/z 0

The function f is continuous and is holomorphic at every point z with f(z) f0 . By the theorem of Rad6-Behnke-Stein-Cartan (cf. [Hei56] ) f is everywhere holomorphic. But then the zero set N(f) is discrete. We can consider f as a meromorphic function on M with exactly one pole of order one at z = 0. So f can have only one zero (cf. [Fo81 ] , Section 1 .4, Proposition 4.24 and corollary), and M must be the Riemann sphere lP' 1 , the well-known one-point compactification of C. In higher dimensions one can find several closures of e n . In [Bie 33] an injective holomorphic map (3 : e2 ---+ e2 is constructed whose functional determinant equals 1 everywhere and whose image U = (3 (e 2) has the property that there exist interior points in e 2 - U. We can regard U as an open subset of lP'2 . Then lP'2 is a closure of e 2 rv U, but this closure contains interior points in its boundary. We want to avoid such "pathological" situations, even if they may be interesting for those working in dynamical systems. Definition. A closure of e n is a triple (X,U, ) where: 1 . X is an n-dimensional connected compact complex manifold. 2. U c X is an open subset with U = X . 3. : e n --+ u is a biholomorphic map. The closure (X,U, ) is called regular if for any polynomial p the holo morphic function p o - 1 extends to a meromorphic function on X .

7.4 Proposition. If (X, U, ) is a regular closure ct en , then Xoo : = X - U is an analytic hypersurface in X . PROOF: Let fi be the meromorphic extension of Zi o - 1 , for i = 1 , . . . , n . The polar set Si of f; is a hypersurface in X . We set P : = P1 U . . . U Pn and claim that P = X - U. One inclusion being trivial, we consider a point x0 E X - U. There is a sequence of points xv E U with limv oo x, = xa . If we write 4 := - 1 (xv ) as 4 = (z iv1 , . . . , zi;'l ) , there must be an index k with l zk"l l ---+ oo . So l fk (xv) l ---+ oo for v + oo . This means that x0 E Pk C P. • --+

--

Example The manifold X = lP'n is a closure of e n . As open subset U with U = X we can take the set U : = Uo = {(zo : . . . : zn ) I zo f- 0}. Then 0 : en --+ U with o(h , . . . , tn ) : = ( 1 : h : . . . : t n ) is biholomorphic. We have seen in Section 5 that any polynomial on U extends to a meromorphic function f on Jpm with

246

IV.

Complex Manifolds

polar set H0 = { (z0 : . . . : zn ) I zo = 0 }. So (lPm , Uo , o) is a regular closure of e n , with Xoo = Ha . Another example is the Osgood space I(Jin , which contains e n via (a :

(h , . . . , t n ) H ( ( 1 : h ) , . . . , ( 1 : tn ) ) .

Also, (I(Jin , e n , (a) is a regular closure, but here

� = ({ m } x lP'1 x . . . x lP' 1 ) U · · . U (lP' 1 x . . . x lP' 1 x {m}) is an analytic hypersurface with singularities. The complex Lie group Gn @* X . . . X e* (n times) acts in a natural way on e n and is contained in e n as an open orbit of this action. Definition . A complex manifold X is called a toric variety or to rus embedding if G n acts holomorphically on X and there is an open dense subset U c X biholomorphically equivalent to Gn by g H gx for each x E U. Remark. In algebraic geometry the group G n is called a torus. This is the reason for the notion "toric variety." A "variety" is an irreducible algebraic subset of en or lP'n . It may have singularities. Here we do not allow singular ities, but we allow arbitrary manifolds that do not have to be algebraic. More information about toric varieties can be found in [Oda88] , [Fu93] , and [Ew96] . We wish to restrict the closures of en as much as possible. If the axes of en can be extended to the closure, then the action of Gn should also extend to the closure. Definition . A closure ( X ,U, (a) of en is called a to ric closure if the action of G n on e n extends to an action on X such that (g · z) = g · ( z ) for g E Gn and z E en . Examples 1 . The projective space Jpm is a toric variety with g ( zo : . . . : Zn ) := ( zo : gz 1 : . . . : gzn ) ·

and G = (e* )n c en = U0 c lP'n. Since the embedding en = U0 '---+ lP'n n is "equivariant," lP'n is a toric closure of en . 2 . A further simple example is the Osgood space I(Jin. Here the to ric action is given by 1 . . ) ) .. = ( (z01 .. gz 1 ) , . . . , (zon .. gzn ) ) . g . ( ( z0 . z 11 ) , . . . , ( z0n . z n 1 1 1

7.

Modifications and Toric Closures

247

3 . Let Ad c N0 be the set of multi-indices 11 with !11! :::; d and define d : Gn ---+ lP'N by d(z) := (z" : 11 E A d ) . Then Xd := d(Gn ) C lP'N is the image of lP'n under the Veronese map vd. n with

(

)

Vd , n (zo : z 1 : . . . : zn ) = zg - lv l z v : 11 E Ad . Obviously, Xd is a toric closure of C'

7.5 Prop osition. Let ( X ,U, (a) be a regular closure oj en and f : Z ---+ X a proper modification with center S c X,. Th en (Z, f- 1 (U) , f - 1 o (a) is again a regular closure. PROOF: It is clear that (Z, f -1 (U), f - 1 (a) is a closure. Now let p be a polynomial. Then the holomorphic function p o - 1 : U ---+ e has a mero morphic extension fi on X . But then fi o f is a meromorphic extension of p u- 1 0 )- 1 = (p o - 1 ) 0 f. • o

0

Now let ( X ,U, (a) be an arbitrary regular toric closure of en. The meromor phic extensions fi of zi o- 1 define a meromorphic m!lpping f = (h , . . . , fn ) : X ----" on and therefore a generalized modification X of X in on . We know that X is an irreducible component of the analytic set �

A = { (x; z1 , ·

. .

, z,) E X X

where we write zi E fi = (!& : !1) .

lP' 1

o n : zb f: (x) - zUMx) = 0 for i = 1 , . . . , n } ,

in the form zi = ( zb

z� ) and fi in the form

X

The two projections 1r := pr 1 l x : X ---+ X and q : = pr 2 l x : ---+ 0" are proper holomorphic maps. It may be that X has singularities over the indeterminacy set S of f. Nevertheless, we consider X as a closure of en. We take fJ : = 1r-1 ( U) as dense open set, and the biholomorphic map (<; := Jr- 1 0 (a : en ---+ D . This is possible, since flu = - 1 is holomorphic and therefore S c X U. �

�

-

X-

fJ is a nowhere dense analytic subset, and The preimage 1r - 1 (X - U) = the meromorphic map f o 1r : X ----" on coincides on U with the holomorphic map q. By the identity theorem it follows that f o 1r = q is holomorphic everywhere on Since (fo 7r) lu = - 1 o 1r = - 1 , every polynomial extends to a meromorphic function (without points of indeterminacy) on Jt . So the new closure X is regular. Since q l u = - 1 maps fJ biholomorphically onto en , the map q : x ---+ on is a proper modification of on (in a generalized sense, since X may have singularities) . �

�

X.

�

Let us now consider the q-fibers. If Zo E on is given, the fiber q- 1 ( Zo) contained in the set

is

248

IV. Complex Manifolds

F : = (fl u ) -1 (zo ) U {x E X -

U

: 3 z v E en with (zv ) --+ x,

z,

--+ zo } ·

For Zo E en we obtain q - 1 (zo ) = { (zo ) } , whereas for Zo E on - en the fiber is a compact subset of the hypersurface X - U. It is easy to check that X is even a toric closure. One uses the fact that X and on both are toric and that flu is
Let X = ]pm be projective space and carry out the above construction. The functions fi : Jpm 1P' 1 are given by --'

fi : ( Zo : Z1 : . . . : Z,) H ( Zo : Zi ). These are meromorphic functions with polar set H0 and indeterminacy set H0 n Hi . In the inhomogeneous coordinates wo , . . . , wj , . . . , Wn with wk = zk/ zJ the function fi is equal to wi/wo for i f. j (respectively to 1/wo for i =

j) .

Now

�

X is given by

X

=

{ (w; Zl ' . . . Zn ) E lP'n X on : zbwi - Z�Wo = 0 for i = 1' . . . ' n } '

This is built from lP'n by a finite sequence of quadratic transformations, and therefore regular. The space X is a regular toric closure of en in the original sense. �

There are many toric closures, but lP'n and on are the most important ones. More information about this topic is given in [BrMo78] and [PSSS]. Exercises

1 . Calculate the strict transform of X = {(z, w) : w2 = z 2 (z + 1 ) } when blowing up e2 at the origin. 2 . Show that KJPn rv 0( - n - 1 ) and Ky 0 ( d - n - 1 ) for every regular hypersurface Y c lP'n of degree d. 3. Let L: 1 be the projective bundle associated to the vector bundle V = OJP' EB OJP' (1) (cf. Exercise 5 . 1 0) . Show that the bundle space of L: 1 is biholomorphically equivalent to the blowup Qp (1P'2) with p = ( 1 : 0 : 0). 4. Consider the meromorphic function f ( z , w) := w / z on e2 and show that its graph G f c e2 x 1P' 1 is equal to the blowup of e2 at the origin. 5. Let X be a complex manifold and L a holomorphic line bundle over X . Show that any set of linearly independent global sections so, . . . , sN E N f (X, L) defines a meromorphic map : X --+ lP' by rv

(x) := (so (x) : . . . : sN (x)) .

What is the exact meaning of this definition?

7.

Modifications and Toric

Closures

249

6. Let X be a compact connected complex manifold. A set of meromorphic functions h , . . . , fm E .4(X) is called analytically dependent if f(Gr) f. I(Jim , where f (h , . . . , fm ) is the second projection Gr C X x I(Jim --+ I(Jim . Let P be the union of the polar sets of the fi and show that the following properties are equivalent: ( a) h , . . . , fm are analytically dependent. (b) The Jacobian o ff : X - P --+ em has rank less than m. (c) There is a nonempty open subset U c X - P, a domain G c em with f ( U) c G, and a holomorphic function g on G such that g( JI (x) , . . . , !m (x)) 0. The functions JI , . . . , fm are called algebraically dependent if there is a polynomial p f. 0 such that p(h (x ) , . . . , fm ( x)) 0 on X - P . Prove that every set of analytically dependent functions is also algebraically depen dent. (Hint: The map Gr --+ I(Jim is proper. Use without proof Remmert 's proper mapping theorem, which says that the image of an analytic set under a proper holomorphic mapping is also an analytic set. Then use Chow's theorem.) 7. Let : G-2 --+ lP'3 be defined by (z1 , z2 ) := ( 1 : z1 : z2 : z1z2 ) . Show that X (G-2) c lP'3 is a closed submanifold that coincides with the image 1 of the Seg re m ap (]1,1 : 1P' 1 x 1P' --+ lP'3 with (]1 , 1 ( (zo : z i ) , (w0 : w i ) ) 2 (zowo : Z1 Wo : ZQW1 : Z1 w! ) . Prove that X is a toric closure of C . =

=

C h ap ter V

Stein T h eor y 1.

Stei n M anifold s

Introduction Definition . A complex manifold X is called holomorph ically spread able if for any point xo € X there are holomorphic functions h , . . . , fN on X such that x0 is isolated in the set N(h , . . . , JN ) = {x E X : JI (x) = · · . = fN (x) = 0 } .

It is clear that the holomorphic map f : = ( JI , . . . , fN) : X ---+ eN is discrete at x0 . In [Gr55] it is shown that if n = dim(X), then there is a discrete holomorphic map 1r : X ---+ en . Thus (X,1r) is a branched domain over en . Furthermore, it follows that the topology of X has a countable base. Note that if X is holomorphically separable, i.e. , for any x , y E X with x f- y there exists a holomorphic function f on X with f(x) f- f(y) , then it is holomorphically spreadable. If X is holomorphically spreadable, A c X a compact analytic set, and x0 E A, then there are holomorphic functions h , . . . JN on X (which then must be constant on A ) such that x0 is isolated in N (h , . . . , JN ) . So x0 must be an isolated point of A, and it follows that every compact analytic subset of X is finite! Definition . A complex manifold X is called holomorphically convex if for any compact set K c X the holomorphically convex hull

K

:=

{x E X

:

l f (x) l < s�p l f l for every f E O (X)

}

is likewise a compact subset of X . Every compact complex manifold is holomorphically convex. A complex man ifold with a countable base is holomorphically convex if and only if for any infinite discrete subset D C X there exists a holomorphic function f on X such that sup D I f I = ()() . The proof is the same as in en . Definition . A Stein m anifold is a connected complex manifold that holomorphically spreadable and holomorphically convex.

is

252

V. Stein Theory

A Stein manifold of dimension n > 0 cannot be compact. In the converse direction, Behnke and Stein proved in 1948 that every noncompact Riemann surface is a Stein manifold (see [BeSt48] ). For n > 2, a domain G c en is Stein if and only if it is holomorphically convex. By the Cartan-Thullen theorem this is the case exactly if G is a domain of holomorphy. This remains true for (unbranched) Riemann domains over en , since Cartan- Thullen is also valid for such domains. Hence for n > 2 there are many noncompact domains in e n and over en that are not Stein. Let f : X ---+ Y be a fin ite holomorphic m ap between complex manifolds. If Y is a S tein manifold, then X is also S tein.

1 . 1 Pr op osition.

In particular, every closed submanifold of a S tein manifold is S te in.

PROOF: It is necessary to show only that X is holomorphically convex. For this let K c X be a compact set and note that f(K) is also compact. We consider an arbitrary point x E K. For g E O(Y) we have g o f E O (X) and l g o f (x) l < supK ig o f l , so f (x) E f(K) and x E f- 1 (f(K) ) . Since Y is holomorphically convex and f proper, f - 1 ( f(K) is compact as well. As a closed set in a compact set, K is compact. �

--

-

�

Since en is Stein, every closed submanifold of en is also Stein.

•

Example

Every affine-algebraic manifold is isomorphic to a closed submanifold of en . Therefore, it is a Stein manifold. On the other hand, we shall see later on that there exists an algebraic surface that is Stein, but not affine-algebraic. 1 . 2 Pr op osition.

If X is a S tein manifold and f

is S te in.

E

O(X),

then X

-

N(!)

PROOF: If X is a Stein manifold, then it is immediate that X N(!) is holomorphically spreadable. Let D be an infinite discrete set in X N(!) . If it is discrete in X , nothing remains to be proved. If it has a cluster point N (!) and unbounded on xo E N (!), then g : 1 / f is holomorphic on X D. • -

-

=

-

Fundamental Theor em s . The following important theorems cannot be proved completely within the constraints of this book. One needs deeper tools, including sheaf theory.

1 . 3 Theorem A. Let 1r : V ---+ X be an analytic vector bundle over a S tein manifold X . Th e n fo r any xo E X there are global holomorphic sections s , . . . ,sN E r (X, V ) w ith the following property: 1

1.

If U = U (xo) c X

Stein Manifolds

253

is an open neighborhood and s E f(U, V ) , then there exists a n open neighborhood V V(xo) C U and holomorphic functions JI , . . . o n V such that s l v = fi · + . . · + f ·

JN

=

N SN .

S1

1 .4 Theorem B . Let 1r : V --+ X be an analytic vector bundle over a S tein manifold X . Then H1 (X, V ) 0. =

Moreover, if A C X is a n analytic set, %' an open covering of X and � E Z 1 ( %' , V ) such that �v �t i Uv nA = 0, then canfind a cochain 'f) E C0 (%' , V ) with TJv l uvnA = 0 and Or] ( ,

we

=

The first part of this theorem will be proved at the end of this chapter. 1 . 5 Oka's principle. 1.

2.

Let X be a S tein manifold.

Every topologicalfiber bundle over X has a n analytic structure. If two analytic fiber bundles over X are topologically equivalent, they are also analytically equivalent.

distribution on C consists of Cousin- I Distr ibution s. A a discrete set { zv : 11 E N} in C together with principal part s h, of Laurent

Mittag-Leffler

series at each z, . A solution of this distribution is a meromorphic function on C with the given principal part s and no other pole. We can express the situation in the language of Cech cohomology. Define Uo = C - { zv : 11 E N} and ho := 0, and let Uv be an open neighborhood of zv that contains no zit with J-L f- 11. Then hv l uv is meromorphic, and h, - h, is holomorphic on Uvw A solution is a meromorphic function f such that f - h , is holomorphic on Uv . =

Definition. Let X be a complex manifold. A Cousin-] distribution {U< : L E I } together with on X consists of an open covering %' meromorphic functions f< on u< such that f< - !I< is holomorphic on u
A solution of the Cousin-! distribution is a meromorphic function f on X such that f - f< is holomorphic on U<. If a Cousin-I distribution is given, then a cocycle � E Z1 ( %' , 0) is defined by �<�< : = (!"' - f< ) I u, K . If the distribution has a solution f, then TJ< : ( !< - f) I u, defines a cochain TJ E C0 (%' , 0) with TJ"' - TJ< �<�< · In other words, OTJ � Conversely, if there exists an TJ with OTJ � ' then f lu, : = f< - TJ< defines a meromorphic function f on X with f< - f TJ< on U< . This means that f is a solution. =

=

=

=

=

The following result is a consequence of Theorem B for the trivial bundle Ox = X x C.

254

V.

Stein Theo ry

1.6 Propo sition.

On

a

Stein manifold every Cousin-! distribution has

a

solution.

In particular, on an open Riemann surface every Mittag-Leffler distribution has a solution. Remark. The condition "Stein" is not necessary. If X is a complex mani fold with H 1 (X, 0) = 0 (for example X = lP'1 ) , then every Cousin-! distribu tion has a solution. And even if H 1 (X, 0) f- 0, then the cohomology class of a given distribution may be zero. This implies that this special distribution has a solution.

Cousin-I I Distributions . Recall the Weierstrass theorem: Let { z., : v E N} be a discrete set in C , and ( n, ) a sequence of positive integers. Then there exists a holomorphic function f on C that has zeros exactly at the z,_, , and these are of orders nv· The distribution ( z., , nv ) vEN is a divisor D on C with D > 0 and div (f) = D . A divisor D on a complex manifold X is called a principal div isor if there is a meromorphic function m on X with div(m) = D. If D > 0, then m is holomorphic. So we are interested in conditions on X such that every divisor is principal. From the exact sequence .4(X)* � 5?(X) � Pic(X) it fol lows that Ker ( 6) = 51( X) is a necessary and sufficient condition. This is, for example, the case if every analytic line bundle on X is trivial. Due to Oka's principle, if X is Stein, it is sufficient that every topological line bundle be trivial. One can prove that the topological line bundles on X are classified by elements of H 2 (X, Z). Here we will show directly that the vanishing of H2 (X, Z) is sufficient for the solvability of the generalized Weierstrass prob lem. Definition. Let X be a complex manifold. A Cousin-1 1 distribution on X consists of an open covering 'P£ = { U, : L E I } together with meromorphic functions f, f- 0 on U, such that on U"" there are nowhere vanishing holomorphic functions g,"' with f, = g,"' . f"' . A solution of the Cousin-Il distribution is a meromorphic function f on X such that flu, = h-;: 1 . f, with a nowhere vanishing holomorphic function h, on U, . A Cousin-Il distribution is therefore a meromorphic section in an analytic line bundle L that is defined by the transition functions g,"'. If L is trivial, then there are nowhere vanishing holomorphic functions h, on U, with g,"' = h,h"' 1 , and fl u, : = h-;: 1 . f, defines a solution. The Cousin-Il distribution uniquely determines the cocycle Z 1 ('P£ , 0* ). If f is a solution with flu, = h-;: 1 . j, , then

'Y

=

(g,"')

E

1 . Stein Manifolds

255

The following shows that the converse also holds.

Let 'Y E Z 1 (%' , 0 * ) be the cocycle of a Cousin-11 distri distribution is solvable if and only if 'Y E B 1 (%' , 0 * ) .

1 .7 Propo sition.

bution.

The

If H1 ( X, 0 * ) = 0, every Cousin- II distribution on X has

a

solution.

Unfortunately, Theorem B cannot be applied to H1 (X, 0 * ) . So even for Stein manifolds we need an additional topological condition.

Chern Class and Exp onential Sequence . Let X be an arbitrary n-dimensional connected complex manifold. We choose an open covering %' = { U, : L E I } of X such that the U, and all intersections U"" are simply connected and construct an exact sequence of abelian groups:

H 1 (%', Z) � H 1 (%', 0 ) � H1 (%' , 0 * ) � H2 ( %', Z ) 1 . If � = (�u<) E Z1 (%' , Z), then also � E Z 1 ( %' , 0 ) . If � is an integer-valued coboundary, then it is also a holomorphic one. Thus j is well defined. 2. Let f = (!"" ) E Z 1 ( %', 0 ) be given. Then we define e([f] ) : = [exp(27T i fu, )].

Obviously, e is well defined, and e o j ( [W = [1] . On the other hand, if e([ JJ) = [1] , then there are nowhere vanishing holomorphic functions h, with exp (211 i fu, ) h,h " 1 . Since the U, arc simply connected, there are holomorphic functions g, with exp(g, ) = h, . Then exp(g, - g") exp(27T i f,") , and it follows that there are integers �'"' with =

This means that [!] = j ([W . 3. Here we define c : H 1 (%' , 0 * ) ---+ H2 ( %', Z). Let h = ( h,") E Z 1 (%', 0 * ) be given. Since the U," are simply connected, there are holomorphic functions g," on U," with exp(211 i g,") = h," . They are determined only up to some 211 i �' " with �'"' E Z, but TJvfN2 := (g/N2 - gv e + gvft ) l uv ,e

uniquely defines a cohomology class

and

[TJ]

E H2 (%' , Z ) , since

256

V. Stein Theory Let c( [h] ) : = [T)] . If [h] e ( [!] ) , then there are nowhere vanishing holomorphic functions h, with

where the g, are suitable holomorphic functions on U, . We have coe( [!] ) = [T7] with T/v1N2 = (!1N2 - fv12 + fvp J i uv , e = 0 , since f is a cocycle. Now let [h]Je given with c( [h] ) C1 ( %' , Z) with

=

0. Then there is an element � = (�u ,J E

where exp(27r i g,�<) = h,�< . So f,�< == g,�< - �'�< is a cocycle with values in 0 , and exp(27ri f,�<) = h,�< . This means that e( [f] ) = [h]. If the covering %' is fine enough, everything is independent of the covering. By construction it is acyclic with respect to z. Definition . Let h = (h,�<) E Z1 ( %' , 0* ) be a cocycle defining an analytic line bundle L E Pic ( X) = H1 (X, O * ) . Then c(h) := c( [h] ) = c(l) E H2 (%' , Z) = H2 (X, Z) is called the Chern class of h (or of L).

Let us return to the Cousin-Il distributions.

1.8 Proposition .

Let X be a S tein manifold. Then a Cousin-11 distribution o n X is solvable if and only if its Chern class vanishes.

If H2 (X, Z) = 0, every Cousin- I I distribution is solvable.

PROOF: If X is Stein, then H 1 (X, O) = 0 and c : H1 ( X, 0* ) ---+ H2 (X, Z) is injective. Let h be a Cousin-Il distribution on X . Then h is solvable if and only if [h] = 0, and that is the case if and only if c(h) = 0.

If in addition H2 (X, Z)

=

0, then also H1 (X, 0* ) = ( 1 ) .

•

Remarks As in the case of Cousin-! distributions, here it is sufficient that H1 (X, 0) = 0, and X need not be Stein. On the other hand, there are examples of simply connected Stein manifolds where Cousin-I I is not solvable. The condition H2 (X,Z) = 0 is essential, and for the solvability of any special distribution the vanishing of the Chern class is necessary.

In the Stein case, using higher cohomology groups one can show that c : H1 ( X, O*) ---+ H2 (X, Z) is even bijective. For a noncompact Riemann surface X it is always the case that H2 (X, Z) = 0. Therefore, in this case the Weierstrass problem is always solvable.

1 . Stein

Manifolds

257

E xtension from Submanifold s. Let X be a complex manifold and A c X an analytic set. A continuous function f : A ---+ C is called holomorphic on A if for every x E A there is an open neighborhood U = U (x) c X and a holomorphic function f on U such that flunA = ! / unA · One sees easily that this notion is well defined, but the local continuation f is not uniquely determined. �

1 .9 Theorem . Let X be a S tein manifold, A C X an analytic set, and

f : A ---+ C a holomorphicfunction. Then there exists a holomorphicfunction f on X with !/A = f . �

�

PROOF: In the differentiable category such a theorem would be proved with the help of a partition of unity. Here in complex analytic geometry we use cohomology. It is a typical application of Theorem B. We can find an open covering %' = (U,),EI of X and holomorphic func tions h on U, with h / u,nA = f / u,nA · Then �"" := J: - � defines a cocy cle � E Z 1 ( %' , 0) , with �"" /u, K nA = 0. By Theorem B there is a cochain TJ E 0° ( %' , 0) with rJ, / u,nA = 0 and OrJ = �. Then

J, - fl< = TJ �

-

!<

- TJ,

on u,l< ,

�nd jiu, : = J: + TJ, defines a global holomorphic function [ on X . Obviously, f/ A = f. •

Unbr anched Domains of H olom orphy. Let X be an unbranched domain of holomorphy over C n . Obviously, X is holomorphically spreadable, and since Cartan-Thullen holds for such domains, X is holomorphically con vex. It follows directly that X is a Stein manifold, and all theorems on Stein manifolds are applicable in this case. We consider now an unbranched domain of holomorphy p : X show that it is also a Stein manifold.

---+

P". We will

Let 1r : c n+ l - (0) ---+ Jpm be the canonical projection. Then we obtain a new manifold X : = { (x , z ) E X x (Cn+ l - ( 0 )) : p(x) = 1r ( z ) }

together with two canonical projections 7r : X ---+ X and p : X ---+ c n+l - {0} . Since ( p ' 7f ) : X X ( c n+l - { 0}) ---+ ]pm X ]p>n is a submersion, the fiber product X = (p, 7r) -1 ( �ll'n ) is in fact an (n + I )-dimensional complex manifold, and we have the following commutative diagram:

-

X

�

cn+ l _ { 0}

X

----+

Jpm

7f t

p

._l- 7r

258

V. Stein Theory

For a point P0 := (x0 , z0) of X there exist open neighborhoods U = U(x o ) c X and V = V(1r(z0)) C ]pm such that p : U --+ V is biholomorphic. Then 1r - 1 (V) is a neighborhood of z0 , 1T - 1 (U) = (U x ('Cn+ l - {0} )) n X a neighborhood of P0 , and p : ?T- 1 (U) --+ 1r-1 (V) a biholomorphic map with p- 1 : z >-+ ( (Pl u ) - 1 ( 7r(z) ) , z) . So p : X --+ M := en+ l - {0} is an unbranched Riemann domain. It is fibered over X by 1T with fibers isomorphic to @*. By hypothesis X is a domain of holomorphy. Therefore, it is pseudoconvex. Now we use notation and results from Section 11.8. If X has only removable boundary points, then X has locally the form U - A, with an open set U c e n and an analytic hypersurface A. But then X also has such a form and is pseudoconvex. If X has at least one nonremovable boundary point, then X (which is something like a cone over X ) has only non-removable boundary points over 0 E en+ l . The set of these boundary points over 0 is thin in ax. Outside the origin X is pseudoconvex, since it locally looks like X X c. By a theorem of Grauert/Remmert (see Section 1 1 .8) it follows that..X is pseudoconvex everywhere. From Oka's theorem we know that then X is a Stein manifold. �

�

�

�

�

�

To show that X is also Stein, we have first to show that X is holomorphically convex. If not, we would have an infinite sequence D = {xi : i E N} in X without a cluster point such that every holomorphic function f on X is bounded on D . We have that f5 == 1T - 1 (D) is an analytic subset in X , and flii-l( x,) := i defines a holomorphic function f__.on D. Since X is Stein, there exists a holomorphic function f on X with f l v = f . On every fiber ?T- 1 (x) ""' e * we have a Laurent expansion �

fi rr-l (x) =

00

L

v = - <Xl

�

a v (x)z v .

Since 1T is a submersion, we can define a holomorphic function g on X by g(x) == a 0(x ) . Then g(x i) = i , and this is a contradiction.

In the same way we can show that for x, y E X with x -I= y there exists a holomorphic function f on X with f(x) -I= f(y) . Thus X is holomorphically separable and therefore Stein.

The Embeddin g Theor em. By a theorem of Whitney every differen tiable manifold can be embedded into a space Jll1. N of sufficiently high dimen sion. In general, this is false for complex manifolds, since, e.g., in e N there are no positive-dimensional compact complex submanifolds. However, Stein manifolds can be embedded.

1.10 Theorem. If X i� n+'l n- dimension al en · an embedding j : X Y

Stein manifold, then there exists

1.

Stein Manifolds

259

The proof is due to Remmert (see his Habilitationsschrift and [Re56] ) , Narasimhan ( [Nar60] ) , and Bishop ( [Bi6 1]). 0.

Forster ( [Fo70] ) proved the following theorem:

1 . 1 1 Theorem. A n y 2 - dimension al Stein manifo ld can be embedded into e4 , but there is a 2 - dimensional S te in manifold that cannot be embedded into

e3

In general, for n > 2 every n - dimension al Stein manifold can be embed ded into e2 n , and for n > 6 even into e2n -f( n - 2 l/3l . For arbitrary n there always exists a proper imm e rs ion into e2n - l _ But there are examples of n dimension al Stein manifolds X that cannot be embedded into eN (and not immersed into eN- l ) for N : = n + [�] .

At the end of his paper Forster conjectured that every n-dimensional Stein manifold can be embedded into eN +l and immersed into eN.

In 1992 J . Schurmann showed that Forster ' s conjecture is true for n > 2 (see [Schue92] , [Schue97] ). In 1970/71 Stehlk had given an embedding of the unit disk � into e2 (see [St72] ), and in 1973 Laufer an embedding of certain annuli into e 2 . At the moment it is an open problem whether every noncompact Riemann surface can be embedded into e 2 . There has been quite a bit of progress on this by the work of J . Globevnik and B . Stens12lnes (see [G1Ste95] ) , but no final result.

The Ser r e Pr oblem . Above it was proved that if f : X --+ Y is a finite holomorphic mapping and Y a Stein manifold, then X is also Stein. In 1953 Serre posed the following problem: Is the total space of a holomorphic fiber bundle with Stein base Y and Stein fiber F a Stein manifold? ( [Se53] ) . It is easy to show that the answer is positive in the case of an analytic vector bundle. Between 1974 and 1980 it was proved that the answer is positive in the case of 1-dimensional fibers (Siu, Sibony, Hirschowitz and Mok). In 1976 Siu generalized this to the case where the fiber is a bounded domain G c e n with H 1 (G , C) = 0, and in 1977 Diederich and Fornress showed that the Serre conjecture is true if the fiber is a bounded domain in e n with C2 smooth boundary. In 1977 Skoda ([Sko77] ) gave the first counterexample, a fiber bundle with e2 as fiber and an open (not simply connected) set in e as base. Demailly improved the example; he used the complex plane or a disk as base. Finally, in 1985 G. Coeure and J . J . Loeb presented a counterexample with e* as base and a bounded pseudoconvex Reinhardt domain in e2 as fiber. Over the years a number of positive examples have been found, for example by Matsushima/Mor imoto, G. Fischer, Ancona, Siu, and Stehlk. Examples where

260

V . Stein Theory

the fiber is a so-called Banach-Stein space play an important role, but we cannot go into the details here. Exercises 1 . Show that the Cartesian product X x Y of two Stein manifolds is Stein. 2 . Usc Theorems A and B to prove that the total space of a vector bundle over a Stein manifold is Stein. 3. Give an example of a Cousin-! distribution on C2 - { 0} that has no solution. 4. Let Uj == {z E C 3 : Zj -=/= 0) and 'P£ = {UI , u2 , U3 } · Show that every Cousin-! distribution with respect to 'P£ has a solution. 5. Let X be a complex manifold, 1r : L --+ X a holomorphic line bundle, and Z c L the zero section. Prove that if L - Z is Stein, then X is a Stein manifold as well.

2.

T h e L evi For m

Covariant Tangent Vect or s . Let X be an n-dimensional complex

manifold and x E X a point. The elements of the tangent space Tx (X) (abbreviated by T ) are the (contravariant) tangent vectors at x . If z1 , . . . , zn are local coordinates at x, then every tangent vector can be written in the form Of

course, T has a natural structure of an n-dimensional complex vector space.

Now we consider the space F = F(T) of complex-valued real linear forms on T. For example, if f is a local (real- or complex-valued) smooth function at x, then its differential ( df) x E F is defined by ( df) x ( v) : = v [f] . It is uniquely determined by the germ of f at x. In local coordinates we get (df)x (v) =

L Vv fzv ( x ) + L Vv fzv ( x ) . v

v

It follows that

(d})x (v) = (df)x (v). In particular, we have the elements dzv , azv E F defined by dzv (v ) := v [zv]

and

azv (v) : = v [zv l ·

Then dzv = dxv + idy, and azv = dzv = dx v - i dyv , for v = 1 , .. . , n . The space F is the complexification of the 2n-dimensional real vector space T* = Homrn:. (T, JR) . Therefore, F = T* ffi i T* is a 2n-dimensional complex vector space, with basis

2. The Levi Form

261

We call the elements of F complex covariant tangent vectors or complex forms at x . In general, a complex covariant r-tensor at

x

1-

is an JR:-multilinear mapping

cp : T X · · · X T --+ C . cp 0

The tensor product given by ( cp 0

r-times

7/J of an r-tensor and an s-tensor is the ( r + s)-tensor

7/J) ( v l , . . . , vr, Vr+l , . . . ,Vr+s )

:=

cp( v i , . . . , vr) . 7/J ( vr+J , . . . , vr+s)·

The set of r-tensors carries the structure of a complex vector space, and the assignment ( cp , 7/J) >-+ cp 0 7/J is C-bilinear. For example, (dz, 0 dz �" ) ( v , w )

=

Vv . Ww

H er mitian Forms. Let X be an n-dimensional complex manifold. The notion of a plurisubharmonic function in a domain G c e n was already defined in Chapter II. Of course, a plurisubharmonic function on a com plex manifold is a real-valued function that is ('i!f = ) differentiable and pluri subharmonic with respect to all local coordinates belonging to the complex structure of X . The notion of plurisubharmonicity is invariant with respect to holomorphic coordinate transformations. So in order to prove plurisub harmonicity at some point of X it is enough to prove it with resepct to an arbitrary coordinate system. However, here we wish to express the notion of plurisubharmonicity in invariant terms. We do this by Hermitian forms. Defin ition .

A Hermitian form at

H

:

T,,

X

x0

E X is a Hermitian form

Tx0 --+ C.

The form H is called positive semidefinite if H ( v , v ) > 0 for all v , and it is called positive definite if H(v , v) > 0 for v -j. 0. A Hermitian form has a unique representation

H= where H := equation H

H

(

hij

=

i, j

=

n

""' L......t

i,j = l

)

ht) dzt

IV> '
azJ ' ·

1, . . . , n is a Hermitian matrix; i.e. , it satisfies the

H t . In the following we suppress the symbol 0 and write

= L;i ,j hi1dziazj ·

262

V. Stein Theory

With respect to the local coordinates can associate to any tangent vectorn v = Li vi a I azi + Li via I azi the corresponding vector = (V I ) . . , Vn) E e and write H ( v, w ) L hijvi wj · H · w t . i,j=l we

v

n

.

= v

=

F Assume that G --+ B is a holo morphic map of domains in en given by equations wk = fk (z · . . , zn ), for k = 1, . . . If g is a differentiable function in B , by the complex chain rule it follows that (g F)z, = L(9wk F) . (!k)z, and (g F)z, = L(9wk F) . ( ik)z, · :

C oor dinate Tr an sformations.

, n.

h

0

0

k

0

0

k

The transformation of a (contravariant) tangent vector (i.e. , a derivation) � is given by F * (�) [g] = �[g o F ] . This means that n n a a F*(�) L �[ik l awk L �[ik l awk k=l k=l n n n n a a � h �i . (fk)z, awk + � h �i . (fk)z, Owk ' or, if � e = (6 , . . ) �n), then F *� ( l:: �i . (h)z, . . . ) L �i . Un)z, ) = e . Ji . +

(

rv

)

(

)

.

rv

1

1

Now, a covariant tangent vector cp at w E B will be transported in the opposite direction: F * cp (� ) := cp (F*� ). In particular, F* (df)w(�) = (df)w (F*�) = (F*�)[J] = �[f o F] = d ( f o F)z(�) for w = F(z). This gives us the formula F* ((df) F(z)) = d( f o F)z · Therefore we also write cp o F := F * cp for arbitrary covariant vectors cp . If H i'> a Hermitian form at w F(z) E B, then we can define a Hermitian form F*H at by F* H( � , JT ) : = H (F*� ' F*rJ). If J = JF is the Jacobian of F at and H = L; k ,l hk, l dwk diVt , then F* H(� , Tl) = (e · J ) . H (ry · J ) = e · (J . H . J ) So F*H is given by the Hermitian matrix J t . H · J. =

z

z

t

- t'

�� t '

.

t

-

.

7J

-t

.

2 . The Levi Form

263

Plurisubhar m onic Fun ction s. Assume that p is a real-valued ('i&' 00 ) differentiable function in a domain B c e n and w E B a point. We consider at w the Hermitian form

given by the Hermitian matrix

H (p, w) : = ( a a'lpa _ (w) Wk

k, l = l , . . . , n

W[

If F : G ---+ B is a biholomorphic transformation with calculation shows that

F

)

.

F (z ) = w , then a direct

H (p o F, z) = JF (z) t · H (p, w) · JF (z ) .

Therefore, * HP will be described by the Hermitian matrix F* Hp = Hpo F ·

H (p o F

, z ),

i.e.,

Now let X be an n-dimensional complex manifold, p a real-valued smooth function on X ' and X E X an arbitrary point. Assume that cp : u ---+ B c en is a local coordinate system at x. Then a tangent vector � at x is uniquely determined by a pair ( cp, e), e E en . We define the Hermitian form Hp : Tx (X)< Tx ( X )---+ e by

If 7/J is another COOrdinate system at

X

and � rv ( 7/J , e), then

and H(po

o

o

We see that the definition of HP is independent of the local coordinates. Defin ition . Assume that p is a real-valued smooth function on the complex manifold X. Then L ev (p) (x , �) == HP ( �, �) is called the Levi fo rm of p at x . It is the quadratic form on Tx (X ) associated with Hp, and it does not depend on local coordinates. The function p is called p lurisub harmonic on a subset M c X if the Levi form of p is positive semidefinite at any point x E M . If the Levi form is positive definite at every point, we say that p is stri ctly p lurisub harmonic.

264

Stein Theory

V.

Example

Assume that h ' . . . . fm are holomorphic functions on X . We show that p := '5:."(; I fk . is plurisubharmonic in X . In fact, we have

fk HP

LPz,:z1 dzi dzj i,j f (I) fk ) zi (fk ) z1 dzi dZj ) k=l t,) ·

�

(� ( fk) zi dzi) ® (� (fk ) z1 dZj )

That means that

Lev

m

(p)(x,�) = Hp(�, �) = L L (fk) zi (x) �i k=I i

2

> 0.

The M aximum Pr inciple. A nonconstant plurisubharmonic function does not take on a maximum. 2. 1 Theor em . Assume that A c X is a compact connected analytic set and that p is a p lurisubharmonic funct ion on X . Th e n is constant.

PI A

PROOF: where

PI A

We may assume that A is irreducible. If there is a point takes its maximum value c, we shall prove:

xo E A

(*) Th e function p is identically equal to c in a small neighborhood

x

(

d

p( x)

xa.

If we know * ) , then we consider the set K of all points E A with = c. The open kernel K0 is not empty. If K0 =f A , there is a boundary point X I of K in A . The function p also takes its maximum at X I · So by (*) it follows again that = c in a neighborhood of X I , which is a contradiction.

p(x)

So we have only to prove ( * ) . We may assume that A is an analytic subset of e n and Xo 0 . If the codimension of A is equal to d, then there is an (n - d)-dimensional domain G' c en - d with 0' E G' and pseudopolynomials w ! (zi ; z ) , . . . , wd(zd; z') over G' such that A is an embedded-analytic subset of the joint zero set of the in a neighborhood of 0 , and 0 is the only point of A over 0' . =

'

Wi

We take a ball B c G' around 0' and restrict everything to an arbitrary complex line £ c B through 0 ' . The restriction AI£ is denoted by A '. Let z be a linear coordinate on £ with origin 0 such that the embedding of £ in B

2. The Levi Form

, n -d)

n- d Z)

265

l

(a1z, . . . , a is given by z >-+ (z1 , . . . z . The restriction wi e may contain multiple factors. We throw away such superfluous factors, so we can assume that every wi e is free of multiple factors. =

l

Let us first assume that the union of the discriminant sets of the pseudopoly nomials over Ji consists only of the point 0' and that A' is irreducible and has s sheets over £. Then A' is the Riemann surface of V'z. We write A' in the form A ' � ( ( t ;a l z, . . .

,an-d Z) : z = t8 }. ,an-dt8 ) we have a localt parametrization of A '. Then By F(t) == (tll 1 t8 , (p o F)a (t) = H(p o F, t) is given again by J H (p, F(t)) . J, where = ( 1 ;a1 , . . , an- d) t denotes the Jacobian of the holomorphic map F . The proof is the same as in the case of a biholomorphic map F . So ( p F)a > and •

•

J

•

.

.

:= PI A'

p o F is a subharmonic function of t. We get p'

-

o

c on A '.

0,

The same is true if A' is not irreducible but has 0' as the joint discriminant set, since 0 is the only point of A ' over O'. Now assume that the union of the discriminant sets is general. Every point in A' can be connected with 0 . We introduce the subset K of all points x E A ' with = c. If K0 # A ', there is a boundary point of K in A '. We know that A' is an embedded-analytic set. Then there is a neighborhood U ! C A' that is embedded-analytic over a disk B' (zr ) C Ji around a point z1 E B such that over z1 the only point of U is x1 and the union discriminant set consists of z1 only. Then we get c (by the same argument as above), which is a contradiction to the property "boundary point." So p' - c follows.

p' (x) (x )

x1

p' l u

This holds for all £, and therefore p c over the whole ball B ,which is in a full open neighborhood of in A . So we have * ) . •

x0

(

_

Exercises 1 . Assume that p is a real-valued smooth function on the complex manifold X . If � � e is a tangent vector and r.p a complex coordinate system at E X , then define

xo

n

(8p) xo ( � ) : =

L (P

v=l

0

r.p- 1 ) zv (r.p(x o ))

·

�v .

Show that (8p)x0 : Txo --+ C is a complex-valued real linear form that does not depend on the local coordinates. Prove the following formulas: (a)

Lev(p q)(x, �) ·

p(x) Lev(q)(x, �) + q(x) . Lev(p)(x, �) + 2 Re ( (8p) x ( � ) (8q)x ( � ) ) . ·

·

266

V. Stein Theory (b)

Lev ( h o p)(x, �) == h" (p(x)) . I (8P)x (�) 1 2 + h' (p(x)) Lev (p) (x, �). ·

2. Let G cc X be a relatively compact domain with smooth boundary. Show that there is an open neighborhood U of 8G in X and a real valued smooth function cp on U such that G n U = { x € U : < 0} and (8cp)x -=/= 0 for X E ac. Show that

cp (x)

Hx (8G) = {� E Tx (X) : (8cp) x ( � ) = 0 } is a well-defined subspace of Tx (X) that does not depend on the boundary function cp. Show that if for every X E ac there is a local boundary function 7/J such that is positive definite on Hx (8G), then cp can be chosen as a strictly plurisubharmonic function. 3 . Let G cc X be a relatively compact domain with smooth boundary, and cp : U (8G) --+ l!l1. a global boundary function. If ( cp) has for every X E ac at least one negative eigenvalue on Hx (8G) , G is called pseu d o concave.

Lev ('I/J)

Lev

Show that if X is connected and there is a nonempty pseudoconcave domain in X , then every global holomorphic function on X is constant.

3.

P seu docon v exity

Pseudoconvex C omplex M an ifold s. If X is an arbitrary complex manifold, then there exists a sequence of compact subsets Ki E X with the following properties: 1. The set Ki- l is always contained in the open kernel (Kit of Ki . 00

i=l If X is holomorphically convex, then the Ki can be chosen in such a way that the holomorphically convex hull Ri =

{ x E X : I J (x) l < s�; I J I for all f E O(X) }

always equals Ki . (One uses the same proof as for domains in en . )

x I !I 1! 1

f

Therefore, for every point E X - K, there is a holomorphic function in > 1 and < 1 on K, . By passing over to a multiple and X such that a power of we can make arbitrarily small on K, and arbitrarily big in a fixed neighborhood of

If (x) I f,

x.

Since Ki + - (Ki + I t is compact, there are finitely many holomorphic func 2 tions , tr;) in X such that for Pi : = L,;:'i 1 2 the following hold:

J? ), . .

.

1 1 Jt)

3.

Pseudoconvexity

2fj7

1 . supKi Pi < 2 - " . 2. Pi > i on Ki +2 - (Ki+l t . Of course, p, is a nonnegative plurisubharmonic function in X .

The sum 2:::: � 1 Pi converges compactly on X to a nonnegative function p , with < c } CC X for every c > 0. EX :

{x

p(x)

If f is a holomorphic function on a domain G c e n , and G' c c G a subdo main, then we have the Cauchy estimate · sup l fl , for z E G ' . c' Using this estimate in the intersection of a local coordinate system for X with (Ki t , one shows that all derivatives of Li Pi converge compactly in X to the corresponding derivatives ofp . So p is 'tf'00 and again a plurisubharmonic function. One can even show that p is real-analytic (see [DoGr60] ) . Definition . A complex manifold X is called pseudoconvex if there exists a nonnegative smooth plurisubharmonic exhaustion function p on X ( i. e . a 'ff'= function p with EX : < r CC X for all r > 0 such that the Levi form of p is everywhere positive semidefinite) .

{x

,

p(x)

}

If we can find for p a strictly plurisubharmonic function in X , then X is called stricly pseudoconvex or l-comp lete 1 . If p is strictly plurisubhar monic only outside a compact set K c X , then X is called ] -convex or strongly pseudoconvex (at infinity). In the literature a pseudoconvex manifold is often called weakly complete.

1-

Above it was shown that

every holomorphically convex comp lex manifold is pseudoconvex. We shall prove later on that every 1-complete complex manifold is holomor phically convex, and even Stein (solution of the Levi problem) . Also, strongly pseudoconvex manifolds are holomorphically convex. But in general this is not true for weakly 1-complete i . e. , pseudoconvex) manifolds.

(

Examples. Strict pseudoconvexity ( ! -completeness) is one of the most important notions in the analysis of complex manifolds. Many constructions can be carried out only in the strict pseudoconvex case. Let us consider some examples. Example 1 : The theory was inaugurated by the following result: 1 An n-dimensional complex manifold X is called q - complete if it has an exhaustion function p such that at every point of X the Levi form of p has at least n - q + 1 positive eigenvalues.

268

v. Stein Theory An unbranched doma in G over en is strictly pseudoconvex it is a domain of holomorphy.

if

and only if

For proofs see [Oka53] , [Br54] , [No54] . E xam p le 2: Every unbranched Hartogs convex domain over en is strictly pseudoconvex. See [Ri68] , where a smoo lhing procedure for slriclly pseudo convex functions is introduced. E xamp le 3 : The proofs

of

the following statements are elementary.

Every compact complex manifold is pseudoconvex. The Cartesian product of finitely many (strictly) pseudoconvex complex manifolds is (strictly) pseudo convex. Any submanifold of a (strictly) pseudoconvex complex manifold is (strictly) pseudovonvex . E x amp le 4: Assume that G c en is a domain of holomorphy and that A c G is an analytic hypersurface. Then G - A is Hartogs convex (and therefore strictly pseudoconvex) . For the proof we just take a Hartogs figure

in the unit poly disk P " = { t : lti I < 1 for i = 1 , . . . , n} and a biholomorphic mapping F : H --+ G - A. The mapping F extends to a holomorphic mapping pn --+ G. If A F- 1 (A) c P" is not empty, then it is an analytic hypersurface in P" - H . Some lines L(t') { t (t 1 , t') : t1 E e } will intersect it in a compact subset of P " - H ; for other t' the intersection is empty (see Figure v. 1 ). =

=

<E-

Figure

=

L(t')

V . 1 . Hartogs convexity of G - A

There is a limit tb with c < ltb l < 1 such that the intersection of L(tb) with A is not empty but in arbitrarily small neighborhoods of tb there are points t' for which the int<M"section is empty. If t E A n L(tb ) , then there is a neighborhood U where A is given by a holomorphic equation f = 0. The function tt >-+ j(t1 , t0) has isolated zeros in U n L(tb ) , and by the theorem

3.

Pseudoconvexity

269

t1 f(t1 , t') Thus F(P n ) c G - A, and the domain G - A is Hartogs pseudoconvex. Example 5: Let A c ]pm be an analytic hypersurface. We know that there is a homogeneous polynomial such that A = { (zo : Z1 : . . . : Zn ) E lP'n : w(zo , . . . , Zn ) = 0}. of Rouche >-+ also has to vanish at some points of U n L (t' ) for t' near to t � . This is a contradiction.

w

Let

w

be homogeneous of order s. Then

p(zo : . . . : Zn ) := log

l (zo · . . , znW } w

,

lP'n - A.

is a well-defined smooth exhaustion function for the affine algebraic manifold We calculate the Levi form in local coordinates tv Zv / 0 v 1 , . . . , n using the properties of the logarithm. Since the Levi form of + vanishes for any holomorphic function it follows that =

z, f f

=

f, Lev(p) (to, e) s . Lev (log(1 + l i t 1 2 )) (to, e) S · ( ( 1 + ;L I 2 ) 2 · l (to , e) l 2 + 1 + l�tol l 2 · l e 11 2 ) 2 · l l + ( l t o l 2 · l e l 2 - l (to , e) l 2 )) , e ( 1 + l :o ll 2 ) 2 ( and this expression is positive for e cft 0 . So p is strictly plurisubharmonic everywhere, and X = lP'n -A is 1-complete. In this case we can show directly _

that X is holomorphically convex:

Every function f

(zo

:

. . . : z . .)

:

zo80 · · · zn _ w(zo, . . , Zn )

n

Sn

= -

.

'

with

l: si = s,

a=O

values of A.

all these is holomorphic in X , and the maximum of the absolute functions tends to infinity as approaches So K c c X for any subset K c c Consequently, X is holomorphically convex, and it is even Stein, as one can see from the following theorem.

(zo : . . . : zn )

�

X. Let X be a holomorphically convex connected complex man

3 . 1 Theorem . ifold that contains n o compact analytic subset is a Stein manifold. PROOF:

of positive

dimension. Then X

E X be an arbitrary point. Then the set A { E X : f ( ) f ( o) for every f E O(X)}

Let

x0 :

=

x

x

=

x

270

V.

Stein

Theory

is a closed a�ic subset of X . Clearly, it is contained in the holomorphi� < convex hull Since for all has to be compact, is likewise compact. This is possible only consists of isolated points. Then there exists an open neighborhood and in such that holomorphic functions h ,

{xo} f E O(X)}. {xo} = {x E X : lf(x) l lf(xo )l A if A U = U(xo) . . . JN U {xo} = A n U = rx E U : h(x) = . . . = fN (x) = O}.

So X is holomorphically spreadable and therefore Stein.

3 . 2 Proposition .

•

A ]-complete complex manifold cannot contain compact

analytic subsets of positive dimension. PROOF: Let p be a strictly plurisubharmonic exhaustion function in the manifold X . Then p is plurisubharmonic, and by the maximum principle it must be constant on any compact connected analytic subset A c X . If A has positive dimension, then there is a point and an open neighborhood c X such that A n is a submanifold of of positive dimen �ion. Tpe function PlAnu is strictly plurisubharmonic and constant. That is Impossible. •

U = U(x)

U

xEA

U

3 . 3 Corollary. Let X be a ] -comp lete manifold that is holomorphically convex. Th e n X is a Stein manifold.

At the end of this chapter we will see that the condition "holomorphically convex" is not necessary. 3 .4

Corollary. Let

A C ]pm be an analytic hypersurface. Then ]pm - A is a

Stein manifold. Every analytic subset B C ]pm of positive dimension meets A in at least one point.

Example 6: There is a famous theorem by H. Cartan:

A domain G c C2 is a domain of holomorphy if and only if the first

Cousin problem is always solvable.

The solvability is also true for higher-dimensional domains of holomorphy, but there is a greater class of domains with this property. Take, e.g . , the domain G c pn c en (with n > 3) that is the union of the three open sets

. .

ui - {z u2 - { z {z u3 . · -

E pn : l zii > EJ , E P" : hi > E) , E P" : I (Z3 , . . , Zn ) I < c}. ·

3. Pseudoconvexity

Figure

V.2.

Cousin-I in

a

27 1

domain which is not Hartogs convex

The domain G = U1 u U2 U U3 is not Hartogs pseudoconvex (see Figure V.2). But all three covering elements Ui are domains of holomorphy. Therefore, we have H 1 (Ui , 0) = 0 for i = 1 , 2, 3, as will be proved in Section V.5. So '2/ = { U1 , U2 , u3) is an acyclic covering for 0, and every Cousin-! distribution in G can be given by a cocycle f E H 1 ('2/, 0 ) pp to a coboundary. Such an f consists of holomorphic functions fiJ in Uij with f1 2 + f2 3 + !3 1 = 0 on U1 2 3 · This implies that the Laurent series of /1 2 (around the origin) in U1 23 contains no powers zf z; with i < 0, j < 0. By the identity theorem this is true on the whole set U1 2 . Therefore, we can subtract a coboundary J{g 1 , g2 , 0} from our cocycle such that thereafter f1 2 0. Then the new !2 3 , fi 3 coincide on U1 23 . Together they give a holomorphic function h in (U1 U U2) n U3 that extends holomorphically to U3 . Then the new cocycle f is equal to the coboundary 6(0, 0, h ), and hence the old cocycle also cobounds (i.e. , is a coboundary) . Consequently the Cousin-! problem is solvable. Example 7: Assume that X is an n-dimensional complex manifold and that x0 E X is a point. We can blow up X in x0. Then we obtain an n-dimensional complex manifold X , an ( n - I )-dimensional complex submanifold A c X that is isomorphic to ]pm - l , and a proper holomorphic map 1r : X --+ X that maps A to x0 and X Abiholomorphically onto X - { x0 }. We have a strictly pseudoconvex neighborhood U around x 0 . We can lift the strictly plurisub harmonic exhaustion function p on U by 1r to 1r- 1 (U). This is a strongly pseudoconvex neighborhood of A that is not, however, strictly pseudocon vex. �

�

�

-

Example 8: For a similar example consider the analytic set

2 A := {z E C n : ZI Zn+j - Zj Zn+ l = 0 for j = 2 , . . n }. ·

,

O,

Outside of the origin A is regular of dimension n + 1. For example, if zi # then Zn+ J = zi . Zn +d zi for j = 1 , . . . , n . So A is there parameterized by z 1 , . . . , zn and zi + l · It follows that dim( A) = n + 1.

272

V.

Stein Theory 0"

We have a meromorphic map from C 2" to the Osgood space wi = zi / Z n + i , i = 1 , .. . , n. Its graph is the set

X = { (z, w)

E

C 2" X ((])"

:

given by

WiZn +i - Zi = 0 for i = 1, . . . , n } .

A

We denote by X the closure of the part of X that lies over - {0}. Let be the restriction of the canonical projection from x 0" 1r : X --+ to Then 1r maps a ! -dimensional projective space onto 0 and the rest biholomorphically onto - {0}.

A

A.

A

A

The space X is an (n + I)-dimensional complex manifold, locally given by the equations Zi = Wi Zn+i for i = 1 , . . . , n, for i = 2, . . . , n . Wi = W 1

A

{

zr /zn+ l = · · · - Zn /z2n = s } is an The set E ( s ) : = (zr , . . . , 2n) E n-dimensional plane for every s E JID1 , and is the union of all these planes. Since E ( s! ) n E ( s 2 ) = {0} for s 1 -=/= s2 , is singular at the origin. It follows that X is a vector bundle of rank n over 1r - 1 (0) "" lP'1 .

A

A

Now we use the function p(z) = Li ZiZi on C2" . It induces a strictly plurisub harmonic function on the complex manifold - {0}. The (n+ I )-dimensional complex manifold X is strongly pseudoconvex by p o 1r, but not 1-complete, since it contains the compact analytic subset 1r - 1 (0) .

A

Examp le 9: Consider the covariant tangent bundle T' of JID" . If � is a global holomorphic vector field on lP'" , then a plurisubharmonic function Pr; on T' is defined by We consider local coordinates in a set Ui c JID" , for example tv : = zv / zo for v = 1 , . . . , n in the case i = 0. Then every w over Ui can be written uniquely in the form w = L v wvclfv. So t 1 , . . . , t,, w 1 , . . . , w, are local coordinates in T' over ui · If � = Lv �!/a;azv , then

We have the following n + n2 holomorphic vector fields over Ui that extend to lP'" :

a atv

- q(O,

a . . . , � , . . . , ) and ll v at

O

uth place

t

= q(O, . . . ,

�

, . . . , 0) ,

uth place

where q : O( l ) ffi · · · ffi O ( l ) --+ TlP'" is the canonical bundle epimorphism in the Euler sequence. Plurisubharmonic functions p� , respectively p�ll' are defined in local coordinates by WvWv , respectively WvWv · t!ltW By adding them all we obtain the plurisubharmonic function

3. Pseudoconvexity

273

0,

if

Finally, we add the Pi constructed for the various Ui , i = . . . , n , we obtain a function p. If Z denotes the zero section in T', then p vanishes on Z and is strictly plurisubharmonic and positive outside of Z. It tends to oo for llwll --+ oo . So the complex manifold T' is strongly pseudovonvex, but not 1-complete. Example 10: This example is probably due to J.P. Serre.

Assume that E is an elliptic curve (i.e. , a compact Riemann surface of genus 1 ). Then E is a !-dimensional torus, given by a lattice of periods (1, O, where the imaginary part of � is positive. We may write the elements of E as real linear combinations = s . 1 + . �' with s, 1]. The first cohomology group H 1 (E , 0) is equal to C (this will follow from results of the next chapter, but it is also a very classical result in the theory of Riemann surfaces) .

t E [0,

t

z

We have a covering '2/ of E consisting of the two elements

u, u2

{ z = s + t� 0 < s < 1/2, O < t < l}, {z = s + t� I /2 < s < 1 , O < t < l}. Denote by C c E the circle {z = 1 + t � 0 < t < 1 ) and define there the function f 1 (and 0 on the other component of U1 n U2 ). Then f 1 -

.

:

is a cocycle in Z ('2/, 0). It yields a nonvanishing cohomology class, since otherwise we would obtain a nonconstant bounded holomorphic function on C. We construct a fiber bundle above E with fibers = C by gluing above C the point X C. We denote this X C with point in the bundle by [ z,

(z, w) E U2 w]

A

A

A,

(z,w - l)E U1

•

Now, is topologically trivial (since we can find a continuous function g on that coincides with f on C such that g} = f) ,but it is not analytically trivial, because it defines a nontrivial cohomology class. It follows from the construction that the notions of real lines, planes, and convexity are well defined in

U2

6{0,

A.

A

Let A be the bundle with typical fiber JID1 that is obtained from by adding the point at infinity to each fiber. Then A is compact and has the infinite cross section D = D= over E. We put X = A - D. Then X does not contain any compact analytic set of dimension 1. Otherwise, there would be a number b such that the analytic set meets every fiber in exactly b points. We could pass over to their barycenters and would obtain an ordinary holomorphic cross section in This would imply the triviality of �

:=A

A.

A.

We consider the real 3-dimensional surface

S = Sr = {[z, w]E A z = s + t� , W = s + r exp ( 27ri :

.

B

) and

O < < l }. B

274

V. Stein Theory

>0

Here r is a very big number. The surface S bounds a tube G, which converges to A as r tends to oo . We look at the holomorphic tangent H in any point of S . We can consider H as a real plane in A . By subtracting s in the fiber above s + we pass over from A to E x C. The hypersurface S is transformed diffeomorphically into a convex cylinder over E and H into a contacting plane that docs not enter the interior of the cylinder. So H does not enter the interior of the tube G. If H were contained in S , then the transformed H would lie in the boundary of the cylinder and therefore be compact. So H itself would be a compact analytic set. We saw that such a set does not exist. So there remains only the possibility that H contacts S of first order; i.e. , the intersection H n S is a real line. We can choose a complex coordinate ( = x + i y in H such that our real line is exactly the z-axis.

t�

If(! is a smooth defining function for S, then (! behaves on H like the function y2 . It follows that (QIH ) (( This means that the Levi form of (! is always positive definite on the holomorphic tangent H. Now we can construct a strictly plurisubharmonic function {j in U D (where U is a neighborhood of D in A) whose level sets are the manifolds Sr such that {j converges to oo It follows that X is strongly pseudoconvex. when approaching

> 0.

�

-

D. 2

Later on we shall prove that a strongly pseudoconvex manifold is holomorphi cally convex, and since X contains no positive-dimensional compact analytic subset, it is a Stein manifold. So there are many holomorphic functions in X . Assume that there is a meromorphic function on A that has poles of order m on D only. Then the coefficient of the highest polar part off is a holomorphic cross section ry in the mth tensor power of the normal bundle of D . Because this normal bundle is topologically trivial, ry cannot have zeros on D. So f = const} meets D. tends to infinity approaching D, and no analytic set This is a contradiction, since A is not analytically trivial. Every holomorphic function on X must have essential singularities on D. �

{!

Analytic Tangents . Let G be a domain in en with n > 2 and p a strictly plurisubharmonic function in G. Denote by X the set {z E G : p ( z) < 0}. Let B c c G be an open subset and w E 8X n B an arbitrary point. The expansion of p in

w

is given by

p(z) = p (w) + 2 Re Q(w, z - w) + Lev(p)(w, z - w} + R(w, z - w}, where

1 t w) + L h · Hess (p) (w) . h t, Q(w, h ) = h . \lp (

2 We may assume that g is globally defined and

{!

=

unbounded strictly monotonic smooth function h very large. The function {i := h o g will do the job.

r

:

on

Sr .

Then we choose an � -+ � such that h" /h' is

3 . Pseudoconvexity

with the complex Hessian Hess (p) ( w) =

.

��

(Pz, z1

R (w, h ) llhll 2

275

)

( w ) i , J = 1 , . . . , n , and

= o.

The map h >-+ Q(w, h ) is a holomorphic polynomial of degree The assignment w >-+ Q(w, h ) is smooth.

2,

for every w .

Let G' c c G be an open neighborhood of B. We can find constants c, k > 0 such that

Lev (p) (w, z - w ) + R(w , z - w ) > kllz - w ll 2 - cllz - w l l 3 for z E G' and arbitrary w .

B if

We say that a real number c > 0 is suffici ently small with respect to every ball U(w) with center w E ax n B and radius c belongs to G', and

is valid on these U(w) . Assuming that this is the case, for w E ax n B we define the analytic set

A(w)

:�

{z

E U(w) : Q(w, z - w ) = 0 }

in U ( w) . On A(w) we have p( z) > 0. Therefore, A(w) - { w } is outside of X . Since w belongs to the boundary of X , this implies that A = A(w) has codimension 1 . We call A an analytic tangent (or, in German, a Stiitzfiache) for X at w . 3.5

Prop o s ition. Let G, p, X, B , G' be as above. There exists a differen tiable family A(w) qf analytic tangents to ax at the points w E B n ax with A(w) n X { w} . =

Here "differentiable" means that the defining quadratic polynomials for A(w) depend smoothly on w •

Exercises

1. Let X be a complex manifold and Y c X a closed complex submanifold. Construct a nonnegative smooth function f : X --+ l!l1. with the following properties: (a) Lev ( f) ( x, �) > 0 for every x E X , � E Tx (X). (b) For every x E Y there is a linear subspace Px C Tx (X) such that Px + Tx (Y) = Tx (X) and Lev ( f) ( x, �) > 0 for � E Px , � # 0 . 2 . Let X be a 1-complete complex manifold and fl , . . . , fq E O(X). Show that X - N(fi , . . . , fq) is q-complete. 3. Let G cc e n be a strictly convex domain with smooth boundary. Con struct the differentiable family A(w) of analytic tangents t o ac.

276

V.

Stein Theory

4 . Let cp be a strictly plurisubharmonic smooth function defined in a neigh

borhood U U (O) c @" with cp(O) = 0 , and 7/J an arbitrary smooth function in U. Prove that there are r > 0 and > 0 sufficiently small such that cp + . 7/J is strictly plurisubharmonic in Br (O) and =

c

c

{

z

E Br ( 0) : ( cp +

c · 7/J) ( ) < 0} z

is a Stein manifold.

4.

Cuboids

Distinguished Cuboids. If Q is a closed subset in en , then in this section we say that something is defined on Q if it is defined in a small neighborhood of Q. Two objects are called equal on Q if they coincide in a small neighborhood of Q. So actually we consider "germs" along Q. Definition .

A cuboid is a closed domain

where ai < bi are real numbers for i = 1, . . . , 2n. If there are partitions

for

i

=

1 , .. . , 2n,

then we denote by A the system of sequences

a. 1• .1

'

=

1 , . . . , 2n,

j = 0, . . . , nzi.

A closed covering 'PEA of Q is defined by the system of cuboids Qj . . . ,hn = 1 ,

{

z

: ai i

1- < Xi < a{i , for i

=

1 , .. . , 2n}.

For any open covering of Q there is a closed cuboid covering which is finer. In the following it will be not enough to have a covering of Q. Additionally, we need a system of complex submanifolds Q = Xo � X1 �

· · · � Xi- 1 � Xi � · · · � Xs , s < n, - such that there is a holomorphic function fi

where xi has dimension n i ' in Xi _ 1 vanishing everywhere on Xi to first order (and maybe also vanishing at points of Xi - 1 - Xi) · Definition . A distinguished cuboid is a cuboid that is equipped with a system { ( Xi , fi) : i = 1 , . . . , s } . The number n - s is called the manifold dimension of the distinguished cuboid.

4 . Cuboids

277

Vanishing of Cohomology. As described above, everything on a closed cuboid is assumed to be defined on an open neighborhood of the cuboid. Therefore, we can consider cocycles and coboundaries with respect to closed cuboid coverings.

4. 1 P r op osition. Assume that Q is a cuboid and %'A a closed cuboid cov ering d Q. Th e n every cocycle .; E Z 1 ( %'A, 0 ) cobounds.

PROOF: First we consider a very simple system A. We just take the case where m 1 = 2 and all other mi = 1 . Then

has the minimal possible number of elements. The cocycle .; is given by one holomorphic function .;0 1 on Qo n Q l = { z E Q : Zl = a n . Let c be a small positive number. Then in the z1-plane we can choose two continuous paths aj from a� + i (a�+l - r:: ) to a� + i (a;+ l + E) (for j = 0 on the left side and for j = 1 on the right side of the line x 1 = ai, see Figure V . 3 ) and define rt = {ry0 , ry 1 } by the Cauchy integral .;o r (w , Z2 , . . . , Zn ) W - Z1

6

d

Then we get ry = .; .

z1-plane

Figu r e V . 3 . Cohomology of

a

cuboid

The next step is an induction on the number m = L:i mi . We just han dled the case m = 2 n + 1 . Now we assume that m > 2 n + 1 and that the proposition already has been proved for any number smaller than m. We put io : = max{i : 1 < i < 2 n and mi > 2} and define

278

V. Stein Theory

By the induction hypothesis, we have � � %' ' = J(r/) and � � %'" = J(r/') with cuboid coverings C1/ ' for Q' and C1/ " for Q". Hence, � - 17', 17" } is a cocycle with respect to the covering C1/ that is given by a holomorphic function g on the surface Q' n Q" { z E Q : Xi0 a:·o - 1 } ·

6{

=

=

As above, by integration we obtain holomorphic functions 170 on Q' and 171 on Q" with g = "11 - 11o . So we have a cochain (}' = 17' + TJo, 17" + "11 } on Q with J(J' = � (since the functions 1]j are already cocycles) . This completes the • induction.

{

V anishing on the Embedded M anifolds. We consider a distin guished cuboid Q in e n with manifold dimension n - s, and for i = 0, . . . , s we prove the following:

Every cocycle of Z 1 (Cf/.A n Xi, 0) cobounds, and every holomorphicfunction on Xi+ I can be extended to a holomorphicfunction on

4.2 Proposition.

xi .

PROOF: We carry out an induction on i. In the case i = 0 the proposition has been proved already. The induction hypothesis now states that it holds for some i, 0 < i < s - 1 , and we prove the extension property for i + 1 . Assume that g is the holomorphic function on Xi + I · Define g ( z) = 0 on the other connected components of N(fi +1 ) c Xi . If the covering Cf/.A is sufficiently small, we can extend this g to a cochain 17 E C0 ( 0&'.A n Xi , 0) . The coboundary 6 (17) vanishes on N (fi+ 1 ) . Therefore, there is a cochain a E C1 (Cf/.A n Xi , 0) with 6(17) = a · fi +1 . It is clear that a is a cocycle, and by the induction hypothesis there is a 1 E C0 (Cf/.A n Xi, 0) with a = J (r) . Since J(ry - fi + I . y) = 0, we get g = TJ - fi+ 1 . 1 as a holomorphic extension of g.

Now we prove that any � E Z 1 (Cf/.A n Xi + l , 0) co bounds . As in the proof of the preceeding theorem we have to show this or.ly in the case where m 1 = 2 and mi = 1 for i > 1. So � is simply a holomorphic function g on (Q 1 , 1 , ... , 1 n Q2, 1 , . . . , 1 ) n Xi + l · We have to find holomorphic functions f' on Q 1 , 1 , ... , 1 n Xi+ l and !" on Q2, 1 , ... , 1 n Xi+l with !" - f' = g. For that we first extend g to ( Ql , l , . , 1 n Q2,l, . . . , r n Xi , construct f', !" for Xi (induction hypothesis), and • then restrict them to Xi+ l · That completes the proof.

)

..

Cuboids in a Complex M anifold. We assume that X is an n dimensional complex manifold and p a smooth real-valued function on X with p(x) ---+ 2 when X ---+ aX . 3 We assume further that p is strictly plurisub3

This means t h at for every E > 0 there is a compact subset p(x) > 2 E for E X K.

-

-

K CX

such t h at

4.

Cuboids

279

harmonic on {p(x) 1 ) and define Y : = {x E X : p(x) < 1 ) . Then Y is a relatively compact strongly pseudoconvex open subset of X . =

For an n-tuple z

=

(z1 , . . . ,zn) we use the norm l z l supmax(IRe(zi )l , l lm(zil). :=

1

For an open neighborhood G c X of 8Y we want to apply the results from the end of V . 3 . (It does not matter that there are no global coordinates on G.) We consider a relatively compact open neighborhood B B(8Y ) C C G and choose a G' with B C C G' CC G. =

For any point x0 E 8Y there is a compact cuboid U* with center x0 in a coordinate neighborhood S around x0 . If a real number c: > 0 is given, we can choose every U* so small that U* c U(w) c c S for every point w E U * , where U(w) is the ball with center w and radius c: (with respect to the local coordinates) . Then for every w E U* n 8Y we have the analytic tangent A(w) given by a quadratic polynomial fw in U(w) . It follows that A(w) n U* n Y w , and the function p i U* n (A(w) z) is positive. This is illustrated by Figure V.4. =

-

For an open subset Y' c c Y we have the following proposition.

There is a distinguished cuboid Q* the following property: 4 .3 Propositi on.

=

U*

x

Qc

eN with

For s n the submanifold Xs is projected biholomorphically onto a N compact set U' c U* with Y' n U* C U' C Y . =

-

U*

F igur e V .4. Projection of the distinguished cuboid

en

and assume that x0 is the origin. We use the PROOF: We may work in differentiable family of analytic tangents A(w) N(fw) with w E 8Y n U* . =

280

V. Stein Theory

Take a positive number r so big that we always have 1 1 /fw(z) l < r for w E 8Y n U* and z E Y' n U * . In a small neighborhood of w we have 1 1 /fw(z) l > r. Therefore, by compactness there are finitely many of these fw , say /I , . . . , fs , with maxi l 1 / fi (z) l > r for all z E 8Y n U * . For a fixed i we denote by Q� the cuboid {z' = (zn+l , . . . ,zn+i) : lz' l < r} and by Q�' the cuboid {z" = (zn+i+l , . . . , zn+s) : lz" l < r}. We put Q* = U* x Q: x Qr The submanifolds Xi c Q* are obtained in the following way: Consider the graph of the i-tuple ( 1 //I , . . . , 1 /fi) in U* x QL take the union of those connected components that contain points over Y , and multiply this union by Q�' - The manifold Xi has dimension N - i. In Xi- l we have the holomorphic function Zn+i . fi (z) - 1 , which vanishes on Xi to order 1 . Finally, the projection U ' of X , contains no point of 8Y. Hence, it is contained • in Y. Since U* n Y' is contained in U', the proposition is proved. 4.4 C or ollar y.

In the above notation, H1 (U' , O) = 0 .

E � nlarging U' . We use the same notation as before and construct a set U' c U* n Y that is bigger than U' where the vanishing theorem still holds. For that purpose we take an open set Y ' instead of Y' with Y' CC 8' C C Y � and U' c Y'. We need a bigger r > r such that we still have 1 1 / fw (z) l < r on 8' for w E 8Y n U* . But now ffiaKi 1 1 / fi (z) l > r no longer is true on 8Y n U * . So we add some functions to the old ones, !s+l , . . . , fs , such that we get the old situation again.

� ' We get the Q� and the Q� , for a = 1 , .. . , s. But we� write � � Q� instead. of Q� , Q�' � � ,.., Jii Wit h N < N . for Q:' ' and xi for xi . Then we have Q* = U* X Qi' X Qi" c "" The projection of X-s is a compact set D' c Y n U* with Y ' n U* again we have the vanishing of the cohomology.

c

D'.

So

The following statement is proved in the next paragraph. ( * ) Every holomorphzc functzon o n U' can be approxzmated arbitrarily well by holomorphic functzons on U'. Suppose all cochains are given with respect to a covering '2/ of U' X-s. If .; E Z1 ( '2/ , 0) is a cocycle over D', we have .; = ( ry) over U' and .; = (if) over D', with cochains ry and if with respect to the coverings UnU' and unD'. Then ij - ry is a holomorphic function over�U' , which we approximate by a sequence of holomorphic functions [}j E O (U'). We may replace the cochain ij by ij1 : = ij - [}j and obtain that ij3 approximates TJ over U' .

6

Now we can prove the following theorem.

""

6

4.

Cuboids

281

If x0 E 8Y and U* is a small cuboid in a local coordinate system around xo, then H 1 (Y n U* , O) = 0 .

4.5 Theorem.

PROOF: Take a fixed open covering '2/ of Y n U* and exhaust Y by a sequence of domains Yi with Yi c c Yi+ I c c Y . We get a sequence of compact sets UJ c U* n Y with Yi n U* c UJ.

u;

If � E Z1 ( '2/ , 0) is a cocycle, for every i we get a cochain 'r/i on such that � = J(ryi ) on As we have seen above, we may assume that the difference of 17·i + I and 'r/i on is as small as we want. Then the sequence 'r/i converges to some ry E C 0 ( 'P/ , O) with J(ry) = � . •

u;,.

u;

Ap pr oxima tion . We first prove the following simple result.

Assume that Q is a cuboid in CN. Th en any holomorphic function ! o n Q can be approximated arbitrarily well by polynomials. 4.6 P r op osition .

PROOF: We make the induction hypothesis that f is already a polynomial in the variables z' = ( z 1 , . . . , zs) . Let /c; be the boundary of the rectangle with

corners (as + l - c, a N +s + I - c) , (bs+ l + c, aN + s+l - c) , (bs+ l + c, b N + s + l + c ) , and ( as+ I - c, bN +s + l + c) in the (xs + l , X N + s+ l )-plane. We have the integral representation

f(z ' , zs + I , z " )

=

� 2Jrl

j

rs

f(z ' , �, z " ) d� � - Zs+ l

It is necessary to approximate 1/(� - Zs +I) with a fixed � E Ire I by a poly nomial. We consider only the case where Re( <) = bs+ I + c . In this case there exists a disk D (with a center far to the left) such that � E 8D and the

rectangle with corners ( as + l , aN +s + l ) , (bs + l , aN + s+ l ) , (bs+ l , bN +s + I ) , and (as + l , bN +s+d is contained in D. Then we expand 1 / (� - Zs+ d into a power series on the disk and get the desired approximation. So the induction step is complete. Since nothing remains to be shown for s = 0, the proof is finished.

•

We have to consider the cuboids Q = U* x Q� and Q = U* x Qi. For this denote by M the Cartesian product of the last 8 - s factors of the Cartesian product Qi. It follows that Q� x M c Qi, and S := (U* x Q� x M ) n X8 is projected biholomorphically onto X , and onto U'. Every holomorphic function on S is the restriction of a holomorphic function in U* x Q� x M and can be approximated arbitrarily well � polynomials, in particular by holomorphic functions on the larger set U* x Qs-. Therefore, every holomorphic function on U' can be approximated by holomorphic functions on U' , and we have proved (*) . �

282

V. Stein Theory

Finally, we wish to replace U* by its open kernel (U* t . For this we exhaust U* by relatively compact cuboids u; with u; c c UJ+ 1 , and choose the functions /i , i = 1 , . . . ,8 (from the "enlarging" process), independently of j . � Then we can approximate cochains on Uj c Uj�by those on Uj c u; , and we can approximate these again by cochains on Uj + 1 . So for any 1-cocycle � on (U*t n Y we obtain (as a limit) a cochain ry on (U*t n Y with Jry = �· Then we have the following theorem. 4.7

Theorem. If x 0 E 8Y and U* is a small cuboid in a local coordinate system around x0 , then H1 (Y n (U*t , O) = 0. Exercises

1. Let G cc en be a strictly convex subdomain with smooth boundary. Use coverings with distinguished cuboids to prove that H 1 (G, 0) = 0. 2 . Prove for the same domain G that every holomorphic function in G can be approximated by polynomials.

5.

Sp ecial C over ings

Cub oid C over ings. Again let X be an n-dimensional complex manifold and p a smooth real-valued function on X with p ( x) ---+ 2 for x ---+ 8X . Assume that p is strictly plurisubharmonic on {p(x) = 1 } , and let Y := {x E X : p (x) < 1 } . We also consider open sets Ya = {x E X : p(x) < l + a } , where 0 < a < c and c is a very small positive number such that also Ya is strongly pseudoconvex. Since 8Y is compact, it can be covered by finitely many cuboids Ut around points xi E 8Y which are always contained relatively compactly in a local coordinate neighborhood such that xi is the origin of the coordinate system. If Ut = { z : lzl < rJ and 0 < ti < ri , then we may also assume that the open sets Ui = {z : lzl < ti } cover 8Y. By adding more cuboids, the ti can be chosen as small as necessary. � � We also need these properties for the sets 8Ya , and for sets 8Y, where Y is an open set between Y and Y,. For that we have to move the centers of the cuboids a little bit. If a is very small, the Ui still cover 8Y. There may exist a compact set K c Y that is not yet covered by the Ui · Then we construct a covering of K with additional cuboids Ui c c Ut c c Y Finally, we have the following result. on. For 5.1 Propositi � a small c > 0, tor a ny a with 0 < a < c and any � open set Y with Y c Y C Ya there exist coverings '2/* = {Ut : i = 1 , . . . , m}

5. Special Coverings

and '2/ = {Ui hold:

:

i

=

1, .

283

�

.

.

, m} of Y with Ui c c Ut such that the following

1 . Every Ut is a compact cuboid in a local coordinate system around a point xi that corresponds to the origin in this coordinate system. The set Ut is small enough in the sense d Sections V. 3 and V.4 to have families
:

l z l < ti }

and

Ut = {z

l z l < ri }

:

in the chosen coordinates. 3. For each i , either Xi E Y or Ut c Y . 4 . For each i , the "star " si = Uj: u n U; # 0 uj is contained in Ut . �

We call ( '2/ , % * ) a special pair of coverings for ( Y,Y) (cf. Figure V .5)

U* '·

Figur e V. 5 .

Special pairs

of

coverings

The Bubble M eth o d . Let a small c > 0 be fixed in the sense of the pre ceding paragraph. We choose a special pair (0?/ , 0?/*) of coverings for ( Y,p ) . Since H 1 (Ui n Y, O) = 0, it follows that '2/ n Y is acyclic, and every coho mology class of H1 (Y, 0) can be given by a cocycle < E Z1 ('2/ n Y, O).

Let x0 E 8Y be an arbitrary point and i an index such that x0 E Ui. Then we have a cochain rt = { 'r/j } E C0 ('P/ n Ut n Y, 0) with J(ry) = � I (Ut n Y ) . We change 'r/j to 0 if U1 rt- Si . Then ry is a cochain over Y, and the cocycle � - J(ry) represents the same cohomology class as � · But � - 6(17) vanishes on ui n Y. Now we use the bubble method. By altering the strongly plurisubharmonic functionp a little bit on some compact subset ofUi (let us say by subtracting a smoo.th function with comnact rugoortlwee.nlame_Y..a little bit to a stronglY. pseuaoconvex open suoset·x =:J r wnn y :..._ r c c�ui. Tnen every conomo1crgy

284

V.

Stein Theory

class represented by some � E Z 1 ( '2/ n Y,O) extends to Y . So the restriction H1 (Y, 0) -+ H 1 (Y, 0) is surjective. We assume that for Y (after a small movement) the conditions for Ui , Ut are still satisfied.

xa

In the next step we go to another E Y and apply the procedure to Y . Continuing in this way, after finitely many steps we have the following result. 5.2

Propo sition. There is afundamental system of strongly pseudoconvex neighborhoods Ya of Y such that the restriction H 1 (Ya, 0) -+ H 1 (Y, 0) is surjective. Frechet Spaces . The following theory can be found in the famous multi

graphed notes [CS53] . See also [GuRo65] . Assume that V is

a

complex vector space.

p:

A seminorm in V is a map Definition. V the following properties: l. p ( a . f) = for a E C, E V. if Jr , f2 E V. 2. + +f2 J

-+

{x E l!l1.

:

x > 0) with

f l a l .p(f) p(fr < p(fr ) p( /2 ), If a sequence of seminorms Pi , i E N, is given in V , we can define in V. If c > 0 and m E N, we call the set

U� m ) (O) =

{f E : Pi (!) < c V

an ( r:: , m)-neighborhood of 0. As course, become smaller.

m

for i = 1 , .

a

topology

. . , m}

increases the (c, m)-neighborhoods, of

A subset W c V with 0 E W is called a neighborhood of 0 if there is an (r:: , m)-neighborhood dm l (O) c W. If E V is an arbitrary element and W a neighborhood of 0, then

fa

fo

fa.

tW

:= {fa t f : f E W}

is a neighborhood of Finally, a set M c V is called open borhood of any E M .

f

(fk )

if

it is a neigh

f

in V is called convergent to if for any Definition. A sequence neighborhood W ( ) there is a number ka such that !k E W for k > ka .

f The sequence (fk ) is called

there is a k, l > ka .

Cauchy sequence if for all c > 0 and m E N - fk) < c for i = 1 , . , m and number ka such that a

Pi (fl

.

.

5 . Special Coverings

285

Definition. A Fre 'chet topology on a complex vector space V is a topol ogy defined by a sequence (p i) of seminorms, with the following proper ties: 1 . For every f E V - {0 } there is an i with Pi (!) > 0 ; i.e., the topology is Hausdorff. 2. The space V is complete in this topology: Every Cauchy sequence converges. A Fre 'chet space is a complex vector space with a Frechet topology. Every Frkchet space possesses a metric that defines its topology. The vector space O(X) of holomorphic functions on a complex manifold X is a Fnkhet space. We just take an increasing sequence of compact subsets Ki with Ki C (Ki +I) 0 and U i Ki = X , and we define pi (f) := sup/f(K ) / . If (!k) is a Cauchy sequence of holomorphic functions on X , then fk / ( Ki) o converges uniformely to a holomorphic function on (Ki) o . Hence (fk ) con verges compactly to a holomorphic function f on X . This means that (!k) converges to f in the topology of O(X). Since every Frechet space V is a metric space, a subset M c V is compact if and only if every sequence (fk ) in M has a convergent subsequence with limit in M. A linear map v : E -+ F between two Frechet spaces Definition. is called compact or completely continuous if there is a neighborhood c E such that is relatively compact in F .

U (O)

v(U)

Obviously, every completely continuous map is continuous. We have the following famous theorem of Schwartz. 5 .3

T heorem of L. Schwartz . Assume that E , F are two Fre'chet spaces and that u, v : E -+ F are two continuous linear maps with the following properties:

1 . u is surjective. 2. v is completely continuous. Then the quotient Fj (u + v) (E) has finite dimension. A typical example of a completely continuous map is given as follows: Let X be a complex manifold and Y c c X an open subset. Using the increasing sequence of compact subsets from above, we have Y c Ki for some i. Now, : Pi (!) < 1} is a neighborhood of 0 in : = { f E O(X) E := O (X ) Let v : E -+ F := O(Y) be defined by v ( f ) : = f / y . Then

U

.

v(U)

c .'JlJ

: = {g E

O (Y) : snp/g/ < 1).

286

V. Stein Theory

Since !!& is closed and bounded, it follows from Montel's theorem that it is compact. This shows that the restriction map v is completely continuous. Remark. It is clear that we can replace the space of holomorphic functions by the space of holomorphic cross sections in an analytic vector bundle.

Finiteness of C ohomology. We wish to apply Frkchet space theory to our standard situation. Let X be an n-dimensional complex manifold and Y = { x E X : p ( x ) < 1 } , where p is a smooth exhaustion function on X with sup(p) = 2. Assume, as usual, that p is strictly plurisubharmonic on a neighborhood of 8Y. Let a small real number c > 0 be fixed such that Y"' is strongly pseudoconvex for 0 < a < c . We have shown that the restriction map H1 (Ya, 0) -+ H1 ( Y, 0) is surjective for sufficiently small a. We start with a special pair ('2/ , % * ) of cuboid coverings for (Y, Y ) , choose numbers � with ti < � < ri and define Ui = { z : lzl < � } - This gives a covering c¥/ = { Vi : i = 1, . . . , m } of Y a · a

�

The spaces C0 ('P/ n Y, 0), Z1 ('P/ n Y, O ) , and Z1 ('P/ n Y" 0) are Frkchet spaces, as is Z1 ( '2/ n Ya , 0) ffi C0 ( '2/ n Y, 0) . Since the cohomology can be extended from Y to Ya , it follows that the map �

�

u : E = Z1 ('P/ n Ya; O)

ffi

C0 ('P/ n Y, O) -+ F = Z 1 ('PE n Y, 0),

with u(�, rt) : = � l %' nY + o(ry) , is surjective. The map v : E -+ F is defined as the negative of the restriction map from Z1 ( '2/ n Ya , 0) to Z1 ( '2/ n Y, 0) , i.e. , by v(�, rt) := -�l%'nY· Then v is completely continuous. �

The map u + v is the coboundary map r5 : C0 ('P/ n Y, O) -+ Z1 ('P/ n Y, O ) , since the first summand goes to 0. So the quotient Fj(u+v) (E) is the first co homology group of Y. Its complex dimension is finite by Schwartz's theorem. Thus we have proved the following result.

If Y is a strongly pseudoconvex relatively compact subset of a complex manifold X , then dime H1 (Y, 0) < oo.

5 . 4 Theor em .

H olomorphic C onvexity. We consider the same situation as above and choose an a > 0 such that Y cc Ya and: 1 . dime H1 (Ya, 0) < 00 . 2. For every x E 8Y we have an analytic tangent A(x) c Ya to 8Y, with A (x) n Y = { x } , given by a holomorphic function f in a neighborhood U of A(x). x

5.

Special Coverings

287

Let x E 8Y be an arbitrary point, f = fx, A = A(x) , and U a suitable neighborhood of A . Then 'llx = {UnYa, Ya - A} is a covering of Ya consisting of only two elements. So Z1 ('llx, 0) = O(U n Ya - A ) ,and every function f - k 1 belongs to Z ( 'llx , 0) .Since H 1 ( Ya , 0) is finite-dimensional, for m ::;» 0 there are complex numbers a 1 , . . . , � not all zero, such that the cocycle given by the function m ) ( Un Y A ) g = ( at . r 1 + · + a, .

.

.

J I -

"

-

is cohomologous to 0. Therefore, we find holomorphic functions h 1 in Ya - A and h2 in U n Ya such that h 1 - h2 = g. The function h

..

_

{ hht + 2

g

on Ya - A , on U n Ya ,

is meromorphic in Ya with poles in A . So hi y is a holomorphic function, and for y -+ x it tends to oo . If K is compact in Y, then the holomorphic convex hull K does not approach x . This is true for every x E 8Y. So K is compact. As a consequence we have the following result. �

If Y is a strongly pseudoconvex relatively compact subset of a complex manifold X , then Y is holomorphically convex.

5 . 5 Theor em.

Moreover, we have a solution of the Levi problem in this special case. 5 . 6 Theor em. Assume that Y = { x E X : p (x ) < 1 } , where p(x) is a smooth function in X with p (x ) -+ 2 tor X -+ ax that is strictly plurisubhar monic o n 8Y. IfY contains no higher-dimensional compact analytic subsets, then Y is a Ste in manifold. PROOF: In Section V . 3 we showed that every holomorphically convex man ifold that contains no compact analytic subset of positive dimension is a Stein • manifold. 5 . 7 Corollary. If p is a strictly plurisubharmonic exhaustion function of X , then Y = { x E X : p (x ) < 1} is a Ste in manifold.

This is clear, since under the assumption there are no compact analytic sub sets of positive dimension in X .

Negative Line B undies. Assume now that Z is a compact n-dimensional complex manifold and that F is a holomorphic line bundle on Z. Let 1r : F -+ Z be the canonical projection. Definition . The line bundle F is called negative if there is a strongly pseudoconvex neighborhood Y c c F of the zero section of F .

288

V. Stein Theory

We identify Z with the zero section in F , and we call Y a tube around Z. For an arbitrary holomorphic vector bundle V of rank s on Z we consider the pullback bundle 1r* V on F. If U c Z is an open set, then the holomorphic cross sections in 1r * V over 1r - 1 (U) are holomorphic maps f : 1r- 1 (U) -+ V such that for any z E U the restriction of f to the fiber Fz has values in V� The trivial line bundle on F is denoted by Op . Its local holomorphic cross sections are the local holomorphic functions on F . If VI u rv u X c s ' then the holomorphic cross sections of 1r* V over 1r - 1 (U) are s-tuples of local holomorphic functions. Since the cocycles in our theory are given by lo cal holomorphic functions, we may replace OF by 1r* V on F and get the same results for X = F and the strongly pseudoconvex set Y . So we have dime H 1 (Y, 1r* V) < oo. Among the holomorphic maps f : 1r - 1 (U) -+ V that correspond to sections in 1r*V we have the maps f that on every fiber Fz are homogeneous of degree m (with values in Vz) · We denote the space of these maps by Om ( 1r- 1 (U) , V). It can be identified with the space of holomorphic cross sections in the tensor product V ® p- m (see Section IV.2) .

1,

}

�

We choose a finite open covering '2/ = { Ui i = 2, . . . , l of Z. Let '2/ be � the covering of F given by the sets Ui = 1r- 1 (Ui) We denote by z;, ('2/, 1r*V) the vector space of cocycles �m = with E Om (7r- 1 (Uij ), V ) . These are homogeneous of degree m on the fibers of F. We also call them cocycles of degree m. Every finite sum � = Li �i of cocycles of distinct degrees mi is con tained in Z 1 (F, 1r* V) and can be restricted to Z 1 (Y, 1r*V). It is a coboundary if and only if all i are coboundaries. Since the cohomology H 1 (Y, 1r*V) is �

{�Ij}

:

�0

�

finite, we obtain the following theorem: 5 . 8 Theor em. Assume that

F

is a negative line bundle and V an arbi trary holomorphic vector bundle o n the compact manifold Z. Th en there is an integer mo such that H 1 (Z, V ® p- m ) = Ofo r all m > mo . Remark. A line bundle F on Z is called positive if its complex dual F' is negative. Since (F' ) ' = F , the above result can be reformulated as follows: If F is a positive line bundle on Z, then H 1 (Z, V ® F ) = O fo r m > mo . "

Bundles over Stein M an ifolds. Now we again consider a general n-dimensional complex manifold X and assume that we have a strictly plurisubharmonic function in X with -+ 2 when X -+ ax. Then is called a special Ste in manifold. < Z ={ EX :

x

p( x) 1}

p

p(x)

We use similar methods to those in the last paragraph. Now let F be the trivial line bundle X x C on X and the strictly plurisubharmonic function p

6.

The Levi

289

Problem

on F be defined by

p(x, w ) :=p(x) + w w . The manifold Y = { (x , w) E F : x E Z and p(x, w ) < 1 ) is a strictly pseudoconvex tube around the zero section in F l z. However, Z is not compact now; it is only relatively compact in X . The tube extends to a neighborhood of Z. The boundary of the restriction to Z has an "edge" 8Y n (8Z x C).

But the proof of finiteness of cohomology goes through in the same way as if there were no edge. We have to apply the method of analytic tangents to 8Y and 8Z simultaneously. If V is a holomorphic vector bundle on X, we denote the pullback of V to F by 8. It then follows that dime H 1 (Yi z , 0) < oo. But in this situation all the tensor powers of F are again the trivial line bundle. We have V 0 p- m V for every m. From this we conclude the following: ""'

5 . 9 Theorem. If Z is a special Stein manifold (as described above) and V a holomorphic vector bundle on Z, then H 1 (Z, V ) = 0.

Exer cises 1 . Let E be a Frcchet space and let U U(O) c E be a neighborhood such _ that U is compact. Prove that E is finite-dimensional. 2. Let u : E --+ F be a continuous linear map between Frechet spaces. Prove that if u (E) is finite-dimensional, then u is compact. 3 . Let E be a Frkchet space. A subset M c E is called bounded if for every neighborhood U = U(O) c E there is an r0 E l!l1. such that M c r U for every r > r0 . Show that a compact set K c E is closed and bounded. Let X be a complex manifold. Show that every closed and bounded subset K c O (X) is compact. 4. Let X be an n-dimensional Stein manifold and Y c X a closed subman ifold of dimension n - 1 . Prove that the line bundle Nx (Y) is positive. 5. Let F = 0( - 1) be the tautological bundle over JID2 . Prove that F is negative and that =

·

6. Let X c lP'3 be a regular hypersurface and i : X injection. Prove that H 1 (X, i * O(k)) = 0 for k E Z.

6.

'--7

JID3 the canonical

T h e Levi P r ob lem

Enlarging: The I dea of the Pr oof. We wish to show that every strongly pseudoconvex ( i.e. , 1-convex) manifold is holomorphically convex, and that every strictly pseudoconvex (i.e. , 1-complete) manifold is Stein. This solves the Levi problem for complex manifolds.

V.

290

Stein Theory

Recall that an n-dimensional manifold X is called strongly pseudoconvex if there is a smooth exhaustion function p on X that is strictly plurisubharmonic > 1}. on EX

: p(x)

{x

Xs = { x

: p( x)

For s > 1 the open sublevel set < s} is holomorphically EX convex, and its cohomology-with coefficients in 0 is finite. We will construct an open set with X, c c X by extending X, in several steps such that every holomorphic function on X, can be approximated by holomorphic functions on

X

�

X. �

{x

: p(x) xi, Ui Ui (Uit UiA(xo) x0 Cri By disturbing p a little bit in a neighborhood of xi, we get a new function P2 and a bigger set X2 = {P2 2 < s} with Xs c X2 and E X2. We do this carefully enough to get X strongly pseudoconvex again. Since everything happens in a cuboid, we can apply the theory of Section V.4 and obtain the desired approximation property for functions in X = X, and X 2 . Then we repeat this process and construct a bigger set X 2 , and so on. N already cover Since X s is compact, finitely many set X i ' X 2 ' X 3 ' ' X Xs, and we order them such tha.!_Xi+ i is obtained bJ: enlarging Xi in a neighborhood of some point i E X 8 • Finally, X, CC X :=XN, and every holomorphic function on X, can be approximated by holomorphic functions In the first step we choose a point Xi E s } , a local coordinate EX system around a sufficiently small compact cuboid around x i in that coordinate system, and an open concentric cuboid cc such that every analytic tangent with E - X, is defined in =

Xi

i

.

•

•

X

�

on X .

Enlar ging: The Fir st Step . We choose an s > 1 . Then p is strictly and also on X - X,. We put plurisubharmonic in a neighborhood of s } and get EX

8Xs = {x

8Xs, 8Xs C 8Xs.

:p(x) = Assume that E axs is a point that is the origin in a local coordinate system, and that Ui = {z : lzl < r} is a compact cuboid in that coordinate system. Furthermore, let U1 Ui be a concentric open cuboid such that every analytic tangent A(x0) with x0 E Ui - X, is defined in Ui- On A(x0) we have p(x) > s outside x0, and there is an c > 0 with p (x) > s + c on 8Ui n A(xo). We assume that p is strictly plurisubharmonic on Ui6 . 1 P r op osition . We can add to p a function < 0 with support U i such Xi

cc

h

that h(xl) < 0, p + h is still strictly p lurisubha rmonic, and the following approximation property is sati5fied:

{x

: p(x)

+ h(:�;) < s}, then every holomorphic function (or If X : = E X cross section in a holomorphic vectorbundle) over X, can be approximated over (i. e . , on every compact subset of X, ) arbitrarily well by holomorphic functions (respectively holomorphic cross sections) over

Xs

X. �

6. The Levi Problem

29 1

PROOF: Assume that K c X, is compact. We choose an R » 0 and con sider the family A(x ) of analytic tangents to the points of U 1 n axs . There is a finite set of points yi , . . . , y:r,1 E U 1 n axs such that for the defining functions ft of A( yi ) and the corresponding vector g* = = ( 1/ fi , . . . , 1/ /:;. 1 ) we have the estimate

lg* ( x ) l := maxl 1/ft (x) l > R on aXs n U 1 . Now we replace the points Yi by points Yi very near to Yi , with p( yi ) > s, and the functions ft by fi (the defining function for A(yi)) such that we still have l g (x ) l > R on aXs n U l , for g := ( 1//I , . . . , 1 / /m 1 ) . We choose R and

m1

so big that

l g (x ) l

< R in K n Ui .

Now we add-a smooth function h with support U 1 that is strictly negative on U1 . We let X := {x E X : p(x) + h(x ) < s} and may assume that the points � � Yi are not contained in X and that l g (x )l > R on X - X . We can make h and its derivatives so small that p + h is still strictly plurisubharmonic in Ui (see Figure V .6). ,

�

X A (yJ )

u; Figur e

V. 6 .

The enlarging process

Next we take an R > R and add further functions fx with X E ax , sax fm1 + 1 , . . . , !m , such�that for g = ( 1 / /1 , . ;..: , 1 / fm ) it follows that l g (x) l > R at every point of ax n Ui , and l g (x ) l < R in K n Ui . Then we define

fJ = {x E X n Ui : lg (x) I < R }.

292

V.

We define

Stein Theory

U'

�

by replacing the number R by R in the first

m,

inequalities.

We use the enlarging method of Section V.4 and prove in the same '!ay that every holomorphic function in U' can be approximated by those in U. �

Now we let U and U' increase by making the number m biggcr (i .e., adding new functions) and, simultaneously, by also making R and ..R bigger with R < R. So we can construct sequences Uj , UJ where the U1 converge to Ui n X and the Uj to Ui n X, . Hence, we have the approximation of the holomorphic functions in X, n Ui by those in X n Ui . �

�

�

Let 1f/ be the open covering of X consisting of the two elements W, = X , and W2 = Ui n X . On the intersection W1 2 we can approximate every holomorphic function by those on W2 . Since the cohomology of X is finite, there are cocycles 6 , . . . , �k E Z1 (11/, 0) such that the corresponding cohomology classes � 1 , . . . , � k form a basis of the image of Z 1 (11/, 0) in H 1 (X, 0) . The mapping cp : c k X C0 (1f/, 0) -+ Z 1 ( 1f/ , 0) defined by �

�

cp(a , , . . . ,a k , rt)

:=

a , 6 + · · · + ak � k + J(ry)

is a surjective mapping of Frechet spaces. By the open mapping theorem (see [Ru74] ) cp is an open map.

If g is a holomorphic function in X, , then g can be approximated on W1 2 by a sequence of holomorphic functions fj E O(W2). Since cp is open, the cocycle g - !1 (which is a small function on W1 2) can be written as a sum fj +6(rtj ) , with a small function fj E O(W12) and a (small) cochain 'r/j · Replacing fj - fj by a suitable approximation f, and 'r/j by a suitable (and also small) cochain { g, , Q2 } , we get g - f = Qz - el, with Q1 E O (WI ) and Q2 E O(W2 ) - The function g ( x ) + Q1 (x ) �

g (x ) . .

_

{ f (x ) + Q2 (x )

is holomorphic on X and approximates g on X , as well as we want. �

•

Enlarging: The W.llole Pr oc ess . Let X 1 = X8 , p, = p , and h, = h. Then we define X 2 :=X and p2 := p + h. Repeating the process from above with a suitable point X2 E 8X 2 , cuboids U2 C C U2 , and a suitable function h2 we obtain a new set X = X3 that is defined by a new function p3 = p2 +h2 , and every holomorphic function on X 2 can be approximated arbitrarily well by holomorphic functions on X 3 . Continuing in this way we can always choose the correction functions hi so small that the strict plurisubharmonicity is never disturbed. After finitely many steps we obtain the following:

6.

The Levi Problem

293

There is a strictly pseudoconvex complex manifold X with X, c c X such that every holomorphic function o n X , can be approximated over X , arbitrarily well by holomorphic functions o n X . 6.2 Pr op o sition .

�

We have the analoguous result for cochains.

Assume that 1f/ is an open covering of X and that � E Z1 (1f/ , 0) is a cocycle with � X = 6(rts), � = J(ry) . Th en, by changing ry, the difference ry - rts can be made arbitrarily small o n X, .

6 . 3 Pr opo sition .

I

s

�

Remark. We can replace X by a set Xs +c:, with an c > 0, since such a manifold is contained in X. �

Solution of the Levi Pr ob lem . Let X be an n-dimensional strongly pseudoconvex manifold that as above is endowed with a smooth exhaustion function p that is strictly plurisubharmonic on {p > 1 ) . The solution of the Levi problem will be given in three steps. First we prove an approximation theorem for X , and X .

Eve ry holomorphic func tion in X, s > 1 , can be approxi mated by holomorphic functions in X . 6 . 4 Theorem.

PROOF: There is a maximal s with s < s < 00 such that every holomorphic function on X , can be approximated by those on Xs. We have to prove 8 = oo . If this were not the case, we could approximate each holomorphic function on X8 by those on Xs+c: with r:: > 0. This would imply the approximation of the holomorphic functions on X, by those on X8+c: and contradict maximality. • Assume now that K is a compact set in X, s > 1 . Then K is also contained in Xt for t > s. We denote the holomorphically convex hull of K in Xt by Kt. In the second step of our proof we show that Kt = K, for every t > s. �

�

�

�

It is always the case that Ks C Kt , and, due to the approximation results, Kt n X, = K8 • Suppose that there exists a t > s such that Kt is bigger than K8 • Then we can find a minimal t' > s with Kt c X t' · It follows that t' < t (since Xt is holomorphically convex), and there is at least one point xo E Kt n 8Xt' · We choose a very small number c > 0 and consider the set �

�

�

�

�

-

�

Xt' + c:-

Let A(x o) c Xt' + c: be the analytic tangent to Xt' at x0. We may assume that there is a neighborhood U = U(x0) and a holomorphic function g in U such that A(x0) = {x E U n Xt' + c: : g(x) = 0}, and we find a meromorphic function f on Xt' +c: with poles only on A(xo ).

294

V.

Stein Theory

xo

Then we construct a sequence of points � outside of X t' converging to and meromorphic functions on Xt ' + c: with poles only on This can be done in such a way that converges to f with respect to the Fubini metric in C (which will be described in Section V l . 3 ) . Then is holomorphic in a neighborhood of Xt' , takes arbitrarily large values, and supK i f stays bounded. By approximation we can fin<:!_ a holomorphic function f* on rt Kt . This is a contradiction. Xt such that > supK i f* l , i.e. ,

A(xv)·

fv fv l!v(xo) l

fv

l f*(xo) l

vl

xo

Let us give some more details for the construction of f and fv · To begin with J , we use the open covering '11 = {U1 , U2 } of Xt'+c: given by U1 = U n Xt'+c: and 1 u2 = Xt'+e - A(xo ) . Since dim H (Xt'+e) 0) < oo , there are complex numbers a! ' . . . ' am such that the function a == alg� 1 + . . . + arng- rn is a cocycle on ul 2 that is cohomologous to zero . Then a = h - !I , with !I E O(UI ) and h E O(U2 ) . So we get the meromorphic function f that is equal to a + f1 on U1 , and equal to h on U2 . It has A(xo) as po lar set. We can find a holomorphic family of analytic tangents A(z) , always given by one holomorphic equation 9z = 0, with z arbitrarily near to xo. If m is big enough, then 1 for every z we have a vector space Pz of principal p arts p = a1g-; + . . . + arng-;=

1

and a linear map 'Pz : Pz -+ H (Xt'+c: , O) . Let E, C Pz be the kernel of t.pz . Then dim(Ez) > 1 , and there is a minimal dimension mo < m for Ez that will occur at generic points. Taking E, only at generic points and forming the closure for z -+ xo , we get a regular holomorphic family of vector spaces. If we always take merom orphic functions fz with principal p art in Ez , we get a continuous family (fz) and the desired convergence fz -+ f for z -+ xo.

The last step of our proof is simple: Let K c X be any compact set. Then K is contained in some K, s > l , and from above we know that Kt = Ks for every t > s. Let x be an arbitrary point of the holomorphically convex hull K . Then E Xt , for some t > s . I ff is a holomorphic function in Xt , then there is a holomorphic approximation f E O(X ). Since we have < < supK i f l , we can conclude that supK as well. So x E Kt , and K = K, . This proves the following theorem. �

�

�

I !I

�

l[(x)l

�

x

l f(x) l

�

6 . 5 Theorem . If X is strongly pseudoconvex, then X is also holomorphi cally convex. It is a Ste in manifold if and only if there are n o compact analytic subsets in X ofpositive dimension.

In addition, we have the following result. If X conta ins a compact analytic subset A of positive dimension, then already A c X 1 .

6 . 6 Proposition .

�

�

The PROOF is the same as that for K = K1 . We also o b t ain t h e following

theorem.

6.

The Levi Problem

295

6 . 7 Theorem . If X is a strictly pseudoconvex (i. e., ]-complete) manifold and V a holomorphic vector bundle o n X , then V ) 0.

H 1 (X,

=

PROOF: This has already been proved for every distinguished Stein mani fold X, s > 1 . By approximation of coboundaries we get the result for X .

•

The C ompact C a se. Let X be a compact complex manifold, and V a holomorphic vector bundle on X . We can find a pair of finite open cc coverings '2/ , % * ) for X consisting of concentric cuboids such that V) 0 for all i. Then C0 ('P/, V), Z 1 ('P/, V) and V) Z ( '2/ * , V )are Fnkhet spaces. The finiteness of the coverings is essential here!

( 1 (Ui , H 1

=

Ui

H 1 (Ut,

Ut

=

Again we consider the linear mappings

such that u is surjective, v is completely continuous, and Im( u + v ) = B 1 ('P/, V). Since the coverings are acyclic, it follows from the theorem of Schwartz that dime ( X, V ) < oo.

H1 All restriction maps f(Ut, V) f(U, V) are completely continuous. From this it also follows that id f(X, V) f(X, V) is completely continuous, and f ( X, V ) is finite-dimensional. So we have the following theorem. :

-+

-+

6 . 8 Theorem . If X is a compact complex manifold, then dime

f(X, V ) <

oo

and

dime

H 1 (X, V ) <

oo .

Exercises

1 . Apply the results of this section to domains in en . What do strong and strict pseudoconvexity mean in this case? 2 . Let X be a strongly pseudoconvex complex manifold. Prove that there is a compact subset K c X such that every irreducible compact analytic subset A c X of positive dimension is contained in K . 3 . Let X be a Stein manifold and V a holomorphic vector bundle over X . Prove that for every E X there exists an open neighborhood = V ) such that for c X and holomorphic sections 6 , . . . , �N E Vx is generated by 6 . . . , �N every E

U(xo)

xo

x U,

(x),

(x).

f(X,

U

Chapter VI

K a h ler Manifolds 1.

Differ ential For m s

The Exter ior Algebr a. Let X be an n-dimensional complex manifold and x E X a point. We consider complex-valued alternating multilinear forms on the tangent space Tx (X) (abbreviated by T) .

W e assume tha t the reader is fam iliar with real multilinear algebra! Defin ition . A complex r-form (or r- dimensional diffe rentialfo rm ) at x is an alternating R-multilinear mapping cp : T x · · · X T --+ C .

r-times The set of all complex r-forms at

x

is denoted by F '.

Remarks 1. By convention, F0 = C. p i = F(T) is the complexification of the 2n dimensional real vector space r; = HomJR (T, R) . 2. Since T is (2n )-dimensional over l!l1., every alternating multilinear form on T with more than 2n arguments must be zero. So F = 0 for r > 2n. 3. In general, F is a complex vector space. We can represent an element cp E F uniquely in the form cp = Re( cp) + i Im( cp) , where Re( cp) and Im(cp) are real-valued r-forms at x . Then it follows directly that

dimc F

=

C;) .

4. We associate with each element cp E pr a complex-conjugate element cp E F by setting cp( V I , . . . , Vr ) : = cp( VI , . . . , vr ) . We have: (a) cp = Re(cp) - i Im( cp) . (b) Tp = cp. (c) ( cp + 'lj;) = cp + '!j;. (d) cp is real if and only if cp cp =

Now let cp E F and '!j; E ps be given. The wedge product cp A '!j; E pr + s is defined by

VI. Kahler Manifolds

298

Then cp A V' (v, w) = cp(v) 7/J(w) - cp(w) 7/J(v) for 1-forms cp, 7/J , and in general: ·

·

1 . cp A 7/J = ( - lt87/J A cp (anticommutativity) . 2. (cp A '1/J) A w = cp A (7/J A w) (associativity) .

In particular, cp A cp = 0 for every 1-form cp. We also write 1\ r F instead of pr . With the multiplication space

DO

"A"

the vector

2n

r

1\F := ffi l\ F = ffi r r=O r=O becomes a graded associative (noncommutative) C-algebra with 1 . It is called the exterior algebra at x . For the moment let Wj : = dzj and Wn+J generated by the elements IN,,

:=

d:Zj for j

=

1 , .. . , n. Then F is

A . . . A Wvr ' with 1 < VI < . . . < Vr < 2n.

The number of these elements is exactly (2rn ) . So they form a basis, and every cp E F has a uniquely determined representation

cp =

I
with complex coefficients a,, ... vr .

Forms of T yp e (p ,q) . Now we consider the influence of the complex structure. Definition . Let p, q E N0 and p + q form of type (p, q) if

=

r.

1 . 1 Prop osition. If cp E pr is a nonzero fo rm are uniquely determined.

A form cp E F is called a

d

type (p , q ) , thenp and q

PROOF : Suppose cp is of type (p, q) and of type (p ', q '). Since cp # 0 there exist tangent vectors VI , . . . , Vr such that cp( VI , . . . , Vr ) # 0. Then

1 . Differential Forms

cp(cv l , · . . , cvr )

{

=

299

p-q . cp ( V! , . . . , Vr ) , p -q . cp ( V1 , . . . V ) . r

C C

C

I

C

I

,

Therefore, cPcq = cP1 cq1 for each c E C . If c = ei t , then e i t (p - q) = e i t (p � - q� ) , for arbitrary t E R That can hold only when p - q = p' - q' . Since p + q = p' + q' = r, it follows that p =P and q = q '. • 1 . 2 Prop osition. 1.

If cp is of type (p, q) , then cp is of type (q, p) . 2. If cp, '!j; are both d type (p, q) , then cp + '!j; and A · cp (with A E C) are also of type (p, q) . 3. If cp is afo rm d type (p, q) and '!j; d type (p ', q ' ), then cp A '!j; is of type

(p + p' ' q + q' )

0

We leave the PRO O F to the reader. Example We have dzv (cv) = (cv) [zv] = c · (v[zv ]) So dzv is a form of type (1, 0) .

=

c dzv (v) , since Zv is holomorphic. ·

From az v = dz, it follows that dz v is a form of type (0, 1 ) .

Then dzi, A . · · A dzip A azj, A · · · A dzjq (with 1 < i1 < · · · < ip < n and 1 < j1 < . . . < jq < n ) is a form of type (p , q) . 1 . 3 Theorem. A n y r-fo rm cp has a uniquely determined representation

'P = 2:: cp(p.q) where cp (p,q) E pr are fo rms PROOF:

d

p+ q =r

type (p, q ) .

The existence of the desi red representation follows from the fact that the forms dzi, A . . . A dzip A dzj, A . . . A dzjq constitute a basis of F'. For the uniqueness let cp

Then

L

p +q=r

=

2::

'!j; (p,q) = 0

cp(p,q) =

2::

cp (p.q) ·

for '!j; (p , q) := cp(p , q) - cp (p,q) .

It follows that

0=

L

p + q =r

'!j; (p,q) (cv 1 , . . . , cvr ) =

L

cPcq · '!j;(P , q) (v 1 , . . . , vr) ·

300

VI.

Kahler Manifolds

For fixed (v 1 , . . . , vr ) we obtain a polynomial equation in the ring C [c, c] , and all coefficients '!j; (p, q) (v 1 , . . . ,vr ) must vanish. Since we can choose v 1 , . . . , v, • arbitrarily, we have cp (p,q ) = cp (p, q ) for all p, q.

B undies of Differential Forms. Let X be an n-dimensional complex manifold. We denote by F(X) the complexified cotangent bundle T* (X) ®C. It has the spaces F(Tx (X)) = HomJR (Tx (X), C) = Tx (X)* ® C of complex covariant tangent vectors as fibers, so it is a (topological) complex vector bundle of rank 2n.1 It even has a real-analytic structure, but not a complex analytic structure. If E is a (topological) complex vector bundle of rank m over X , then for 0 < r < m we can construct a bundle 1\ r E of rank (';) over X such that (1\r E)x = 1\ r (Ex ) for every x E X . If E is given by transition functions 9ij ,

then 1\r E is given by the matrices g2j·l whose entries are the (r of 9ij ·

x

r) minors

2

In particular we have the vector bundles 1\ r F(X) of rank ( n ) , with a real r analytic structure. Defin ition . An r-fo rm (or an r- dimensional differentialfo rm) on an open set U c X is a smooth section w E f(U, 1\ r F(X)). So an r-form w on U assigns to every x E U an r-form w, at x . If z1 , . . . , zn are local coordinates in a neighborhood of x, then Wj := dzj and wn +J : = dz1 form a basis ofthe 1-forms on this neighborhood, and there is a representation w, = where x

>--+

a i .. . ir ( x ) are smooth functions. ,

Henceforth, the set of all (smooth) r-forms on U will be denoted by szfr (U) , and the subset of all forms of type ( p , q) by szf (p, q) (U). If f is a smooth function on U, then its differential df E s4' 1 (U) is given by x >--+ ( df )x . In local coordinates we have

dJ =

n

n

v=l

v= l

2:: !zv dzv + 2:: hv azv .

A (smooth) vector field is a smooth section of the tangent bundle T(X) . So in local coordinates it can be written in the form

1

The tangent bundle T(X) is a complex-analytic vector bundle of rank n . Here we denote its r ea l dual bundle by T* (X) . It is a real-analytic bundle of rank 2n (over �). The complex dual bundle T' (X) of T(X) (with fiber T� (X) = Homc (Tx ( X ) IC) ) is a complex-analytic bundle of rank n over IC. ,

1 . Differential Forms

301

where the coefficients �v are smooth functions. Then we can apply df to such a vector field and obtain n

n

v=l

v=l

dJ( � ) = L �vfzv + L �Jzv · For any open set U the differential can be generalized to the Poincare map d = du : Jt1r (U) ---+ Jt1r+ I (U) in the following way:

If w =

""' /

a . · w· A �1

"'

. .

1-r

'L l

1
du (w)

...A

wir

is the basis representation in a

:=

One can show that this definition is independent of the choice of the local coordinates and that d has the following properties: 1 . If f is a smooth function, then df is the differential of f. 2 . d is @ -linear. 3. d 0 d = 0. 4. dv (w lv) = (duw)lv, for w E Jt1r (U) . 5. If cp E Jt1r (U) and 7/J E JZfs (U) , then c£..(_p A 7/J ) = dcp A 7/J + ( -1 r cp A d'ljJ . 6. d is a real operator; that is, dcp = dcp. In particular, dcp = d(Re cp) +

i d(Im cp) .

The differential dw is called the total derivative or exterior derivative of w . Now we consider the decomposition of an r-form into a sum of forms of type (p,q) . We use some abbreviations. If I = (i 1 , . . . ,i p ) and J = (j1 , . . . ,jq ) are multi-indices in increasing order, we write

instead of

ai, ... ip,j, . ..jqdzi,

A . . . A dzip A azj, A . . . A dzjq ·

So a general r-dimensional differential form w has the unique representation

w = L L aiJ dz1 A dzJ , p +q =r III=p I JI= q

and the

Poincare

differential of w is given by

302

VI.

Kahler M anifolds

dw =

L L

p+ q=r lll=p I J I= q

da i J A dz1 A d:ZJ .

If f is a smooth function, then df = a f + af, with n

n

v=l

v= l

Here af has type ( 1 , 0) , af has type (0, 1 ) , and df type ( 1 , 1 ) .

1 . 4 Prop o sition . If ep is a form of type (p, q) , then dep has a unique de .composition dep = aep + aep with a ( p + 1 , q) -form aep and a (p, q + 1) -form aep.

T

I, J

T

I . J

Then dep = aep + aep is the unique decomposition of the ( p + q + 1)-form dep • into forms of pure type. For general r-forms the derivatives with respect to z and z are defined in the obvious way, (In the French literature one writes d' ep instead of aep and d" ep instead of aep.) 1 . 5 Theorem. 1.

a and a are C-linear operators with d = a + 8 . 2. aa = 0, aa = 0, and aa + aa = 0. 3. a, a are n o t real. We have aep = aep and aep = aep. 4· If ep is an r-form and 7/J is arbitrary, then a( ep A 7/J) a(ep A 7/J)

11,

+ ( - 1 r ep A ·tj.; , aep A 7/J + ( - 1 r ep A a'ljJ. aep A

PROOF: It suffices to prove this for forms of pure type. Then the formulas can be easily derived from the corresponding formulas for d and the unique • ness of the decomposition into forms of type ( p q) . ,

Remark. Sometimes the operator d" := i(8 - a) will be used. Then deep = de ep, so d" is a real operator with dede = 0. We have dd" = 2 i aa. A smooth function f is holomorphic if and only if a f = 0. Correspondingly, it follows for a (p,O)-form ep = l.: I I I= a1 dz1 that aep = 0 if and only if all P coefficients a 1 are holomorphic. Hence we make the following definition:

2 . Dolbeault Theory

303

Definition . Let cp be a pform on the open set U c X . 1 . cp is called holomorphic if cp is of type ( p, 0) and 8cp 0. 2 . cp is called antiholomorphic if and only if cp is of type (O, p) and 8cp 0. The set of holomorphic pforms on U is denoted by f1P (U) . =

=

Clearly,

cp is antiholomorphic if and only if cp is holomorphic.

Exercises

1. Let X

be an n-dimensional complex manifold, f a smooth function on X , and w a smooth form of type (n - 1, n - 1 ) on X . Prove that

2. A real differential form of type (1, 1 ) is called positive if i w(v, v) > 0 for every tangent vector v -1- 0. Prove that i88llzll 2 is a positive form on e n . 3 . Consider the (n, n - I)-form ry0 on e n defined by

1]o

:=

n k ( - 1) n ( n- 1 )/ 2 L(-1) z k dz1 A · . . A dz, A dz 1 A · · .A di;., A · . · A az n k=l

and calculate 8ry0 . Let f be a holomorphic function on an open neigh borhood of the closed ball Br (O) , and w : (f(z)/ llzll 2n ) · ry0 . Show that dw 0 4. Let X = = lPm . On Uo : {(z0 : z1 : . . . : z,) E ]pm : zo -1- 0} we use the holomorphic coordinates t, zv / z0 . Then w0 dt 1 A · · · A dt, is a holomorphic n-form on U0 . Show that a holomorphic n-form on an open subset U c X is the same as a holomorphic section over U in the canonical bundle Kx . Prove that there is a meromorphic section s in Kx with s lua w0 and div(s) - (n + 1)Ho , where H0 {z0 = 0} is the hyperplane at infinity. =

=

.

=

=

=

2.

=

=

=

Dolb ea u lt Theor y

Integr ation of Differ ential Forms . We recall some results from real analysis. Let G c e n "' lll1. 2n be a domain. A closed subset M c G is called an r-dimensional submanifold of class � k if for each z E M there exists a neigh borhood U of z in G and a � k map f : U + lll1. 2n - r such that: 1. u n M r - 1 (0). 2. The rank of the (real) Jacobi matrix o ff is equal to 2n - r. =

VI. Kahler Manifolds

304

It follows from the (real) implicit function theorem that for any z E M there is a neighborhood U of z , an open set W c Jll1.r , and a � k map cp : W ---+ G such that:

1. cp maps W homeomorphically onto U. 2. The rank of the Jacobian matrix of cp is equal to r.

Then

cp

is called a local parametrization of M.

An r-dimensional differential form w = Lp+ q=r L I , J afJ dz1 A az J is called an r-form of class � k if all coefficients are functions of class k. Then dw is an ( r + 1 )-form of class �k - 1 . We always assume that k > 1 . (a

---+ B2 is a �k map between domains B1 c em and B2 c en , then every r-form w of class �k on B2 can be lifled back Lo B1 by

If

: B1

( * w)z ( v1 , . . . , vr )

W ( z) ( * v1 , . . . , * vr ) · If u 1 , . . . ,u 2n are real coordinates in B2 then * (L ai, ... ir dui , A . . · A duir ) L(a i, .. . ir o ) d(u i, o ) A " · A d(uir o ). :=

=

* w is again an r-form of class �k . We have: * (w1 A w2 ) = * w1 A * w2 . * ( dw) = d( * w) , in particular * (df) = d(f o (a).

Therefore,

1. 2.

Now let w = a dz 1 A . . . A dz , A az 1 A . . . A d:Zn be an arbitrary (2n)-form. Since dzi A azi = -2i dxi A dyi, it follows that

a · ( -l) n( n - 1 )/ 2 dz1 A az 1 A . . . A dzn A dz n n n 1 2 n dx 1 A dy 1 A · · · A dx, A dy n -2) i . ( )/ . ( a ( - l ) n +1 a · ( - lt ( n ) 1 2 (2i) n dx 1 A dy 1 A . . . A dx n A dy,. We write a := a · (-l) n( n+ 1 ll 2 (2�r · If B c en is a bounded domain and a continuous in a neighborhood of B, then w

l w l a(x1 + iy1 , :=

0 ° 0 ,

Xn + iyn ) dx 1 dY1

° 0 0

dx n dYn·

M c G is an r-dimensional submanifold of class �k , w an r-form of class �k , and K c M a compact subset that is contained in the range of a local parametrization cp : W ---+ M , then we define

If

r w jK

·=

1
The integral does not depend on the parametrization if we allow only changes of parameters with positive Jacobi determinant. If K cannot be covered by a single parametrization, then one uses a partition of unity.

Dolbeault Theory

2.

305

Finally we have the famous Stokes's theorem.

Let G c en be a bounded domain with smooth boundary and w a smooth ( n - 1 -form in a neighborhood of G. Then

2. 1 Stokes' s theorem .

)

1o dw.

r w=

lao

The Inh omogeneous C auchy For mula. We consider a generaliza tion of Cauchy 's integral formula in the !-dimensional case. Let G c c C be a domain with smooth boundary, and U = U(G) an open neighborhood of the closed domain.

Let f be a continuously differentiable (complex-valued)func tion on U. Then, for z E G, we have

2. 2 Theorem.

()

fz

�

{ f (() ( + � d ( 27rl }ao 27rl Z

=

1 (8f/-8()(Z ()

d( A d(.

(

0

PROOF: Let z E G be fixed, choose an r > 0 such that the disk Dr (z) lies relatively compactly in G, and define G, := G - Dr (z). The differential form

w(z)

�

:=

27rl

( z)

=

-� 27rl

·

By the theorem of Stokes we have

1Or w (z) d

If 0 < E < r and on Dr (z) ,

g ( ()

r

}A

Ac: ,r

e,r

(

-z

=

:=

r w(z)

Ja or

d(

V = V(Gr ) ,

is of class 'i&'1 in an open neighborhood ential is dw

!-(()z

(

·

(8!18()(( ) -z (

=

and its Poincare differ

d( A d(.

r w(z) - f

lao

Dr (z) - Dc: (z) ,

/':Dr ( z )

w(z).

then for any continuous function g

d( A d(

2 i lr r 21r g(z + (! e i t ) e - i t dt d(! . c: Jo ·

This integral exists and stays bounded

I dw(z ) Jo

=

I w(z) lao

if

E

·

tends to zero. So

- lim

I w(z) . -�'l };m_!z)

306

VI. Kahler Manifolds

In order to show that the limit on the right side is equal to

r

Ja or ( z )

z

) we estimate

1 f(z + re't ) i dt - -27r1 J f(z) dt 7rl 0 27r 1 - 1 ( J (z + reit ) - f(z)) dt 2 27r

1 -. 2

w(z) - f(z)

<

0

·

0

Jr

It follows that

f(

sup lf(z + re'' t ) - f(z) l ---+ 0 for r ---+ 0.

[0 , 27r]

f(z) = J80 w(z) - fc dw(z).

•

The 8-Equation in One V ariable. The Poincare lemma from real analysis can be formulated as follows:

Let G c e n be a star-shape d (e.g. , a convex) region, cp E and dcp 0. The n there exists a 7/J E dr - 1 (G) with d'ljJ

=

=

We want to prove a similar theorem for the one variable using the integral

JZ/T (G), 'P

r > 0,

·

8 operator. For this we start in

f E 'i&' 1 (G) . Let f E 'i&' 1 (C) be

for bounded functions

a func tion with compact support. Then there is a contin uous ly diffe rentiable function u on C with U z- = f .

2. 3 Theorem.

PROOF: Choose an R > 0 such that supp(f) cc D := DR(O) . Then u : Ch f is a continuously differentiable function on D, and we can assume that the integral is taken over the whole plane. It follows that

=

( ) = 21 '

_

U Z

7r

I

1 f (( + z) dl
I '>

':.

A

dl

'> •

and therefore (applying formulas for the derivative of parametric integrals)

u-z (z)

=

-

J 1a(,( ( + z ) d( d( a 1 ( 27rl � 1 8 ! ;a((() d( d( ( z 27rl

�

A

A

-

{ of jo( ( () A d( d(

� 2 7rl }D ( - Z f( z ) - � r (!(-()z d( 27rl laD

=

f( z ) ,

307

2 . Dolbeault Theory

since !laD

•

= 0.

f

E 'i&' 1 (G) a Let G cc e be a bounded domain, and boundedfunction. Then there exists a continuously differentiable fun c tion u on G with U-z = f.

2.4 C or ollar y.

PROOF:

Again we define u

:= Ch f .

Let z0 E G be an arbitrary point and D . There exists a smooth function (! on e with

1 . 0 < (! < 1 , 2 . Q ID = 1 , 3 . supp(Q) C C G.

f

We define fr == (! · and function on e with !r iD

Dr(z0 )

a disk with D C C G.

h := f -andfr .hThen /1 is a continuously differentiable a function on G with hiD = 0

= !ID, There exists a 'i&' 1 function u 1 on e with (u1 )-z = Jr .

is defined on G, and given by

.

The function u 2 :=

Chh

1 h ( ()

d( A d( . u 2 (z) = � 271" 1 G - D ( - z

D, and holomorphic in z, Obviously, u = u1 + u2 on G, and on D we have U-z = ( u 1 )-z = fr =f .

The integrand is continuous and bounded on G for z E D .

•

A Theorem of H artogs. Now we are able to prove the so-called Kugel satz, also known as Hartogs ' theorem: 2. 5

Theorem. Let G c en be a domain, n > 2, K c G a compact subset. If G - K is connected, any holomorphicfunc tion on G - K has a holomorphic extension ! o n G .

f

�

P ROOF : Choose an open neighborhood U = U(K) cc G and a smooth function (! on e n with supp(Q) cc G and (! lu = 1 . Define

9k : = f · UZlc ;;; , We have For fixed

z"

= ( z2 ,

ogk azz .

•

.

,

=f.

for k

EJ2(! 8zk azz

Zn ) we define

=

1,

. . . , n.

VI. K ahler Manifolds

308

Then U-z1 = 91 (we can apply the previous corollary, since support) , and for k > 2 we get

91

(9 1 ) zk (( , Z2 , . . . , Zn ) d( 1 2 7rl ( - Z1 � 1 (9k ) z1 (( , Z2 , . . . , Zn ) d ( 21r1 (-z

�

IC

1

c

has compact

1\

d(

1\

d(

9k (z1 , z2 , . . . , Zn ) · Therefore, u is a holomor For z rt supp (Q) we have U-zk ( z ) = 0, k = 1 phic function outside s upp (Q) . Since 91 ( (, z' ) = 0 for arbitrary ( and large II z' II, it follows that u ( z1 , z ') = 0 for large II z' I I , and so u is defined everywhere ,

.. . , n .

and is identically zero on the unbounded connected component Z of G - K.

There exists a nonempty open subset V of Z n (G - supp (Q)) C G - K . The function f := ( 1 - Q) . f + u is defined on G and coincides with f on V. In addition, we have �

in G - K, in So f

is

U.

holomorphic in G.

•

Dolb eault' s Lemma. Before provmg the 8 analogue of Poincare's lemma we need the following result.

Let P c c Q be polydisks around the origin in en . Let be a smooth June tion in Q that is holomorphic in zk + 1 , . . . , Zn , fo r some

2.6 P r op osition.

g

k>

1.

Then there exists a smooth function u Zk+ 1 , . . . , Zn such that u-zk = g on P .

in

Q that is again holomorphic

in

PROOF: There are disks E cc D around 0 in the zk -plane such that P = P' x E x P" and Q = Q' x D x Q". Let Q be a smooth nonnegative function on C with QI E = 1 and Q k - D = 0. Then we define

U ( Z1 , . . . , Zn ) . -_ _1_ .

.

2 7rl

1 Q(() · 9(z1 , · · · , Zk-- 1 , (, zk+ 1 , · · · , zn ) d1 C

o.,

( Zk

1\

dl o.,

The function u 1s defined on Q , and the integrand has compact support. Therefore, U-zk (z) = (Q . 9) ( z) = 9(z) on P, and by the rules for parametric integration we get u-z So u is holomorphic in these variables.

,

= 0 for i = k + 1 , . . .

,n.

•

2.

2. 7

en .

Dolbeault Theory

309

Dolb eault' s lemma. Let P c c Q be poly disks around the origin in For each :p E ( Q) with q > 0 and 8:p = 0 there exists a 7/J E with 87jJ = :P IP ·

a'p,q

JL1P. q-1 (P)

PROOF : Without loss of generality we may assume that p = 0. Let k < n be the smallest number such that :p does not involve the differen tials dzk + 1 , . . . , d:Zn . We carry out the proof by induction on k .

For k

= 0 there is nothing to prove,

since

q

> 0.

Now we assume that k > 1 and the theorem has already been proved for k - 1 . Then we write

:p = dzk A + (3, o

where If a = then

a

E J4'0 ·q- 1 (Q) and (3 E !4'0 •q (Q) are forms not involving azk ,

I: jJ I=q-1 aJazJ 0

--

(where

I:* means that azk > . . . , d:Zn do not occur),

-azk A 8a + 8(3 - L azk A azv A

= D:p

v =#

. . . , d:Zn .

(aJ)zv dzJ) + 8(3. ( I JL* I=q-1

= 0 for v > k. By the above proposition there Then it follows that are smooth functions in Q that are holomorphic in Z k +1 , . . . , Zn such that = on P . We define y : = Then

(aJ)zv AJ

(AJ )zk aJ

a,

=

I:jJ I=q-1 AJ azJ. L* L (AJ )zv azv A dzJ I J I=q-1 v !f-J L * aj azk A dzJ + L (AJ)zv azv A cLzJ v
(

)

dz k A a + ry, where ry does not involve dzk > . . . ,d:Zn . We see that also (3 - ry does not involve azk , · · · , dzn · Since (3 - ry = (:p -azk Aa) -(81-azk Aa) = :p -81 , we have 8((3-ry) = 8:p = 0, and the induction hypothesis implies that there is a form (}' E J4'0·q- 1 (P) with {)(}' = (/3 - rt ) IP · Setting 7/J := (}' + y , it follows that

87/J = ((3 - rt ) + (az k A a + rt ) = :p

on P.

This completes the proof. We immediately obtain the following result for manifolds X :

•

3 10

VI. Kahler

Manifolds

If U C X is an open set and cp E pf0·q (U) a smooth fo rm with q > 1 and acp = 0, then fo r every x E U there exists an open neighborhood V = V (x) C U and a fo rm 7/J E JZ1D,q - l (V) with a'ljJ = cp l v . Without proof we mention the following result, which provides more precise information (see [GrLi70] and [Li70] ) . 2.8 Theorem (1 . L ieb). Let G c c e n be a Levi convex domain with smooth boundary and w = L J aJ cLzJ a smooth (0, q) -form on G with aw = 0. Moreover, suppose that there is a real constant M > 0 with

llw ll

:=

ma J x suGp l aJ I < M .

The n there exists a constant k independent on G with a'ljJ = W and 11¢11 < k . M .

ci

w and afo rm 7/J

of

type ( O,q - 1)

Siu and Range generalized this theorem to domains with piecewise smooth boundary (see [RaSi] ) . L. Hormander has given a quite different proof for the solution of the 8-equation with L2 -estimates on pseudoconvex domains using methods of functional analysis (see [Hoe66] ) . In the meantime there exists a big industry of solving 8-equations, but we do not want to go here into detail.

Dolbeault Gr oup s . Let X be an n-dimensional complex manifold and U c X an open subset. Then we have natural inclusions2

Together with the Poincare differential d and the a-operator we get the fol lowing sequences of C-linear maps:

1 . The de R h a m sequence

If U is diffeomorphic to a star-shaped domain, then the sequence is exact, i.e., {f E d0 (U) : df = 0) = C and zr (U) := Ker ( d : Jd'r (U) ---+ Jt1r + 1 ( U)) is equal to Br(U) := Im ( d : dr- 1 (U) ---+ dr(U) ). In general, we have only Br(U) c zr ( U) (because d o d = 0) . 2 . The Dolbeault sequence

We define { cp E dp,q ( U) : acp = 0} for 0 < q < n , - a(dp,q - 1 (U)) for 1 < q < n and BP · 0 (U) Then Bp,q C Zp,q, because a o a = 0 (respectively 0 C SlP(U) ) . 2 Here C stands for the set of locally constant functions.

.·

0.

2.

Dolbeault Theory

311

Definition. The complex vector space Hp,q (U) := Zp,q(U)j Bp ,q(U) is called the Dolbeault (cohomology) group of type (p, q ) of the set U, and the complex vector space Hr(U) := zr (U)j Br(U) is called the rth de R ham (cohomology) group of U.

We have

> 1 the condition Hp, q (U) = 0 means that for every cp E dp , q (U) with flcp = 0 there is a 7/J E dp , q - l (U) with 87/J = cp. From the solution of the 6'-equation (using, for example, [Hoe66] , chapter IV , or Dolbeault's lemma and an approximation theorem due to Oka) we obtain that Hp,q (G) = 0 for every strictly pseudoconvex domain G c en and q > 1. Eor

q

If U is connected, then H 0 (U) = C. If U is star-shaped, then Hr (U) = 0 for r > 0. Now let n� == /'tT' (X) be the holomorphic vector bundle of holomorphic p-forms on X (in particular, 0� = Ox , 01: = T' (X) and flx = Kx ) . Then flP (U) = r ( U, n� ) for every open set U c X. We construct a linear map

For every � E Z1 (X , n� ) there is an open covering CZt = { Ui : i E I } of X such that � is given with respect to this covering. Then we can pass to any refinemenl. So we may assume Lhal CZt is as fine as we wanl.

Let � = ( �ij ) E Z1 ( ezt , 0� ) be given. Then �ij E flP ( Uij ) and �ij +�jk = �ik on Uijk· Let ( ek ) be a partition of unity for the covering CZt . We define differential forms 'Pi := L ek�ki E sz1P·0 (U; ) , for i E /.

kEf

'PJ - 'Pi = I > k ( �kj - �ki ) = I > k�ij = �ij · k

In

UiJ we have

acp1 - acpi = a�;1

k

= o.

Therefore, a global (p, 1 )-form w on X can be defined by w \ u, : = acpi. Now, dw = o. s o w E ZP·1 (X) .

If � is a coboundary, � = Jry for some ry E C0 (CZ? , O� ) , then 'rfJ - 'r/i = �,1 = 'PJ - 'Pi on Uij , and global (p,O)-form 7/J on X can be defined by 7/J\u, : = 'r/i - 'Pi· We get a

3 12

VI.

Kahler Manifolds

8( -7,/J) I ui = 8cpi - 8rti = w I u, ,

since 'Tfi is holomorphic.

D

So the assignment � >--+ w induces a map : H 1 (X , n� ) ---+ HP· 1 (X) by [�] >--+ {w], where [ . . . ] denotes the residue class. One can show that this definition is independent of the covering CZt .

2.9

D ol beau lt ' s

theorem .

isomo rphism.

The map D

:

H 1 (X, O�) ---+ HP· 1 (X ) is an

D

is injective, we PROOF: We use the notation from above. To show that suppose that there is a (p,O)-form (! on X with 8 (! = w. Then 8cpi = 8Qi ui , for every i E I. It follows that 8( 'Pi - (!) = 0 and therefore Ti := 'Pi - (! 1s holomorphic on ui 0

Then T = (7;) is a cochain of holomorphic pform s, with

So the class of � vanishes. Now let an element w E ZP· 1 (X ) be given. Since 8w = 0, we can find a covering CZt = {Ui : i E I } of small polydisks (in local coordinates) with (p, 0)-forms 'Pi such that 8cpi = wl ui . Then 8( 'Pi - 'Pi ) = 0 for every pair (i, j), and �ij := 'Pi - 'Pi lies in flP (Uij ) · Clearly, � = (�ij ) is an element of Z 1 (ezt , n� ) whose cohomology class is mapped by onto the class of w. This shows that is surjective. •

D

D

Dolbeault's theorem has an interesting consequence:

Let X be a Stein manifold. Ifcp is a (p, l ) -form on X with = 0, then there exists a fo rm 7,/J o n X of type (p ,O) with 87,/J = cp.

2. 1 0 Theorem.

8cp

PROOF: By Theorem B for Stein manifolds we have H 1 (X, n� ) = 0 for every p, and the result follows from Dolbeault's theorem. • Remark. Dolbeault's theorem is also true for forms of type (p, q) with q > 1 . For the proof one has to introduce higher cohomology groups H q (X, 0� ), to generalize Theorem B for those cohomology groups and to show that H q (X, 0�) "' Hp,q (X ) .

In an analogous way one can prove de Rham' s t h e or em :

where Hr (X, q denotes the cohomology group with values in the set of locally constant functions (isomorphic to the singular cohomology group with values in C ) . Let CZt be an open covering of X such that all intersections tractible. In the case r = 2 the de Rham isomorphism

u.o

. .

·•k

are con

2.

Dolbeault Theory

3 13

is given as follows: Let w E d2 (X) be given, with f3v E d1 (Uv) with

dw = 0 . On every Uv wluv = df3v ·

Then d(f3�" - f3v) = 0 on Uv w Therefore, on every (complex-valued) function fvl" with

Now,

there exists an element

Uv�" there exists a smooth

d(fl">. - fv>. + fvl") = 0 on Uvl">. · It follows that av J"A : = ( !!").. - !v>. + !vl") l uv,>.

is constant. So we have a = (av�">.) E Z2 ('1f , C) , and the de Rham class {w]E H2 (X) will be mapped onto the cohomology class {a] E H2 ('1£ , q. Exercises

1 . Let ry0 be the

( n ,n - 1 )-form defined in Exercise 1 .3, and

function on an open neighborhood of the closed ball

f a holomorphic B r (O). Prove that

(n - 1 ) ! r f(z) J (O) = (2 7ri ) n Ja sr ( O ) ll zll2 n . 'Tfo (integral formula of Bochner�Martinelli) . 2. Let (rv) be a monotone increasing sequence tending to r > 0, and let cp be a smooth (0, q )-form on p n (o, r) with q > 0 and 8cp = 0. Construct a sequence ( 7/Jv) of smooth (0, q - 1 ) -forms on p n (o, r) such that (a) 87/Jv = cp on p n (o, rv)· (b) 17/Jv +l - 7/Jv l < 2 � v on p n (O , rv � !) in the case q = l, and 7/Jv + l = 7/Jv On p n (O , Tv� I ) for q > 2. Show that Hp,q (P) = 0 for q > 1 and every polydisk P in en . 3. Let X be a complex manifold and 1r : E ---+ X a holomorphic vector bundle of rank q over X . If cp : Elu ---+ U x Cq is a trivialization, there are sections �v : U ---+ E defined by cp o �v(x) = (x, ev) · The q-tuple � = (6 , . , �q ) is called a frame for E . Use the concept of frames to define differential forms of type (p, q ) with values in E locally as q-tuples of forms. Let dp,q(U, E) be the space of (p, q ) -forms with values in E over U. Show that there is an analogue of the Dolbeault sequence for forms with values in E. .

.

3 14

3.

VI.

Kahler

Manifolds

Ka hler

Metrics

Her mitian metrics . Assume that X is an n-dimensional complex man ifold. Definition . A He rmitian metric H on X is an assignment of a positive definite Hermitian form Hx to each tangent space Tx (X ) such that locally in complex coordinates it can be written in the form n

Hx = L 9ij (x) dziazj ,

i ,j =l

with smooth coefficients 9·iJ

If a : [0, 1 ] ---+ X is a differentiable arc, we can assign to it a length

LH (a ) : =

1

1 JHa(t) ( a (t) , a (t)) dt,

where a (t) = a * a;at = 2::: " a� (t)a;azv+ I:" a� (t)a;az" and a = (a1 , . in local coordinates. So

. . , an)

The length is positive if a is not constant

K

ow, for a tangent vector � E Tx (X ) a norm can be defined by

1

Then LH (a) = f0 lla * 8j8tii H dt. There is a uniquely determined differen tiable 1-form a along a defined by

(

(J'a( t) (�) = Ha( t) � lla '

�� )

· H : It followsthat a * a = a d t, with a = a(a * 8j8t) = lla * 8j8t11H · Traditionally, a is called the line element and is denoted by ds, though it is not the differential of a function. We have

LH (a) a * Ha(t ) = lla* 8/8tl17:r dt 2 tian metric H by ds 2 . and

=

e a * ds

fo

=

I. ds, ,.

= (ds) 2 . Therefore, we also denote the Hermi

30 Kahler Metrics

315

Let CZt = (U,),E I be a (countable) open covering of X by local coordinate charts with coordinates (zL 0 0 0 , z� ) Then there exists a partition of unity (Q,) for CZt , and o

ds 2

n

L Q, 0 L dzt dzi

:=

LE I

i=l

is a Hermitian metric on X 0 Thus we have the following result.

3 0 1 Pr oposition 0 nco

0n

every complex manifold there exists a Hermiti an m e t

The Fund amental For m . In a local coordinate system we associate to our Hermitian form H = ds2 = Li ,J. 9ij dziazj the differential form t ,J

(In the literature one often uses w = i /2 0 L 9ij dZi A - i 0 Li ,j 9ijazi A dz, = W So w is a real ( 1 , 1)-formo

azj o)

We have w -

o

Let G = (% I i) = 1 , 0 0 0 ,n) be the (Hermitian) matrix of the coefficients of H, and e, rJ the coordinate vectors of two tangent vectors �, 'rf Then H ( �, rt ) = e c ry t o

o

.

o

If we apply a holomorphic coordinate transformation z = F (w) = (fr (w) , . , 0 , fn (w)) ,

we obtain

This can be written in matrix form as t G (w) = JF (w) 0 G (F (w)) 0 JF (w) ,

which is the transformation formula for Hermitian forms. Thus F*wH = WF· H 0 So the correspondence H +-+ w is independent of the coordinates. The form w is globally defined. In fact,

j _

o

-

(e

-

c

o

ry

t

2 1m ( H ( � ry)) ,

ry

-

o

c

o

-t

e

)

o

The form WH is called the fundamental fo rm of the metric H

VI.

316

Kahler Manifolds

Definition . A Kahle r me tric on X is a Hermitian metric H with dw H = 0. If X possesses a Kahler metric H, the pair ( X ,wH ) is called a Kahler manifold. In this case we also say that X is a Kahler manifold. In general, a Hermitian metric H does not satisfy the condition dwH = 0 . There are many complex manifolds that cannot have a Kahler metric. How ever, on a Riemann surface X every 3-form vanishes. So every Riemann sur face is a Kahler manifold.

Geodesic C oor dina tes. Assume that H is a Hermitian metric on the manifold X and that z = ( z l , . . . , zn ) are holomorphic coordinates around some point x0 E X . Definition . The coordinates z are called geodesic in x0 (with respect to H ) if H = "' L.t,J g'·J· dztazJ· and 1. 9ij ( x0 ) = 6ij , i.e. G ( xo ) is the unit matrix, 2. (9ij)zv ( xo) = O and (gij);Zv( xo) = O for v = l , .. . , n, i .e., ( dgij )x0 = 0. ·

·

We will show that the existence of geodesic coordinates is a characteristic property of Kahler metrics. For that we need a lemma. 3 .2 Lemma. H = Li ,J. 9ij dziazj is a Kahler me tric (9ij ) z,, = (9vj )zi

and

if

and only

(9ij hv = (giv )zJ , for all i, j,

if

v.

( The firs t equation implies the second one.) PROOF: dw

Let

w

= w H be the fundamental form of H. Then we have

( + ?= (9ij )z v azv A dzi A azj ) I · ( 2:) 2:) (9ij ) zv - (9uj)zJ dzv A dzi) A azj + 2::) 2:) (9iv )z1 - (9ij )zJ azv A d:Zj ) A dz; ) . v J I · ?:= (9ijk dzv A dzi A azj �,),V

't 1 ) 1 V

v
J

t

<

Thus dw = 0 if and only if (9ij k - (9vj )z, = 0 and (9iv )z1 - (gij )zv = 0 . 3 . 3 Prop osition. H is a Kahle r me tric coordinates around every point in X .

if

and only

if

•

there are geodesic

PROOF: We consider a fixed point xo E X. If geodesic coordinates exist around x0 , then all first derivatives of the 9ij are zero there, and dw = 0 at xo.

3 . Kahler Metrics

3 17

Assume now that H is a Kahler metric. We take a local coordinate system around x0 such that x0 is the origin of this coordinate system and apply a unitary transformation z = w · u t such that Ut . G (O) . U = En . We denote the new coordinates by z again and the new coefficient matrix of H also by G such that now G(O) = En o Finally, we apply a transformation

Z· - w · + -w · A,· · w t , 2 �

where the A; Then

-

�

1

k, l = 1 , . . . , n

(

)

;- 1 , . . . , n, .

are constant symmetric matrices.

So the Jacobian J of the transformation is given by J (w) = E n + L k A k · Wk , with

)·

1 ,. . . , n · = 1 ,. . . , n

i

=

The coefficient matrix G of H in the new coordinates is therefore given up to first order by �

G(w)

- J (w) t . G (z) . J ( w) En + L Akw k t · G(z) · En + L Azwz

(

k

)

(

l

)

G(z) + L A� . G(z)wk + L G(z) . A1wz. k

l

Since ( z; ) w1 (0) = 6;j , we get ( G o z) w>- (0) = Gz, (0) for all A. So Gw , (0) = G, ( 0 )+ A1 . The first derivatives of the functions g;j have to he zero at 0. We can achieve this by setting Since dw = 0, it follows that the matrices A, are in fact symmetric (use the • equations of the lemma) . So geodesic coordinates exist at x 0 .

Local P ot entials. We assume that H is a Kahler metric on X and that U is a polydisk in a local coordinate system. We have the fundamental form w = i . L i ,j g;j dz; A d:Zj of H with dw = 0. Then by the Poincare lemma we can construct a real 1-formcp with dcp = w in U. We have the decomposition 0 'P = 'P o + 'P I = L ; a, dz; + L; a; ah Comparing types, we get 8cpo = 8cpl =

318

VI. Kahler Manifolds

and ocp0 + ocpi in with oho

u

=

w.

Then, by Dolbeault's lemma, there are functions h0 , hi cpo and ohi = 'PI · It follows that

=

The operator i oo � d dc is real. So we can replace h * : i (ho - hr ) by the arithmetic mean h of h* and its conjugate. Therefore, i ooh = w . The real-valued function h is called a local potential of the Kahler metric H . It is strictly plurisubharmonic, because =

=

iooh = i L hz,zJ dzi A azj

t,j

is the fundamental form of a Hermitian metric; i.e. , the Levi form of h is positive definite. In the other direction, if local potentials h exist for a Hermitian metric H , then w iooh is the fundamental form to H, and dw 0. So H is a Kahler metric. =

=

Pluriharmoni c Functions . A real-valued function h on an open subset U c X is called pluriharmonic if ioo = 0. In this case the 1-form cp ah - ah is closed (i.e. , dcp 0) and purely imaginary (i.e., of the form cp i7,b, with a real form $) . In fact, we have cp - cp. =

=

=

=

If U is simply connected and g by g(x ) : I:o 1)- Then dg f := h + ig. It follows that =

x0

=

a fixed point of U, we can define a function 7,b = - i cp, so dg - i oh and og i oh. Let =

=

of = oh + iog = oh - oh = o, so f is holomorphic. Furthermore, df = of

=

2 dh.

The Fubini Metri c . The complex projective space Jpm is one of the most important examples of Kahler manifolds. Let

7r : ( Zo , . . . , Zn )

>-+

( Zo : . . . : Zn )

be the canonical projection from c n + I - {0} onto lPn . The unitary linear transformations LA (z) z . A t of te n + I (with A E U(n + l) , i.e. , A E GL n (C) and A t · A E,) give biholomorphic transformations cp 'PA : lPn --+ lPn , by cpA (7r(z)) 1r (LA(z)). The set of these is the group PU(n + l ) of projective unitary tramformations. If x0 , xi are arbitrary points in the projective space, then there exists a projective uni tary transformation cp with cp( xo) x i . =

=

=

=

=

On c n - I we have the canonical strictly plurisubharmonic function

3.

Kahler Metrics

319

which is positive outside of 0. It is invariant by uni tary linear transformations Of cn+ I .

x0 E lP'n is a point, we can find a simplyn+connected neighborhood U(x0) c l - {0} for the submersion and a holomorphic cross section f u

If

:

7r.

-+

The cross section is determined up to multiplication by a nowhere vanishing holomorphic function h. We use log(po f) (or equivalently any function log(po ( h f) ) ) as local potential for a Hermitian metric. Since .

i88 log(p o ( h . f) )

=

i88[log(p o f ) + log (h) + log(h)]

=

i88 log(p o !) ,

we obtain a well-defined Hermitian form H on lP'n . Its fundamental form is WH

=

i 88 log(p o !) .

We now show that H is positive definite. Everything is invariant under the group PU ( n + 1 ) .If A is a unitary matrix, = then , and f a holomorphic cross section in a neighborhood of 1 is a section in a neighborhood of S o p o f = (p o f ) o f := LA1 is a local potential of H near It follows that Hx0 (C,, rt) = Hx1 (cp * C,, cp * ry) . Therefore, it is sufficient to prove only the positive definiteness at the point = ( 1 : 0 : . . . : 0).

'PA (x0) x o f o 'PA

xr,

x0.

xa.

'PA

x0

At this point a holomorphic cross section is given by f : ( Zo : . . . : Zn )

>-+

Zn ) ( 1 , -Zr , . . . , . Zn

Zn

Therefore, we may take as potential the function

log(p o where t,

=

f(zo, . . . , Zn ))

log(1 + td1

tdr + . . . + tn fn )

+ . . .+

tnfn + terms of higher order,

z;jz0 are the (inhomogeneous) local coordinates at x0. Therefore,

C,rry 1 + · · · + C.n rtn , and this is positive definite. The metric H constructed in this way is called the Fubini me tric (or Fubini- Study me tric) on lP'n . Since H is given by local potentials, it is a Kahler metric.

H(C,, rt)

=

If X is a Kahle r manifold, then any closed submanifold X is also a Kahle r manifold

3 . 4 Prop osit ion .

YC

.

PROOF: We can restrict the given Kahler metric H on X to the submanifold Y. Then the restricted fundamental form wH I Y corresponds to the restricted Hermitian form H I y , which again is positive definite, and w II I y is also closed. So Y is a Kahler manifold again. • The following is an immediate consequence. 3.5

Theorem . Eve ry projective algebraic manifold

is

a Kahle r manifold.

320

VI. Kahler Manifolds Definit ion . Let X be an arbitrary n-dimensional complex manifold. We say that meromorphic functions !I , . . . , fk on X are analytically de pendent if

rkx ( fi , . . . , fk ) < k

at every point where each of the functions is holomorphic. They are called algebraically dependent if there exists a polynomial p(w1 , , w k ) that is not the zero polynomial such that p(fi , . . . . fn ) vanishes identically wherever it is defined. •

X is called a Moishezon manifold if it is compact and has independent meromorphic functions .

n

•

•

analytically

On compact manifolds algebraic and analytic dependence are equivalent no tions . In this case one can show that the field of all meromorphic functions on X has transcendence degree at most the dimension n of X . For proj ective manifolds there is equality. But a Moishezon manifold is in general not a Kahler manifold. Moishezon's theorem states that

A Moishezon manifold X is projective algebraic if and only if it is a Kahle r manifold (see [Moi67] , cf. also the article "Modifications" of Th . Petemell, VII.6.6 in [GrPeRe94] ). Deformations. There are many compact Kahler manifolds that are not proj ective algebraic. There are even examples of Kahler manifolds where every meromorphic function on them is constant. The Kahler property is invariant under small holomorphic deformations, but not under large ones (see [Hi62] , and also [GrPeRe94] , VII.6.5) . Definit ion . Assume that X is a connected (n + m )-dimensional com plex manifold, Y an m-dimensional complex manifold, and 1r : X ---+ Y a proper surj ective holomorphic map that has rank m at every point. Then ( X ,1r, Y ) is called a holomorphic family of compact n- dimensional

complex manifolds.

Indeed, since 1r is a proper submersion, the fibers Xy = 1r- 1(y) , y E Y, are compact submanifolds of X . When y varies, the fibers may have different complex structures . However, they are isomorphic as differentiable manifolds. 3 . 6 Theorem . If X0 is a Kahle r manifold and 7f : X ---+ Y is a deformation of X0 (i. e. , a holomorphic family with Xo = 1r- 1 (Yo) fo r some Yo E Y) , then there exists a neighborhood U ( yo) C Y such that every fiber over U is again a Kahle r manifold.

On the othe r hand, it m ay happen that allfibers Xy exceptfo r X, are Kahle r manifolds.

3.

Kahler Metrics

321

The first statement can be found in many books on Kahler manifolds (for example in [MoKo71] ). The original proof of the second statement was given by H. Hironaka. Exercises 1 . Let X be a complex manifold and E a holomorphic vector bundle over X . Use local frames as described in Exercise 2 . 3 . A He rmitianfo rm H on E is an assignment of a Hermitian form Hx to each fiber Ex such that (t;i I �j ) : = H(t;i, t;'j ) is smooth for every local frame t; = (6 , . . . ,t;'q ) · Let ht; be the matrix ht; : = ( ( t;i I t;'j ) ) . Calculate a transformation formula of ht; for changing from t; to another frame t;' . We call H a Hermitian scalar product (or a fiber metric) on E if Hx is positive definite for every x E X . In this case, prove that if x0 E X is fixed, then there is a local frame at x0 such that ht; (x0) = Eq and (8ht;)x 0 = 0. 2. For z E e n+ l {0} consider the linear map -

:

'Pz e n+ I

-+

T,,(z) (lP'n ) ,

which has been defined in Section IV.5. Show that the Fubini-Study metric H is given by H (cpz (v) , cpz (w)) = (v l w) . For z E e n+ l with llzll = l let w E Hz with llwll = 1 define

Hz := { w E en+ l

:

(z I w)

=

0) . For

a(t) :=7r((cos t)z + (sin t)w) . Prove that a is a closed p ath of length 1r . Using the fact that every geodesic on lP'n is of this form, prove the formula 1r

1r

dist ( (z) , ( w)) 3 . For decomposable k-vectors A

=

arctan

= a1

j (z w - z w)

(

A · . . A ak and B = b 1 A · . . A b k define

and IAI = (A I Ar . Prove that there is a Kahler form w on Gk, n locally given by w = i88 log/Aj. 4. Assume that X is a connected complex manifold and that A c X is a nowhere dense analytic set. Prove that if h is a bounded pluriharmonic function on X A then it can be extended as a pluriharmonic function to X . 5. Let B c e n be a closed ball. Is there a plurisubharmonic function h on e n B with limz-+B h(z) = 00 ? 6. A Kahler metric H on a complex manifold X is called complete if X is a complete metric space with respect to the associated distance function. Construct a complete Kahler metric on en {0}. •

-

-

"

,

-

-

322

4.

VI. Kahler Manifolds

The Inner P r o d u c t

The Volume Element. Let X be an n-dimensional complex manifold with a Hermitian metric H, and w = WH the associated fundamental form. For x0 E X it is possible to choose coordinates z1 , . . . , zn such that x0 is the origin and 9iJ ( xo ) = r5iJ · Therefore, Hx0 = Li dzi dzi · These are called Euclidean coordinates. We calculate the 2n-form w n in these coordinates at xo . We have

w =

i

. Lv dzv A dzv , and so v= l v= l

n

2"

·

( L dxv A dyv ) u=l

n

.

A simple induction on k shows that

(?n � dxv A dyv )

k

=

k!

�v <

1 < v, <

dXv1 A dyv1 A · · · A dxv, A dy v, .

k n

It follows that and

Remarks

1 . In general coordinates one would get wn

=

2n n! g dx1 A dy 1 A · · · A dx, A dyn ,

where g = det (giJ l i, j = l , .. . , n ) . 2. In the literature one finds other formulas:

� Lv dz, A dZv , then wn = n! dx 1 A dy1 A . . . A dxn A dYn ·

If w is defined by w

=

If, in addition, the wedge product is defined in such a way that cp A 7/J � (cp ® 7jJ - 7jJ ® cp ) for 1-forms cp and 7/J, then w = - � Im H and dx1 A dy1 A . . . A dx n A dy,

=

1

n!

( - 1 Im H) 2

n

=

4. The Inner

Defini tion.

Product

323

The 2n-form dV : =

;

;

w n = nl ( 2 lm H) n l 2 n 2 -

is called the volume element associated to H . The tradi tional notation "dV" does not mean that the volume element is the differential of some (2n - 1 )-form. It is easy to see that this is impossible, e.g. , on compact manifolds. Moreover. in the case of a compact Kahler manifold ( X ,w) it follows that fx wn = 271n1 fx dV > 0 .

4. 1 Pro p o s i tion. If (X, w) is a compact Kahler manifold, then w k defines a nonzero class in the de R h a m group H 2 k (X) fo r 1 < k < n = dim( X) .

PROOF: It is clear that d (w k ) = 0 . Suppose that there is some ( 2 k - 1)-form cp with dcp = W k . Then d( p A w n - k ) = w n , and by Stokes's theorem

This is a contradiction.

•

The S tar Oper ator . Let z 1 , . . , zn be Euclidean coordinates at x 0 E X . We use real coordinates in the following form: .

Then any r-form cp can be written as cp = 2: 1 a1 du1, where the summation is over all ascending sequences I = ( i 1 , . . . , ir ) , 1 < i 1 < · · · < i r < 2n. Definition. The (Hodge) star operator the ([>linear map defined by

where I U I' = ( 1 , . . . , 2n} , I n I' = 0, c J , I'

c J , l'

*

:

dr(X)

-+

JZ12n -r(X)

IS

= ± 1 , and

. du1 A dup = dV.

The star operator depends on the metric (which determines Euclidean coor dinates) and on the orientation (which is determined by the order of the real coordinates), but on nothing else. So it is globally defined and invariant under unitary coordinate transformations (in the sense that * ( cp o F ) = ( * cp) o F for these transformations).

324

4.2

VI. Kahler Manifolds

P r op o sition .

1 . If cp is a n r -form, then * * 'P = ( - 1 t · cp. dV if I = J, A duJ = 2. du1 * . o therwtse. 0 3 . * is real; i. e., * cp *'P ·

{

PROOF:

=

(1) We have * * du1 = cJ' , J CJ , J' du1, and ( ) C] ' , J C 1 , 1 ' ( - 1 t 2n - r dui A dUJ• cJ' ,1( - 1 t dV (- 1r dul ' A du1 .

(2) du1 A * duJ

=

q p du1 A dup

(3) We have * (Re cp + i Im cp)

=

=

0 if I n J'

i= 0 , and =

dV if J' = I'.

*(Re cp) + i *(Im cp) , by definition .

•

If cp = 2::::1 a1du1 and 7/J = I: J bJdUJ, then cp A * 7/J =

L a1 bJdu1 A * duJ = ( L a1b1 J dV.

In particular, this expression is symmetric in cp and 7/J . Now we define * ( cp) : = * cp. This is a complex antilinear isomorphism between dr (X ) and d 2n - r (X ). We get cp A * 7/J = (2:::: 1 a1b1) dV and cp A * cp = ( 2:::: 1 \ a J \ 2 ) dV .

The Effect on (p, q)-Forms. We calculate * cp for (p,q)-forms cp. 4 . 3 Lemma.

If cp

= dz1 A dzJ with III = p and IJI = q, then cp A * cp = 2PH dV.

=

PROOF: We write I M U A and J = M U B, with pairwise disjoint sets A . B , M c N := { 1 .. . , n } such that dz1 A dzJ = co dzA A dzB A WM where W!vf = IT!LEM dz, A dz!l and \ co l = 1.

,

,

Let m : = \ M \ . Then W!vf = ( -2 i ) m IT !LEM du 2!l - A du 2 w The remainder dzA A dzB is a form of type ( p - m, q - m) .

1

If, for example, A

dzA

= {a 1 , . . . , as} (with s = P - m ) , then (du 2a1 �

L

v =O

-1

jS-V

+ i du 2 a1 ) A · · . A (du 2 as - 1 + i du 2aJ

L

I C I =v

ac du a c A du a 'c '

4.

325

The Inner Product

where C c {1, . . . , s } is an ascending sequence of length I C \ , a c the sequence of the 2a; - 1 , i E C, and a� the sequence of the 2ai , i E C; ac is a number equal to ± 1 . So we have 2:51'- = o (�) = 2" summands. The analogue will be obtained for azB · Altogether we see that cp = dz1 A azJ is equal to 2m times 2PH - 2 m different monomials cK duK with \cK I 1 . So =

Again let cp = dz1 A dzJ with III = p, I J I = q and p + q ( 2n - r)-form, and it can be written as

= r.

•

Then

*

cp is a

v +fl=r I K I =n - v I L I = n- !l

with certain constants aKL· Now we apply the unitary transformation

C=

0 0

For I = { i 1 , . . . ,i p } c

N

= {1, .

.

.

F (z) = z · c t , with

ti E

l!l1. for i = 1, . . . , n.

, n } we set t1 := t;1 + ... + tip · Then

F*cp = exp(i (t1 - t.r)) . cp and

Since the star operator is invariant under unitary transformations, we have

It follows that * cp = exp(i (ti - tJ)) .

L

l: aKL . exp ( i (tK - tL)) azK A dz£. v + fl= r K, L

The uniqueness of the representation implies that

326

VI. Kahler Manifolds

Arbitrary t; are allowed. If we choose them all very small, then t1 + tK tJ + tL ·

If there is a pair ( K,L ) with I U K -1- J U L, then for example there is an i E I that is not contained in JU L. Setting t; -1- 0 and tJ = 0 for j -1- i gives a contradiction. Therefore, p + ( n - v ) = q + ( n - JL ). On the other hand, we have v + JL = p + Q. This is possible if and only if q = JL and p = v . * cp is a form of type ( n - q, n - p ) , and we can write

* cp =

L a K L dz K 1\ dzL ·

IKI

n-p L I I=n-q =

We consider an index i E I U J. There are three possibilities:

1 . If i E I and i tf. J , then i must lie in L, since we always have I UK = JUL. We choose t, = c (positive, but very small) and t1 = 0 for j -1- i , and we consider the equation t1 + tK = tJ + h . Since t1 = c , tK > O and tJ = 0, it follows that tL > E, and therefore even t L = E. But then tK = 0. This implies that i E L and i tf. K. 2. If i E J and i tf. I, then analogously i E K and i tf. L. 3 . If i E I n J , then cp contains the term dz; A az; and therefore the term du2; -1 1\ du2 ; - This implies that du 2; -1 1\ du2 ; (and therefore dz; 1\ az;) cannot occur in * cp.

So I U K = J U L = { 1 , . , n } and * cp = a rl:ZI' 1\ dzy , with some constant factor a . We may assume that E: J . J ' = EJ.J' = 1 . .

4.4 P r op osition . In the above notation

*(dzl (\ azJ) = a dz!' (\ azy PROOF:

with a = 2p+q- n i n ( - l ) q(n -p)+n( n +l )/2

For cp = dz1 1\ riZJ we have

dzl (\ azJ (\ (a dz!' (\ dzy ) a( - l ) q(n -p) dzl (\ dzl' (\ azJ (\ azy a( - l ) q(n-p)+n(n - l )/2 (dzl (\ azi ) (\ . . . (\ (dzn (\ d:Zn ) a( -l ) q(n-p)+n(n-1) / 2 ( -2i) n dV a( - l ) q(n-p)+n(n- 1) j2+n 2n i n dV. On the other hand, we know from the lemma that cp 1\ * cp = 2P+ q dV. Com paring the coefficients, it follows that a = 2p+q-n i -n ( -l ) q(n-p)+ n(n+l ) /2 . •

Remar k . The formula is valid only with respect to Euclidean coordinates. And one has to observe the rule that cJ . I' = EJ. J' = 1 .

327

4. The Inner Product

The Glob al Inner Pr odu ct . We work on a compact Hermitian man ifold X . Let % = {UI , . . . , UN } be a finite open covering of X with coordi nates, and ( r,h ) a partition of unity for % . If is any differentiable 2n-form on X , we define the integral

7/J

N

J 7/J : = vL= I J

v 'l/J· [! Uv

X

(In local coordinates Qv 'l/J has the form c(zi , . . . , zn ) dV with a differentiable function c with compact support. We already know the meaning of the inte gral over such a 2n-form.) One can show that this definition is independent of the choice of the partition of unity.

is an r-form, then 7/J r.p 1\ * r.p is 2n-form that locally is equal to a form r.p c dV, with c > 0 and c ( x ) = 0 if and only if 'Px 0. Definition . The inn er product of two r-forms r.p, 7/J on X is defined by (r.p , 7/J) : = i r.p i\ * 7/J .

If

a

=

=

4.5 Proposition.

) , ('PI , 7/J ) + ( 'P 2 , 7/J ) · 'P 7/J = 2 r.p,r.p) 7/J) =r.p'c · (r.p) , 7/J). ( . 7/J = 4 - ( r.p , r.p) > 0, and (r.p , r.p) = 0 if and only if r.p = 0 . 1.

('PI + 2. (c 3. ('t/J ' ·

PROOF:

( 1 ) and (2) are trivial.

7/J A * r.p 7/J 1\ * r.p = r.p 1\ * 7/J . 0 always. If r.p "I 0, then there is a point x0 (4) It is clear that ( r.p, r.p) X with 'Px o "I 0. If x o E Uv and Qv ( x0 ) > 0, then we can find a small neighborhood V = V ( x0 ) C C Uv such that Qv'P = c dV on V, with c > 0 everywhere in V . It follows easily that ( r.p , r.p) > 0. Since ( . . . , ) is a Hermitian scalar product on dr(X) , a norm on this space is defined by II r.p II : ( r.p , r.p) I 1 2 . 4 . 6 P r op osition . Twojo rms different type are orthogonal to each other. PROOF: Let r.p, 7/J be two forms of type (p, q) , respectively ( t ) , such that + n - t). If Then r.p 1\ * 7/J is a form of type ( p + n +t p +q s > p , then q > t and therefore q + n - t > n. If s < p , then p + n n. So r.p 1\ * 7/J 0 in these cases. =

(3) We have

E

>

•

. . .

=

s

=

=

=

r.

of

s, s, q

s >

•

328

VI. Kahler Manifolds

Unfortunately, dr (X) is not complete in the induced topology, so that we cannot apply Hilbert space methods directly. We will show how to construct a closure of dr(X).

Curr ents. Let X be a compact Hermitian manifold. A sequence 'P v = :E r a 1,v dn1 of (smooth) r-forms on X is said to be convergent to zero in dr (X) if for every a all sequences Dcxar ,v are uniformly convergent to zero. Definition. A current of degree 2n - r is an JR-linear map T : dr(X) --+ C such that if ( 'Pv) is a sequence of r-forms converging to zero, then T ( 'Pv) converges to zero in C.

The set of all currents of degree 2 n

-

T

Examples

7/J

is denoted by dr(X)'.

7/J

1 . A current of degree 2n is a distribution in the sense of L. Schwartz. 2. Let be a differential form of degree 2 n - r. Then defines a current T,p of degree 2 n - r by

T,p [r.p]

:= lx 7/J I\ r.p. r

It is easy to see that T,p is, in fact, a current. So .0"2n- r (X) C dr (X)' . 3. Let M C X be an r-dimensional differential submanifold. Then a current TM is defined by TM [r.p] := JM r.p. Therefore, we say that a current of degree 2 n - r has dimension r .

Let ( 'Pv) be a sequence in dr (X) converging to some r-form r.p (i.e., 'Pv - r.p --+ 0 ) . Then also *('Pv - r.p) --+ 0, and therefore T,p [*('P v - r.p)] --+ 0 in C for every 7j; E dr (X). It follows that

v-+ oo 7/J lim (

Definition.

,

'P v)

* 'Pv = lim T,p [ * 'Pv] 7/J 1 T,p [ * r.p] = 7/J * r.p = (7/J r.p) . 1 ' lim

V---7 00

{

X

For T E dr (X)' and

(7/J ,

1\

V ---7 00

1\

r.p E .0"2n - r(X)

r.p) . If In particular, we have (T,p , r.p) = limv-+oo ( 7/Jv , r.p). This motivates the following.

7/Jv --+

we define

7/J , then (T,p , r.p)

5. Hodge

Definition. limv-+ oo 7/Jv E

Decomposition

329

If (7/Jv) is a Cauchy sequence in ,0" 2n -r(X), then T = dr(X)' is defined by

(T , cp)

:= lim (7/Jv , V -+ 00

cp)

The limit T is called a square integrable current. We omit the proof that T is actually a current.

Before defining the notion of a limit of a sequence of currents we note that ( T, cp) - (7/Jv , cp )

T = vlim 7/Jv -+ oo

T[cp] - T,p Jcp]

....-.+

....-.+

0 for all

0 for all

cp E .0"2n-r (X)

cp E dr (X) .

Definition. A sequence (T, ) of currents in d r (X)' is called convergent to a current T E dr (X)' (written as T = limv-+oo T, ) if

lim

V -+ 00

(T[cp] - Tv [cp])

=

0 for all

cp E dr (X) .

We state the following without proof.

Th e space dr (X)' of currents of degree 2n - r is complete; i. e . , every Cauchy sequence converges. Th e set of square integrable currents -2n-r fo rms a complete subspace of dr (X)', which we call the closure d (X) of the space of (2n - r)-forms. 4 . 7 Theorem.

Exer cises 1 . Compute the volume of the complex projective space JIDn with respect to the Fubini metric, and the volume of a p-dimensional linear subspace. 2. Let (X, w ) be a compact Kahler manifold and Y c X a closed subman ifold with dim(Y) = m. Then vol(Y) = c . JY wm for some constant c. Calculate this constant! 3. Try to define the product of a current and a differential form! 4. For T E dr (X)' define dT E ,0" r+ l (X)' by dT[cp] : = ( - l y+1T[d!t?] · Show that dT is in fact a current with d(dT) = 0. Calculate dT for the case that T is a form or a submanifold.

5.

H od ge Decomp osition

Adjoint Op er ator s . A complex vector space with a Hermitian scalar product is called a unitary vector space. Let V1 , V2 be two unitary vector

330

VI.

Kahler Manifolds

spaces, and T : V1 ----+ V2 an operator (i.e., a (>linear mapping). If there exists an operator T* : V2 ----+ V1 with

(T ( ) , w) = ( , T* ( )) v

v

w

for v E

V1 , w E V2,

then T* is called an adjoint operator for T. It is clear that T* is uniquely determined, but in general it may not exist.

If V1 , V2 are unitary vector spaces and T : V1 ----+ V2 is a continuous operator, then it follows from the Cauchy-Schwarz inequality that v f--+ (T( v ) , w) is continuous for any fixed w E V2. If V1 , V2 are Hilbert spaces, then it follows by the Riesz representation theorem that there exists an element T* ( w ) with (T(v) , w) = ( v , T* (w) ) , and that w f--+ T*(w) is continuous. Now let X be an n-dimensional complex manifold

5.1

Lemma.

is compact and r.p a (2n- 1)-form on X , then fx dr.p = 0 .

If X

PROOF: Let %' = {Uv : v = 1 , . . , N} be a finite open covering of X by local coordinates and ( f2v) a subordinate partition of unity. We choose open subsets U� cc Uv with piecewise smooth boundary and supp(f2v) c U� . Then .

by Stokes's theorem. 5.2

The operator o := - Hi* dr+l (X) ----+ dr(X) is the d : dr(X) ----+ ,0"r+l (X). :

Propo sition.

adjoint of

PROOF:

r.p 1\

•

* 7/J

Let r.p E dr (X) , 7/J E is a (2n - 1)-form, and

,0"r+l (X)

be two arbitrary forms. Then

Therefore ,

(dr.p ' 7/J)

L dr.p i\ *7/J = L d(r.p i\ * 7/J ) - ( - 1 Y L r.p l\ rl* 'ljJ ( - 1 y+ I + 2n-r L r.p i\ * * M 'l/J = L r.p l\ *(* rf* 'l/J) ( r.p ' -* ri* 'l/J) . -

•

5 . Hodge Decomposition

331

5.3 Propo sition . _ Th e operator 6 := - * ch : ,0"P+ l . q (X) ....-.+ ,0"P. q (X) , re spectively {) := - * crt. : ,0"p,q+1 (X) + .0"p,q (X), is the adjoint of d , respec tively 8.

PROOF: We consider only the first case. Let r.p be a form of type (p ,q) with P + q = r and 7/J a form of type ( p + 1 , q) . Then r.p 1\ * 7/J is a form of type (n- 1,n) .Therefore, 8(r.p A * 'tP ) = 0 and

It follows that (8r.p ' 7/J)

fx ar.p A * 7/J = l d(r.p A * 7/J) - ( - 1 Y l r.p A 8(* 7/J) ( - 1r+ I +2n -r l r.p i\ * * 8 ( * 7/J ) = (r.p , - * fn 'l/J ) . •

5.4

l. 2. 3. 4-

We have

Propo sition.

o = rJ + rJ , oo = 0, {){) = 0 and {){) = 0, {){) + {){) = 0, rJr.p = rJ( r.p) and rJcp = rJ( r.p) .

The proofs are trivial.

The Kahlerian Case . Now let X be an n-dimensional Kahler manifold. For the moment X is not assumed to be compact. A canonical operator L : dr(X) ....-.+ ,0" r+2 (X) is defined by Lr.p :=

l 2 r.p 1\

w,

where w is the fundamental form. If cp is a (p,q)-form,then Lcp is a (p+ 1, q+ 1 ) form. Since dw = 0 and w is of type ( 1 ,1), we also have dw = ow = 0. It follows that 8Lr.p = -1 8(r.p 1\ w) = � 8r.p 1\ w = L8cp z ?

and

2

[)Lr.p =

In addition, we have

! o(r.p 1\ w) = � {)r.p 1\ w = Lor.p . 2

Lcp = Lip.

332

VI. Kahler Manifolds

5. 5

Proposition. If X is a compact (Kah ler) manifo ld, then the operator A : dr+2 (X) ....-.+ dr (X) ot L is given by

If '1/J is an ( r + 2)-form, then w A

PROOF:

( Lcp , '1/J )

* 'ljJ is a

(272

- r)-form, and

� k cp (\ w (\ * '1/J � fx ( - 1r cp (\ ** w =

_

adjoint

(\

* '1/J

(cp , (-lr * L * 'I/J) .

•

Note also that A is a real operator: Acp 5. 6

Proposition. hold.

=

Acp.

Th e commutativ ity relations {)A

=

A{) and {)A

=

A{)

We leave the proof to the reader.

Br acket Relations. Lel k, l be inlegers, and S : dr (X) ....-.+ dr+k (X), T : dr(X) ....-.+ ,0"r+ 1 (X) linear operators that are defined for every r. Then [S .T] := S o T - T o S : .0"r ( X )

....-.+

dr+ k+ l

is called the bracket or the commutator of S and T. Example

As we have seen above, 6, {) : dr ....-.+ dr- l and A : dr ....-.+ dr -2 have the brackets [rJ, A] = [rJ, A] = 0 (as operators from d ' to dr- 3 ). Before we calculate more brackets, we have to calculate {)cp very explicitely for a form cp of type (p, q) . For that we need geodesic coordinates. So everything that follows is valid only on Kahler manifolds. Let cp = I: 1 J aiJ dz1 1\ azJ be an arbitrary (p ,q)-form. We consider geodesic coordinates a t a point Xo E X such that w = i I:v dzv (\ azv . Then

* cp = L aiJ *(dz1 1\ azJ ) = a (p , q) · L aiJ dzl' 1\ azp , I,J !,1

where a (p , q) = 2p+q-n i n ( -l ) q (n-p)+n(n+ l ) / 2 , and C J , J' = C J ',J' = 1. In a neighborhood of x0 the formula is more complicated, because the coefficients 9iJ of the metric are involved. We have to apply a in this neighborhood, but because we are working with geodesic coordinates, we obtain the exact value at Xo by simply applying a tO OUr representation Of * cp at Xo . It follOWS that

a *

5.

Now we need some notation.

Hodge Decomposition

333

�

If J = (j 1 , . . . , jq ) , then J (v ) := (j1 , . . . , jv , . . . , jq ) for v = 1 , . . . , q. Furthermore, let I* and J* ( v ) be arrangements of I, respectively J ( v ) , such that

cJ '. I *

E (j" .J '). J * (v) 1. Then dzT * = (-1)P (n - p) dz1 and UzJ• (v) =

=

we get

=

(-1) n - q+ q (n - q) + v - l rzJ( v ) , and

q -a(p, q ) · a(n - p, n - q + 1) · ( - 1) n - p L L(aiJ )z1" dz1• 1\ az J* (v) I,J v= l a(p , q ) . a( n p, n - q + 1 ) . ( -1 ) n -p+p (n -p) + n -q + q (n - q ) q L L(-1t(aiJ)z1" dz1 1\ az J (v) I,J v=l q 2 · (-1)P L L(- 1 t (aiJ)z1" dz1 1\ az J (v) · I , J v=l -

x

5. 7

Theorem.

PROOF: with

0n

a compact Kahler manifold [ TJ, L]

In order to calculate TJLr.p and LTJr.p for a L r.p

tv 1\ r.p

(since r.p 1\ w

( � � dzA azA) A

=w

=

i 8.

(p, q )-form r.p

we begin

1\ r.p)

A

� (-1)P ) :

L aiJ dZA I\ dzJ I\ azA I\ azJ . I,J AEI ' n Jt

From that we get TJ Lr.p

with

=

i ( -1)P+ l L L (-1)P+ 1 (aiJ)z" dzA I\ dz1 1\ az J I,J A El'nJ' q v + i (-1)P+ l L L L( - 1)P+ + I (aiJ )z1" dzA l\ dz1 1\ azA 1\ az J(v) I,J A El'nJ' v= l i L L (aiJ)z " dzA 1\ dz1 1\ UzJ + r.po , I,J A El'nJ'

334

VI. Kahler Manifolds q

L(- l ) v (aiJ)z1 v dz>-. l\ dz1 1\ az>-. 1\ azJ(v) · I .J A EI 'nJ' v= l

cp 0 := i · L

L

On the other hand, .l

2

w

-

(\

rJ
( � dz>-. 1\ az>-.) 1\ (- l )P t; t (- l) v (aiJ ) z1v dz1 1\ azJ(v) i

Q

i LL

L

T . J v= l >-.E T'nJ(v )'

(- lt(aiJ ) z1 v dz>-. l\ dzJ I\ az>-. l\ dz J(v)

1

L L (- lt (- lt - 1 (aiJ )z1v dZjv 1\ dzl 1\ azJ I ,J q v + i L L L ( - l ) (ai J)z1v dz>-. l\ dzJ I\ az>-. l\ d:Z J (v) I,J A EI' nJ' v= l - i L L (aiJ )z> dz>-. 1\ dzi i\ azJ + 'Po · I,J A EI ' nJ

i

In

u=l

total we obtain {)Lcp - L{)cp

= i

L L (a1 J )z"

I .J >-. El'

dz>-. 1\ dz1 1\ dz J =

i acp . •

5. B C or ollary.

[ �] = - i a,

[a, A] = - i rJ

[a, A] = i rJ .

and

The proof is an easy exercise.

T he Laplacian . Let X be an n-dimensional compact complex manifold. Definition .

The (real) Laplacian A : dr(X ) -+ dr(X ) is defined by

The complex Laplacian

D

: dp,q (X) -+ dp,q(X) is defined by u p : = (o rJ + rJ IJ) cp .

Since d and 6 are real operators, A is also real. 5.9

Proposition.

On a compact Kahler manifold,

0

= D.

5.

PROOF: We have a {) = - {} a and a {) bracket relations. Therefore,

-i o

=

Hodge Decomposition

335

- {} a, as can be seen from the

- i (a rJ + rJ a) a( - i rJ) + ( - i rJ)a a(aA - Aa) + (aA - Aa)a.

From this one easily derives that

-

i D = i D.

•

A further consequence is

Therefore, operator.

D

io

= a(a A - Ad) + (a A - Aa)a = i (arJ + rJa) .

= arJ + {)a and

Dp

= (a{) + rJa)r.p

= D

5 . 10 P r op osition. On a compact Kahler manifold, PROOF:

r.p;

D =

i.e., D

is a real

� A.

We have A

do + od - (a + a)(rJ + rJ) + (6 + rJ)(a + a) (a rJ + rJ a) + (a rJ + rJ a) 2 0. •

This formula is not valid on general compact manifolds !

Har monic Forms. Let X be a compact Hermitian (not necessarily Kahler) complex manifold. We define

: - d.'Zf r - l (X), : - oJZfr+ I (X), : - {r.p E .'Zfr(X) : �r.p = 0}. The elements of yrer (X) are called harmonic forms. Example For calculating the star operator on (p , q)-forms one needs the number (p , q) = 2P+q-nin ( - 1) q (n-p)+n (n+l ) /2 . 1f n = 1 , then a(1, 0) = 1) = and a(1, 1) = - 2i. It follows that in geodesic coordinates

a

-i, a(O,

��

=

odf

- * d * Uz dz + !z az) - * d( ifz az - i fz dz) - *( 2i fzz dz A az) - 4fzz = - fxx - Jyy ·

i

336

VI. Kahler Manifolds

This motivates the name "harmonic. " 5. 1 1 P r op o sition . 1.

r.p E £r <¢==::} dr.p = or.p = o . 2. The vector spaces :36r, :_gr , and Jt"r are mutually orthogonal. PROOF: ( 1 ) If 6.. r.p = 0, then (r.p , dor.p) + (r.p , o dr.p) = 0 . But ( p , dor.p) ( or.p , or.p ) and ( r.p , o dr.p) = ( dr.p , dp ) . Therefore, dr.p = or.p = 0. (2) We have ( dp , o'ljJ) = (r.p , oo'ljJ) = 0 for all r.p, 'l/J. And if /:17/J = 0, then ( dp , 7/J ) = (r.p , o'ljJ) = 0, and analogously, ( o g , 7/J ) = r Q , d'ljJ) = 0 for all r.p, g . • So the spaces are mutually orthogonal. 5 . 1 2 Hodge theorem.

IfX

is a compact Herm itian manifold, then

We give only a sketch of the proof. For more details see [dRh84] , [Schw50] , or [GriHa78] , and for a very explicit and rather elementary proof see [War71].

PROOF:

( l )If we assume that .r4r is finite-dimensional, then the proof is very easy. We let V be the orthogonal complement of !J6r + :_gr. Then .r4r = :36r EB :_gr EB V, and it follows immediately that 6.. r.p = 0 for every r.p E V . (2) If .r4 r is infinite-dimensional, then we consider its closure, the Hilbert space 22 of square integrable currents. We can define derivatives of currents

by

aT

OUv

[r.pJ

:=

-

T

[ ar.p ] . OUv

So it is possible to speak of harmonic currents (i.e., currents T with t::..T = 0) . Now, the Laplacian is an elliptic operator, the definition of which we now recall. A linear differential operator P has locally (in real coordinates u 1 , . . . , u,) the form oo p = L aa ( u) Da : 'ifoo (U, lR M ) --+ 'if (U, lRN ) ' lal
ap ( u, � )

:=

L lal=k

aa ( u)

C : lR M

•

•

--+

lRN,

5.

Hodge Decomposition

337

which depends on � E JRn {0}. The operator P is called elliptic if ap (u, �) is injective for every u and every � -=/= 0. Since aL:> ( u , � ) = - I I � 11 2 id , it follows that A is elliptic. -

·

We state the r egu larity theorem :

Let P : dr (X) -+ dr(X) be an elliptic operator, and r.p an element of dr (X) . If there is a square integrable current T with P ( T) = r.p, then there is an r-form 7/J with T = T,p . If 11' is the orthogonal complement to !J6r EB �r in 22 , then simple Hilbert space theory implies that 22 = !J6r EB �r EB W, and the sum is orthogonal. Just as in the finite-dimensional case it follows that �(Y') = 0, and using approximation of currents by forms one sees that every harmonic current belongs to W. From the regularity theorem it follows that 11' = Jt"r .

Finally, one shows that if 7/J is an r-form and T,p = T + S + T'P , with a harmonic form r.p and currents T E !J6r , S E �r, then there are also forms T E 86r , CJ E � r such that 7/J = T + a + r.p. • 5 . 1 3 Theorem.

Jt"r (X) .

If X is a compact Herm itian m anifold, then Hr (X) =

PROOF: The de Rham group Hr (X) is the quotient of the closed forms modulo the exact forms. Let r.p E dr (X) be given with dr.p = 0. Then we have a unique decomposition r.p = d'ljJ + o Q + H p , with � ( Hr.p) = 0. Now, (og , og)

-

(r.p , og)

(because of the orthogonality) ( dp , Q ) = 0.

So r.p = d'ljJ + Hr.p , and we define h : Hr (X) -+ Jt"r (X) by h : [r.p] H Hr.p This is well defined, and since r.p = d'ljJ <¢==::} Hr.p = 0, it is injective. If a is harmonic, then d a = 0 and H a = a . So h is surjective. • .

Remark. By the theorem of de Rham we have Hr (X) "' Hr (X, C). There fore, f3r (X) := dimc (£r (X)) is a topological invariant. We have

the rth Betti number. In Chapter IV only not work with higher cohomology groups. 5 . 1 4 Theorem (Poincare duality).

b1

was introduced, since we did

On every n-dimension al compact

Herm itian m anifold we have Hr (X) "' H2n- r (X) .

338

VI. Kahler Manifolds

It is easy to see that % A = � * - Therefore, * : yrer (X) ....-.+ yre2n-r ( X ) is an isomorphism (depending on the metric) . Then we apply • the previous theorem. PROOF:

Remark. The original Poincare duality states that there is an isomorphism Hr (X, Z) ....-.+ H2n - r (X, Z) . Our theorem here is a very weak version of this topological result.

C on sequences .Let X be an n-dimensional compact Hermitian manifold. We define sub spaces of .'Zfp, q (X) by

!J6P,q �p,q yrep,q .

· -

· -

· -

We have Dr.p = 0 <¢==::} ar.p mutually orthogonal.

=

[JJZ(P,q-1 ' {)JZ(P, q+ l ' { r.p E JZfP, q : Dr.p = 0}. {) r.p

=

0, and therefore the three spaces are

5. 15 Decompo sit ion theorem for (p,q)-forms.

We have

The proof is the same as in the case of r-forms. One needs the ellipticity of 0 , which can be shown directly (and is trivially given on Kahler manifolds, since then D = ! �).

Hp,q (X) "' Yep,q (X) fo r all p , q. The proof is the same as for Hr(X). 5. 17 Finiten ess theorem . dimc Hp, q (X) < oo for all p , q. Using the theory of elliptic operators one can prove that yrep , q is finite 5. 16 Theorem.

dimensional. This implies our result. On the other hand, we know al ready from Dolbeault's theorem that HP· 0 (X) rv f!P(X) = H0 (X, n�) and HP· 1 (X) rv H 1 (X, !1�), where n� is the complex analytic bundle of holo morphic p-forms. By the results of Chapter V these cohomology groups are finite-dimensional. This can be generalized to higher cohomology. So we have two different ways to prove the finiteness theorem. 5 . 1 8 Serr e-Kodaira du ality theorem.

On an n-dimensional compact Hermitian manifold Hp, q (X) "' H n - p,n - q(X) for all p, q .

PROOF:

It is easy to see that * D

phism from yrep,q onto yren - p,n - q .

=

0

* Therefore, ·

* defines an isomor

•

As mentioned above, Poincare duality can be proved by topological means. This is not possible for the Serre-Kodaira duality.

5.

Hodge Decomposition

339

Now we consider applications to Kahler manifolds. 5. 19 Theorem.

1. £r (X)

PROOF:

=

If X is an n-dimensional compact Kahler manifo ld, then

EB

p+ q=r

£P, q (X) .

( 1 ) If r.p E £r , then there is a unique decomposition r.p =

2:.:

p+ q= r

r.p (p,q ) ,

with r.p (p, q ) E dp,q . It follows that o =

t::..r.p

=

2:.:

p+ q=r

t::..r.p (p,q)

If X is a Kahler manifold, then A = 2 0 . Therefore, f:1r.p (p ,q ) is a form of type (p,q). The uniqueness of the decomposition then implies that f:1r.p (p,q ) = 0 for all p, q. So r.p (p,q) E £p,q (X). (2) Since D is a real operator on Kahler manifolds, complex conjugation defines an isomorphism from £P,q to £q,p . • 5 .20 Decomp o sit ion theorem of Hodge-Kodair a. Kahler manifold and every r, Hr (X) 1n

rv

EB

p+ q =r

For every compact

Hp , q (X) .

particular,

The theorem of Hodge-Kodaira follows immediately from the decomposition theorem on harmonic forms. We define (3p,q (X) := dime HP,q (X) . Then we have the equations 1 . f3r = :Ep+ =r (3q, r , q 2. (3p,q = (3q, p , 3. (3p, q = f3n -p, n - q , 4 . f3r = f32n- r · 5 . 2 1 Corollary. PROOF :

We have

If r is odd, then f3r (X)

lS

even.

340

VI. Kahler Manifolds

f3r

=

I: (3p, q = I: f3p . q + I:

p+q= r

=2·

=r p+q > p r/2

=r p+q < p r/2

I: (3p,q ·

=r p+q < p r /2

•

5.22 Proposition. If L : dr (X) ----+ dr+2 (X) is the map given by L(cp) �cp i\ w, then Lr ( l ) E Jt'r,r.

PROOF:

We have a(U( l ))

=

Ua( l )

19(Lr ( l )) = (L19 + i a) U -1 (1)

=

=

0 , and

L19U -1 (1) + i u -1 a( l )

Repeating this, we finally have 19(U( l )) momc. 5.23 Corollary.

PROOF:

=

=

U19(1)

= 0.

=

L19U -1 (1) .

Thus U( l ) is har •

lf r is even, then f3r (X) -=/= 0.

We have

with c -=/= 0 for 0 < r < n . Therefore, Jt'r,r -=/= 0. Since Jt'r, r that f32 r (X) -=/= 0.

C

£2r , it follows •

This gives a topological condition for Kahler manifolds. Examples

1. Every Riemann surface X is a Kahler manifold. If X is compact , then g : = �h(X) is called the genus of X. For example, an elliptic curve (a !-dimensional torus) has genus 1, since its homology is generated by two independent cycles.

Now we have (31, 0 + (3o , 1 Le., g

2.

=

= b1 =

2g and (31, 0

dimc H 1 • 0 (X)

=

=

(3o 1 · So (31,0

dimc H 0 (X

,

,

=

f3o , 1

= g,

Kx ) .

The genus is the number of independent holomorphic 1-forms on X. Let H be an n-dimensional Hopf manifold, n > 2 . Then b 1 = 1 , as we have shown in Chapter IV. So H cannot be a Kahler manifold!

Exercises

1. Prove the commutativity relations 19A = A19 and 19A = A19. 2. Prove the bracket relations [19, L] = - i a, [a, A] = - i 19, and i 19'

[a, A]

=

6.

Hodge Manifolds

341

3. Let X be an n-dimensional compact complex manifold. Use the norm 1 11 4? 11 = (cp, cp) 1 2 on dr (X) and prove for a given cohomology class u E Hr(X) that among all representatives cp E dr (X) of u a representative cp0 has minimal norm if and only if J cp0 = 0. 4. Calculate Hp,q (X) for X = JIDn and all p, q .

6.

Hodge Manifolds

Assume that X is an n-dimensional compact complex manifold and that F is a holomorphic line bundle on X. We define a new notion of negativity for F. The new definition is formally more restrictive, and if we would generalize it to arbitrary vector bundles, it would, in fact, be different from our old notion. But it turns out that for line bundles the new negativity is equivalent to the old one. Negat ive Line Bundles .

Let a system of trivializations cp, : F l u ---+ U, x C ( with transition functions guJ be given. A fiber metric on F is given by a system h of positive smooth functions h, such that ,

Then a smooth function

X.h

:

F ---+ lR can be defined by

Definition. A line bundle F over a compact manifold X is called Griffiths negative if there exists a system of trivializations cp, : F l u ---+ U, x C (with transition functions guJ, and a fiber metric h = (h,) on F such that X.h is strictly plurisubharmonic on F - Zp (where ZF denotes ,

the zero section in F).

The line bundle F is Griffiths negative if and only if there exists a system of positive smooth functions [h on U, such that:

6. 1 Proposition. 1.

2.

- log [h is strictly plurisubharmonic on U, .

[h =

l gu, l · f2>< on U, " .

PROOF: Let h be any fiber metric on F and consider a point x0 E X. There is a trivialization cp at x0 such that h'P(x0) = 1 and all first derivatives of h'P vanish at x0 . Then h'P(x) · l w l 2 is strictly plurisubharmonic at every (x0, w ) with w =f. 0 if and only if h'P is strictly plurisubharmonic at xo . Taking such trivializations cp, we have only to set g, : = ( h J - 1 12 . •

VI.

342

Kahler Manifolds

The tube T : = { v E F : Xh ( v) < 1} is a strongly pseudoconvex neighborhood of the zero section in F. Locally, for T, := T n n-1 (U, ), we have The boundary of T is given by the equation ww / f2Z ( x)

=

1.

6.2 Proposition. The line bundle F is Griffiths negative if and only if F is negative (in the old sense).

PROOF: One direction is trivial. To show that every negative line bundle is Griffiths negative one needs rotations in the fibers and a smoothing proce • dure. For details see [Gr62] .

The line bundle F on X is called Griffiths positive if the complex dual F' is negative. Definition.

It is clear that F is Griffiths positive if and only it is positive in the old sense. We assume that F is a positive line bundle on a compact manifold X . Then F' is negative. We always denote the coordinate on the fibers of F' by w . Of course, the variable w depends on the local trivialization. Special Holomorphic Cross Sections.

The strongly pseudoconvex tube around the zero section of F' is locally given by T = {(z , w) E U x C l w l < Q (z ) } , where U is a domain in e n . From the last chapter we know that dime H1 (T, 0) oo . In this situation we prove the following result. :

6.3 Proposition. are valid:

If m is sufficiently large, the following two statements

1.

If x 1 , x 2 E X are two different points, then there are holomorphic cross sections s0, s1 E f(X, Fm ) with s0 ( x i ) -/= 0 and s 0 (x 2 ) -/= 0 such that the quotient sd s0 has different values at these points. 2. If x0 E X is a point, then there are holomorphic cross sections so , S l , .

·

·

,

S

m n E f(X, F )

with so (xo ) -/= 0 and det J( s i fs0 , . . . ,s n /so ) (xo ) -/= 0 .

PROOF: If s o , s1 are two sections in a line bundle with so ( xo ) i= 0, then there is a holomorphic function f in a neighborhood of xo such that s1 (x) = f ( x) s0 ( x) in that neighborhood. The quotient sd s o is defined to be equal t o f near xo. ·

6. Hodge Manifolds

343

We operate in several steps:

(1) Let x0

X be a point. We assume that in local coordinates x0 corresponds to the origin. We call a trivialization of F' around x0 usable if E

i,j in a neighborhood of 0. Here a is a smooth function and ative definite.

:E i . 9iJZdij is neg,J

The existence of a usable trivialization follows easily. We take an arbitrary trivialization and write log(g(z))

= ao + Re ( Q(z )) + L 9iJ ( z )zi Zj + llzll:� · a( z) ,

with a0 E lR and a quadratic polynomial Q(z ) with Q(z ) the trivialization by dividing w by exp (a0 + Q(z )).

= 0. Then we change

(2) Now let x1 , x 2 be two different points in X . We choose usable trivializa tions simultaneously over neighborhoods U1 (xi) and U2 (x2 ) , and, moreover, strictly increasing sequences 0 < r1 < r2 < · · · < 1 and 0 < s1 < s2 < · · · < 1 such that I s i/ri I of. 1 and { w = ri } n T, { w = si } n T are analytic sets in T, contained in F' I U1 (respectively in F' I U2 , cf. Figure VI . l ). We get

W=

Ti

-�>

s� �==s :�� / \

F'

W=

Si

X Figure VI. l .

Construction of special cross sections

principal parts of meromorphic functions 9i = 1/(1 - wjri) in T n (F' I UI ) and gi = 1/(1 - w/ri) in T n (F' I U2 ) , and hi = 1/(1 - wjsi) in T n (F' I U2 ) as well. Because of the finiteness of cohomology there are complex numbers ai that are not all zero such that there are meromorphic functions g, h in T with principal parts I;i ai(9i + gi ) and I; i ai (9i + hi)- We may assume that the first nonvanishing coefficient ai is equal to 1. The functions g, h are holomorphic in a neighborhood of the zero section , and locally they have power series expansions

344

VI.

Kahler Manifolds 00

00

g(x, w ) = L 9rn (x)wrn and h(x, w) = L hrn (x)wrn . m =O m =O The coefficients 9m (respectively hm ) define global holomorphic sections so,m (respectively s1,m ) in the tensor bundle F ". If over x1 the first nonvanishing part of g is aigi = 1/(1 - wjri), then we have

g(x1, w)

=

1 ( r� + Cm ) wm ,

with

Cm

-+

0 for m

-+ oo .

So for large m we have 9m(x1) >:::J 1/(rf'), and also 9m (x2) >:::J 1/(rf'). Analo gously, hm (xl) M 1/(rf') and hm (x2) >:::J 1/(sf' ). We take so = so, m and s1 = s1, m for such a large m. Then the quotient sl/ so at x1 has a value that is nearly l, and at x2 nearly (rdsi) m . These values are different for large m. ( 3) Now we work at some point x0 , and over a neighborhood U of xo we define g, = 1/(1 - w/ri) and hu (z , w) = 1/[1 - (w/ri) · (1 + Ezj)], for j = 1 ,. . . , n . If U and c are sufficiently small, the polar sets of 9i and hi,J are always analytic sets in T. As above, we obtain meromorphic functions g and hj in T, for j = 1 . , n, which have power series expansions g ( x, w) = :Em 9m ( x) w m and hj (x, w ) = :E m hj,m (x)w m in a neighborhood of the zero section. Again let g; be the first principal part with nonvanishing coefficient a, . For large m we have 9m (x0) >:::J (1/ri) m and hj,m(x0) >:::J (1/r;) m (l + m . Ezj)· As above, 9m and hj . m define sections so and Sj in p m with (sj/so) (xo) >:::J 1+ mEZj . So the Jacobian Of (sl/so, . . . , s n /so) at Xo is approximately mn cn and hence is • not zero for m large enough. ,

_

Pr ojective Em bed dings . Let X be an n-dimensional compact complex manifold and F a positive line bundle on X . Then for large m we have many global holomorphic cross sections in pm . If there are sections s0, . . . , s N that do not vanish simultaneously, then

<[>

:= :

(so (x) : . . . : sN(x))

X

-+

JIDn

6.4 Kodaira' s embedding theorem. For sufficiently large m, there are holomorphic c ross sections s0, . . . s N m pm such th at the m ap ,

<[> : X

is an embedding cl X in JIDN

-+

(So ( X) : . . · : S N (X ))

PROOF: In the preceding paragraph it was shown that for any point xo E X there is a holomorphic cross section so in some tensor power ci F that does not vanish at xo.

6 . Hodge

Manifolds

345

The set X 1 = {x E X : s0 (x) = 0 } is a proper analytic set. We decompose it into (finitely many) irreducible components, and for every irreducible compo nent we choose some point in that component and a holomorphic cross section that does not vanish at this point. In this way we obtain finitely many holo morphic cross sections s0 , . . . , sz whose joint zero set X2 has dimension n - 2 at most. We decompose this again into irreducible components and continue the procedure. After finitely many steps we obtain holomorphic cross sections s0 , . . . , sq in some tensor power p mo that do not vanish simultaneously. For any two points x1 -j. x2 E X there are two holomorphic cross sections to, h such that (to (xi) : h (xi)) are homogeneous coordinates of two different points. The set Y1 = { (x, y) E X x X : (t0 (x) : h (x)) = (t0 (y) : t1 (y) ) } has lower dimension. We decompose it into irreducible components and proceed a s above. After finitely many steps we get holomorphic cross sections to, . . . , tp such that x H (t0 (x) : . . . : tp (x)) is injective. For any point xo E X there are holomorphic cross sections u0 , u 1 , . . . ,ur such that u0 (x0 ) -1- 0 and the map x -+ (uo (x) : . . . : ur (x) ) has a nonvanishing Jacobian at x0 . Altogether we have holomorphic cross sections fo , . . . , fN that do not vanish simultaneously such that the map
Hodge M et rics . We consider an n-dimensional complex Kahler manifold X . Let H be the Kahler metric on X , and w =

WH

=

i L 9ij dZi A dzj t ,J

the associated fundamental form. It is a closed real form of type (1, 1). We choose an open covering %' = {Uv : v E N } of X such that all in tersections Uv, , ... ,vk are contractible. If "'/d' is of the same kind and a refine ment of % , and G is an abelian group, then the induced homomorphism H i (%', G) -+ H i (Y', G) is an isomorhism, for i = 0, 1, 2. We choose %' to be fine enough so that H has a potential h, in every Uv : w l uv

=

i aahv .

Here the hv are smooth real-valued functions, and the differences h,, hi" - h, are pluriharmonic in Uvw If we set f3v := - i ahv , then df3v ahvl" + ahvl" (respectively i dchvl" imaginary). The sum

w l uv . The closed form dhv�" = i aahv = ahvl" - ahvl" ) is real (respectively purely =

346

is

a

VI. Kahler Manifolds closed holomorphic form of type (1, 0). 9vl" (x)

:=

l

x

xo

If

x0 E Uv�" is a fixed point, then

i dchvl"

defines a purely imaginary function on Uvw It is determined up to a purely imaginary constant. Then !vi" : = h,, + 9v�" is a holomorphic function (since dfv�" = 2ahv�" ) . We have d(f�">-. - fv >-. + !vi") = 0, and therefore !�-">-. - fv >-. + fv�" E iR Then f = (fij ) E C 1 (%', 0) and o f E Z2 (%', i lR) . On the other hand,

(31" - f3v

=

1 -;- O hvl" I

=

1 -. dfvw 2I

This shows that (1/2 i )of defines a cohomology class c = c(H, %' ) E H 2 (%' , 1R) "' H 2 (X, lR) that corresponds to w under the de Rham isomor phism (cf. the description at the end of Section 2 ) . In the following we do not distinguish between H 2 (%' , Z) and its image in H 2 (92,R) . If a cohomology class of H 2 (92�) lies in H 2 (92f:,), then we call such a class an integral class. Definition . A Kahler metric H is called a Hodge m etric if it is integral. A Hodge manifold is a compact complex manifold that carries a Hodge metric. Clearly, every closed complex submanifold of a Hodge manifold is again a Hodge manifold. 6 . 5 Theorem. Let X be a compact complex manifold. Every Hodge metric on X defines via the de Rham isomorphism a cohomology class in H 2 (X, Z) that is the Chern class of a negative line bundle F on X . PROOF: Let w be the fundamental form of the Hodge metric. We take a finite contractible covering %' = {Uk : k = 1 , . . . , q} that is so fine that in every Uk we have a potential hk for w . As above, we construct the holo morphic functions fkz with dfkz = 2ahkl · Then o{fkz } E Z 2 ( %' , 2 i Z ) , and '/kl : = exp( nfkz) defines a cocycle '/ = ('/kz ) E Z 1 (%' , 0 * ) that determines a holomorphic line bundle F on X . If CfJk : F l uk -+ Uk x
:=

{ (x, w) E Uk

X

and get CfJ k o cp! 1 (Tz l uk, ) = Tk luk, . So we have a tube T around the zero section in F with T l uk = Tk . Since - log( exp( -7rhk )) = nhk is strictly pluri subharmonic, it follows that F is negative. •

6.

Hodge Manifolds

34 7

The above procedure can be reversed. 6.6 Theorem. Assume that X is a compact complex manifold, and F a line bundle o n X with a tube T around the zero section such that Tluk

= {(x, w) : l w l < exp(-nhk ) }

with strictly plurisubharmonic functions of a Hodge metric on X .

hk .

The n the

hk

are local potentials

All this finally implies that

Hodge manifolds are just projective algebraic manifolds.

This theorem was first proved by K . KoDAIRA. We shall see in the next section that it contains a generalization of the classical period relations for the n-dimensional complex torus. These were originally derived by completely different methods (see, e.g., [Ko54] ). Example On lP'n we have the Fubini metric H and consider the metric (1/(2n))H with

(

2 + . . . + Zn 2 zo hk = -l- log Zk Zk 21f as local potentials, for k

=

)

0, 1, . . . , n. It follows that

and !k z = !- log(zk /zz) is a holomorphic function with Re(fk z) = hkz · There fore, the transition functions of the associated line bundle F are the functions /'k l = exp( 1r fkz) = Zk j zz , and F is the (negative) tautological bundle. Exercises L

Assume that X is a compact Riemann surface and that x0 E X is a point. Show that the bundle [- x0] (associated to the divisor ( - 1) . xo ) is negative. 2. Compute the embedding of lP'n into JIDN defined by a basis of the vector space f(JIDn , 0( 2 )) and determine the image. 3 . Let X be a compact complex manifold and L -+ X a negative line bundle. Prove that f(X, L) = {0} . 4. Let X be a compact complex manifold and 1r : E -+ X a holomorphic vector bundle. Assume that there are trivializations 'PL : El u, -+ UL x
348

VI.

Kahler Manifolds

for every x E UL , and h " = gLt" . h, . g LK on ULK · Prove that there is a smooth nonnegative function Xh : E -+ lR with Prove that if x0 E U" ' then r.pL can be replaced by a trivialization '
r.p : E l u -+ U x eq is a trivialization and r.p( e ) = (x, v ) , then the tangential map r.p. : Te (E) -+ Tx (X) EB eq is an isomorphism. Denote the space (r.p.) - 1 (Tx (X)) by He (r.p) (space of "horizontal" tangent vectors at e with respect to r.p) . Prove that E is Griffiths positive if and only if there is a fiber metric h on E such that for every e E E - ZE with n (e) = x there is a normal trivialization r.p at x such th at Lev( Xh ) (e , � ) < 0 for � E He (r.p) , / -=f. 0 . (c) Prove that T(lPm) is Griffiths positive with the Fubini metric. (d) Let E be a Griffiths positive vector bundle and F C E a subbundle. Show that E j F is Griffiths positive. What does this mean for the normal bundle of a submanifold of pn ? 5 . Let X be a compact complex manifold and L -+ X a positive line bundle. Show that there is an open neighborhood U = U(ZE) C C E such that E - U is strongly pseudoconvex.

7.

Ap p lication s

Per iod Relation s. Recall the definition of the n-dimensional torus. We take 2n vectors wl ' . . . , W2n E en that are linearly independent over R The additive subgroup

acts freely on en by translations. The quotient space Tn = en jf is an n dimensional compact complex manifold and is called a torus. We say that W = (wf, . . . , wi ) E Mn , 2 n (C) is the period m atrix of T". n For z E en we denote its equivalence class in T" by [z] .By the quotient map 1r : en -+ Tn , the affine space en becomes the universal covering space of T ". Since 1r is locally biholomorphic, we can use the affine coordinates z1 , . . . , Zn of en as local coordinates on T ".

7 . Applications

349

Every torus has the structure of an abelian complex Lie group: [vJ + [w] := [v t w] . It is clear that this definition does not depend on the representatives of the equivalence classes. If W E e n is a fixed vector, the translation Tw : e n --+ e n is defined by Tw (z) : = z + w . It induces a biholomorphic map T" --+ Tn by [z] H [z t w] , which we also denote by Tw . It depends only on the equivalence class of w. From topology it is known that H2 (Tn , Z) is a free group spanned by the ( 2;') cycles Cij (s , t) := SW; + tWj , Q < S < 1 , 0 < t < 1, with i < j . Since T" is compact, every de Rham cohomology class [w] E H2 (Tn ) is uniquely determined by the periods Vij : =

j

Ci1

W,

i <j.

On en we have the Euclidean metric ds 2 = dz1 dz1 + . . . + dzn dzn , which is invariant under translations. It induces a Kahler metric H0 on yn with fundamental form n

i Wo =

L dzv

ZJ=l

1\

rlZv ·

It follows that every torus T" is a Kahler manifold. Using the Euclidean volume element, for every differentiable function f on yn we define the mean value M[f] by M[f] (z) : =

l

I

lwETn f o Tw (z) dV

with

I

:=

{ dV.

lrn

It is clear that M[f] is invariant under arbitrary translations and therefore a constant function. 7.1

Lemma. If

w

is the fundamental fo rm of some Kahler metric H on Tn , the n there is another Kahler metric H on T" such that the associated fundamental fo rm w = wH has constant coefficients and the same periods as w . If H is a Hodge metric, then H is also a Hodge metric. PRO O F:

]f w

=

i

� v ,,.. g,,.. dzv A rlZ,.. , then we define 9v,.. := M[gv,..] and W := i

E 9v i-L dzv 1\ ctz,... V ,fL

The form w has constant coefficients. Since the 9v,.. are periodic (i.e. , invariant under translations by elements of f), we have

Kahler Manifolds

VI.

350

Using Fubini we compute V ij

1C J

=

i

W

Now H is a Hodge metric if and only if wH defines an integral class. Since a cohomology class in H 2 (Tn) is uniquely determined by the periods, the • second assertion also follows. w;

W 1" th

= w1(i) , . . . , wn(i)) 10r • = 1 ,-

(

l

,

.. . , 2n, we have

and therefore v.!"

- 2Im (L 9v�" wS'1 w � 1 ) = v,!" for G

:=

- 2Im (w ;t . G

. wj ,

)

(9v�") .

Propos ition. Th e torus Tn zs a Hodge manifold if and only if all periods Vij are integral. 7.2

PROOF:

Since every element of Tn has a unique representative

= (u1 , . . . , u2n ) as real coordinates on Tn . We have a special contractible covering %' = {Uk} of rn . For k = (k1 , . . . , k2n ) with 0, 1} we define the open set we can use u

k; E {

Uk = {U E Tn :

; . k; <

U;

<

; . (k; + 1) } (see Figure VI.2) .

7. Applications

35 1

The boundaries of the Uk consist of parts of the "walls" SJ and SJ* ' which are defined by ·

{

Sj = U E yn

:

U� -

�}

and

Sj = { u E yn : Uj = 1 } .

Since 'P/ is contractible, the singular cohomology group H2 (Tn ) is isomorphic s1

s;

' Uu

Uo1 Uoo Figu r e VI . 2 .

u1 0

'

<E---

<E---

s,

s�

A special covering for the torus

Z for the to the cohomology group H 2 ('P/, Z). In particular, H 2 ('P/, Z) torus T1 , which has real dimension 2. The class [du 1 A du 2 ] is integral, and every integral class is an integral multiple of [du 1 A du 2 ]. If we denote by CP£ (n ) the covering of T ", then an easy induction argument shows that a cohomology class in H 2 (CP£ (n ) , Z) is integral (respectively real) if and only if it has the form :E v < ,.. a v ,.. d u' ;\ du,.. , with av,.. in z (respectively lR) : For any v < J.L the class [ du, A du,..] is integral. Assume that the case n - 1 has already been proved. Then yn = T1 x yn-1 , and after subtracting an integral sum of [du; A duj] with i < j < n the cohomology class is given by a covering CP£ ( 1 ) x yn-1 and is therefore a lifting of some integral multiple of du 2n-1 A du 2n · So =

[w] = L V;j [du; A duj] with Vij i< j

=

1

CiJ

w E JR,

and [ w]is integral if and only if all periods v;j are integral.

•

We call T" an abelian variety if T" is projective algebraic. 7.3 Propos ition. A torus T" is an abelian variety if and only if all pe riods are integral (with respect t o some Kahler fo rm w) . More precisely, we have the following result on period relations.

7.4 Theor em. Let T" be given by the period mat rix W ( w f , . . . , w in ) . Th e n yn is an abelian variety if and only if the re is a positive definite Her mitian mat rix G such that Im (Wt . G . W) is integral. =

352

VI.

Kahler Manifolds

The Siegel Up per H alfplane Defin ition . The (generalized) Siegel upper haifplane (of degree n ) is the n (n + 1)/2-dimensional domain Hn : = { Z = X + ..

•

Mn(C)

:

Zt

=

Z , and Y positive definite } .

7.5 Theorem. If Z E Hn , then W = (En , Z) E Mn , 2n (C) is the period m atrix of an abelian variety T"' . PROOF: Then

We take for G the positive definite symmetric real matrix .....

wt . G . w

It follows that

) ( En ) x t iY t + ( En x t + iY t

(

·

·

y - 1 (En , ·

(Y - 1 , y - 1

y- 1 x t y-1 + i En ·

Im(W t G W) ·

·

=

X -

·

X

iY) -

i En )

y - 1 X i En x t . y- 1 . x + Y

(�

·

- En n

-

)

l •

is integral, and the period relations are satisfied. A converse of the theorem is also true, but we do not give the proof here.

Semipositive Line B u n d les. We consider an n-dimensional compact

complex manifold X with a holomorphic line bundle F on X .

Defin ition . The bundle F is called seminegative if there is a tube T around the zero cross section such that T = { (z, w) : l w l < g(z)} in local coordinates and - ln(g) is everywhere plurisubharmonic and strictly plurisubharmonic at at least one point of X . The bundle F is called semipositive if the dual bundle F' is seminegative. We give an example of a seminegative bundle F with dim H1 (T, 0) = oo. The boundary function - ln ( &J) can be chosen to be strictly plurisubharmonic outside a nowhere dense analytic set in X . Here is the construction. We take a compact Riemann surface R of genus g > 1 and a negative line bundle L on R with H 1 (R, L ) -1- 0. The complex dual L' is positive, and the cross sections in L - m , m > 0, are holomorphic '

7. Applications

353

functions on L that are homogeneous polynomials of order m on the fibers on L. We add a point at infinity to each fiber of L and obtain a compact algebraic surface X . We denote by D the divisor of all points at infinity, and by F( m) the line bundle over X , associated to the divisor mD. Then all cross sections in L - m are sections in F(m) over X that vanish on the zero section Z = ZL c L c X of order m. These sections are holomorphic functions on the dual bundle F ( - m ) : = F(m ) ' over X (associated to the divisor -mD) that are linear on the fibers of F ( - m) ,but vanish over Z. Let 9 be the set of all these functions. The function f(x) - 1 on X is also a cross section in F(m) that vanishes on D to mth order, but it does not vanish on Z. It gives a holomorphic function on F( - m )that is linear on the fibers, vanishes over D , but does not vanish over Z. We denote it by h and add it to the system !7. Now, 9 is a set of holomorphic functions on F ( - m )that are linear on the fibers and whose joint zero set is the zero cross section of F ( - m ) . Forming the sum of several products f f, f E 8 .,we construct a plurisubhar monic function '
1. The zero set of 'ljJ is the zero cross section in F ( - m ) . 2. '
So our tube T { ( w , x) : 'z . =

The restriction F ( - m h is trivial. Therefore, the holomorphic functions on (F ( - m ) i z ) n T consist of certain convergent power series I:; a; (x )w " , where w is a coordinate on the fibers. All powers i appear. The first cohomology of Z with coefficients in these power series is infinite-dimensional, since it con tains the direct sum of the cohomology with coefficients in those holomorphic functions that are polynomials on the fibers with fixed degree i. We can ex tend these polynomials to F ( - m )by replacing w by h. If we restrict to these polynomials, we get an injection of the cohomology of Z with coefficients in the polynomials on the fibers into H1 (T, 0). So the cohomology H1 (T, 0) is likewise infinite-dimensional.

Moishezon Manifolds. We previously mentioned Moishezon manifolds (see Section VI . 3 ) . An n-dimensional compact complex manifold is called a Moishezon manifold if it has n analytically independent meromorphic func tions. Without proof we state here a result that was originally called the Grauert-Riemenschneider conjecture (cf. [GrRie70] ) . 7 . 6 Theorem. If a compact complex m anifold X has a semipositive line bundle, then it is a Moishezon manifold.

354

VI. Kahler Manifolds

The conjecture was proved by Y.T. Siu (see [Siu84] and the summary in [GrPeRe94] , Vll.6) . To obtain a result in the other direction we have to blow up along a d codimensional submanifold Y C X with d > 2 (cf. Section IV.7, "Monoidal Transformations"). This leads to a proper modification X of X , where Y is replaced by a 1-codimensional submanifold Y and the map Y -+ Y is a fiber bundle with typical fiber rd- l _ �

�

�

7.7 Theorem. If X is a Moishezon manifold, there is afinite sequence of monoidal transformations with d- codimensional centers with various d that lead to a proper modification X of X such that X is a projective algebraic �

variety.

This statement is proved in [Moi67] . We know that X carries a negative line bundle. By blowing down to X we get a "meromorphic" negative line bundle on X . Now over certain points of X there is a family of lines instead of one line. �

In the case that X is a surface every Moishezon manifold is projective alge braic (see, e.g. , [ChoKo52]). But already in dimension three there are Moishe zon manifolds that are not projective algebraic. The first example was given in Russia (the "Russian counterexample" or "the complex Sputnik," as it was called at Princeton in 1958), but see also [Nag58] or better just the classical example of Hironaka ([Hi60] ) , or [Pe93] . For a proof of the following important result see [Moi67] or [Pe86] . 7 . 8 Theorem . Assume that X is a Moishezon manifold which has a Kahler metric. Th e n X is a projective algebraic manifold. Exer cise s

1 . Prove that the torus T 2 given by the period matrix

W=

( 01

0 1

has only constant meromorphic functions. 2. Prove that the Siegel upper halfplane is contractible. 3 . For W E Mn , 2n ( 0 and T ""' T(En , Z) · 4. Find a compact complex surface X with a seminegative line bundle F such that F is not holomorphically convex. �

C h ap ter VI I

B ou n d ar y Beh avior 1.

Strongly P seu do con vex M anifold s

The Hilber t S p a ce. Assume that X is an n-dimensional complex man ifold that carries a (real analytic) Kahler metric ds 2 = "L, 9ijdzidZj , where the 9ij are real analytic functions of the local coordinates ZI , . . . , zn . We con sider a domain n C C X whose boundary an is 1&'00-smooth and strictly Levi convex. In this chapter such a domain is called strongly pseudoconvex. If x0 E X is a point, we can introduce local coordinates in a neighborhood of x0 such that G (9iJ (x0 )) is the identity matrix. We call them Euclidean coordinates in x0 . They are determined up to holomorphic coordinate trans formations that at x0 are given to first order by a unitary matrix. We have a Euclidean volume element dV in x0 , and also the complex star operator * given by =

r.p A * '
=

2p+q

.

(La IJ · b IJ ,

JJ !

dV,

for forms r.p "L, aiJ dz1 A azJ and '
by

\ r.p (xo ) I '
: = 2 p+ q

L ai J · biJ ·

This is invariant by unitary transformations and therefore independent of the choice of the Euclidean coordinates. Assume now that r.p and '
( r.p' '
L \r.p ( ) I '
X

=

I'd

'
This is a pre-Hilbert space that completes to the Hilbert space L� , L�, ( f!) q q of ( p ,q )-forms with square integrable coefficients. It contains the space ,0"P , q (f!) of 1&'00- (p, q)-forms over n as a dense subspace, and also the space .0"cf'q (!1) of such forms with compact support in R . Every element of L� , is q a square integrable current of degree p + q (or bidegree ( p ,q)) . =

Ope{ a tor s. First, we wall ex �end the operator a : dJ'' q (D) --+ dp, q+l (D) to a bigger domam dom( a) c L , · For th i s we use the fact that ti)ere is a pq uniquely determined adjoint operator {) : .0"J'' q + l (D) --+ .0"J' · q (D) such that

356

VII. Boundary Behavior

The proof for this statement is the same as in the case of compact Kahler manifolds. Now the existence of the extension follows easily. If r.p E L;.q is given, we + define the current ar.p by (ar.p, '<j;) (r.p, 1'J'<j;) for all '
Next we shall extend 19 to the conjugate operator a* on a dense subset dom(AJ C L; . q+l · Assume that '
( )

l (ar.p, '
*

where II'PII = J(r.p, r.p) and c > 0 is a constant that depends only on '<j;. Then - , '
d om (a *)

{ '
:=

:

*

a '
Next we consider the operator D = (}*""tJ +act that maps a form in L;. q onto a current of bidegree (p, q) over n. We define

dom(D) : {r.p E L;. q r.p E dom(a)ndom (a*) , ar.p E dom (a *) , a?p E dom(a) }. =

:

Then D maps dom(D) to L;_ q· For r.p, '
The space of harmonic fo rms is defined to be the space

£P. q := { r.p E dom(D) : D(r.p) = 0 } . , We get very simply that Jt"P q is closed in dom (D) and that D( dom(D)) is q orthogonal to Jt"P . .

2. Subelliptic Estimates

357

Boun dary Con dition s. We now consider forms r.p E dp, q (n). In this case all derivatives or.p, a* r.p and also those of higher order are square inte grable. So dom ( D ) n dp,q (n) consists of those forms r.p for which the equation (*) is valid for r.p and or.p. We assume that in a neigborhood U(on) there is a smooth real function r such that U n n {x E U : r(x) < 0} and that dr(x) -=j 0 for x E U. Then by a computation (partial integration) one has the following: =

A fo rm '
space of smooth differentialforms.

The following is then immediate. A fo rm r.p belongs to dom ( D ) if and only if the follow ing

1. 1 P r op osit ion .

two boundary conditions are satisfied on an : 1 . or /\ * r.p - 0. 2. or A * or.p - 0.

These conditions are called the o-Neumann boundary conditions. For liter ature see [DL8 1] and [FoKo72] . The main result of the next section is the following:

Assum e th at q > 0. Under th e hypotheses given above, th e harmonic fo rms of type (p, q) (i. e., elements of Jt'P,q) represent the elements cl th e cohomology group HP.q (n) . The first proof of this theorem was given in [Ko63] . E xercises

-

1. Give an explicit description of the operator o* . 2. Prove that the boundary conditions for the operator o and �oo functions are always satisfied. 3. Prove that the boundary conditions are never satisfied if q = n and the form is not identically zero on an. 4. Build a theory of harmonic forms for a "cohomology with compact sup port" HJ (n, 0). What are the boundary conditions?

2.

S u b ellip tic E stimates

Sob olev Sp aces. We look for an operator N : dp,q (n) -+ dp ,q (n) n d om ( D )

id. Finding such an operator is the so-called Ne u m ann such that D o N problem. To solve this problem we need Fourier transforms and the so-called subelliptic estimates. =

VII. Boundary Behavior

358

We begin by introducing the notion of a Sobolev space. We start with the vector space !/' of rapidly decreasing % -functions f : JRN -+
lx"'D13j(x)l -+ 0 for IJxiJ -+

oo

and for all o:, (J E NN

Here

We have the Fourier transfomation 3 : !/' -+ !/' given by

F[ f] (x) = f(x) :=

l (27r) N/ 2

J f(y)e- i (x , y) dy.

For every s E lR we define the norm I I · . - l is by

The Sobolev space of index s is defined as the completion Hs (JRN ) of !/' with respect to this norm. Now we work in the space JRN + l with coordinates ( x,r ) = (x 1 , . . . , x , r) , N and consider the half-space JRN+l = { (x, r ) : r < 0} and 'i'00 functions f (x, r ) with compact support in JRN+l _ Then for fixed r these functions are in 9 We take the Fourier transforms with respect to x , for fixed r , and define

Ill Jill: =

r

J - 00

l l f( . . . ' r) ll ; dr.

This norm is called the tangential Sobolev norm. Now let us return to our strongly pseudoconvex domain n c c X . We put N = 2 n - 1, and take any Xo E an and a neighborhood U (xo ) c X with local coordinates Z 1 = X 1 + ix 2 , · . . , Zn- 1 = X 2n- 3 + iX 2n -2 , Zn

=

X 2n- l

+ ir.

Here r is the boundary function for an as in the preceding section. In the following we will work only with respect to these local coordinates. If f E -�?' 0 (Un n), then f is % · · -smooth and has compact support in JRN+I . Hence, we can define the norm 111 1 111; . For r.p E d"rf' q (U n n) we put 1 / r.p //1 ; = ""L, f 1/ l f ll/ ; , where f runs through the coefficients of r.p. Definition . The Neumann problem for ( p, q)-forms is called subelliptic on n if there is an E > 0 SUCh that for every Xo E an there is a COOrdinate neighborhood U(x0 ) and a constant c > 0 such that

2. Subelliptic Estimates

359

1 1 4?11 ; < c . Q ( cp , cp) -* ) . for all cp E d'rf• q (U n 0) n dom(o

-

-

81P) +( o*cp, o*1P) is the so-called energy form Here Q (cp , '
2. 1 Theorem. If 0 is strongly pseudoconvex, the Ne um ann problem is subelliptic fo r all (p,q)-forms with q > 0 . The proof of this theorem is rather difficult, of course. The so-called Kohn Morrey inequality is involved. See [DL8 1], for instance.

The Neumann Oper a tor . There is an old theory on elliptic bound ary problems, beginning with Dirichlet's problem (see the old papers of L . Bers and L . Nirenberg, for instance). Here the Gdrding inequality plays an important role. If it holds for a problem, existence and regularity of solutions of the problem can be proved. In our terminology it can be expressed in the form

ll4?lli < c . Q (cp, cp ) .

If c < 1, then the inequality for subellipticity is much weaker than the Gdrding inequality. But nevertheless, in the papers of Kohn and Nirenberg it was proved that there are results on the smoothness of the solution of the Neumann problem that are similar to those in the old theory and that follow from subellipticity alone. We state a first important result (see [DL81] ): 2 . 2 P r op o sition . Assume that 0 is strongly pseudoconvex and th at the Ne um ann problem is subellipticfo r (p,q)-forms on R . Th en there is a unique operator N : L; , -+ dom(D) with the following properties: q 1 . N(Yt"p,q ) = 0. 2. D o N = id . 3. N(L; ) is orthogonal to Yep , q ,q This operator N is called the Ne umann operator. Under the above conditions the following theorem is true: 2 . 3 Theorem.

1. Th e harmonic space Yep, q isfinite-dimensional. It is contained in d'p, q (0) and fo r cp E £p, q we have or 1\ *

VII. Boundary Behavior

360

For a proof see [KoNi65] and [FoKo72] . Both papers are difficult to read. In the case X = e n there is a simpler proof. A domain !1 c c en with smooth boundary is called regular if there exists a finite number of subsets si c an, , N , such that i = 0, . .

.

1. 0

= SN C SN - 1 C · · · C S1 c So = Of!. 2. If z E Si , but z (j. Si+ 1 , then there are a neighborhood U(z) and a

differentiable submanifold M c U n 8!1 with holomorphic dimension equal to zero such that Si n U c M •

If n is pseudoconvex and regular, then a certain compactness estimate holds for the Neumann problem for ( p,q)-forms with q > 1 (cf. [Ca84] ) . Still using a result of KohnJNirenberg (see [KoNi65] ) , from the compactness estimate the existence and the compactness of the Neumann operator N follow. See also [BoStr99] for further discussion. If n is an arbitrary pseudoconvex domain, q the Neumann operator exists as a continuous linear map L� , -+ dom (D). This is proved in [Ca83] . A strongly pseudoconvex domain n c en is always regular, and therefore the above theorem on the Neumann operator holds in this case. But in en the harmonic space Jt'P . q is always equal to zero.

Real-Analytic Boun daries. We assume again that X is a complex manifold, but we consider only a domain n with real-analytically smooth boundary. Here we only assume that n is pseudoconvex. Then the following holds: 2 .4 Proposition. Th e Ne umann problem is subelliptic on Rfo r (p,q)-forms with q > 1 if and only if all germs of local complex-analytic sets A c an have dimension less than q . Moreover, we have the following result: 2.5 P r o p osition. Assume that n c c X = en . Th e n there is no germ of a local complex-analytic set A c aG ci positive dimension. For a proof see [DiFo78] and [Ko79] .

E xam ples . We will construct two 2-dimensional examples n cc X ,where n contains just one compact irreducible curve Y. From the existence of Y it will follow that the cohomology group H 1 (!1, 0) is different from zero. The cohomology classes are given by unique harmonic forms that satisfy the boundary conditions. We can study these as if they were functions, which may be useful for our investigations. Unfortunately, the forms in £0 • 1 are not in general d-closed. So there is no connection between £0 • 1 and the cohomology with coefficients in e as in the case of compact Kahler manifolds. We will see this more precisely in the examples.

2. Subelliptic Estimates

36 1

Example 1

We take a compact Riemann surface Y of genus g > 1 and a line bundle X on Y with Chern class - 1 . This line bundle is negative. There is a real analytic metric on the fibers of X such that n = { E X : ll x ll < 1 } is a strongly pseudoconvex relatively compact subdomain of X with real analytically smooth boundary. We get a uniquely determined Kahler metric on X by using the metric of constant curvature on Y and the metric on the fibers and requiring that the surfaces { x E X : ll x ll < r} be orthogonal to the fibers.

Over Y we have the vector bundle Oi of homogeneous polynomials cf degree i on the fibers of X . If i is sufficiently large, then H 1 (Y, Oi) vanishes. So the finile-dimensional cohomology group H1 (!1, 0) is generaled by finilely many of these groups H 1 (Y, Oi). The cohomology classes are given by differential forms r.p E m"0 ,1 (Y, Oi) = r(Y, A0 ' 1 (Y) Q9 Oi) . They all may be viewed as differential forms on n whose coefficients are holomorphic polynomials of degree i on the fibers. They vanish on Y to order i. We denote the set of these forms by 2t� ' 1 . Examples for forms of this type are given by the derivatives af' where f is a differentiable function in n whose restriction to the fibers is a holomorphic polynomial of degree i. We take the orthogonal complement of these forms and get £0' 1 = EB :/f;0 '1 , where :/f;0 ' 1 C 2t� ' 1 . This follows by a i direct computation. Thus we obtain a basis of finite length, and the restriction of these forms to an satisfies the boundary conditions of harmonic forms.

Since g > 2, the space of harmonic forms �0 ' 1 has positive dimension. We can obtain a basis by explicit computation. The elements are harmonic forms in n vanishing on Y c R. If there were an isomorphism onto the cohomology with coefficients in
Example 2

We will construct another 2-dimensional example of a completely different type. For this we use compact complex-analytic curves that are no longer reg ular. They are irreducible reduced !-dimensional complex spaces (i.e., topo logical Hausdorff spaces that locally look like analytic sets in some complex n-space) . Here we go a bit beyond the scope of this book, but we hope that the reader can nevertheless get an idea of the method.

We take the projective line JID1 with the points 0 and oo. We identify these two points, obtaining a compact irreducible 1-dimensional complex curve that has a "normal crossing" of !- dimensional domains in a neighborhood of the gluing point 0. This point 0 is the only singular point. �

�

The projective line JID1 is the so-called normalization of Y : There is a sur jective holomorphic mapping 1r : JID1 -+ Y that maps 0 and oo t o 0 and

362

VII. Boundary Behavior

is biholomorphic elsewhere. The first cohomology group HI (Y, 0) of Y has dimension 1 (as can be computed) . We build a 2-dimensional complex manifold X around Y. There is a neighborhood U = U(O) in Y and a holomorphic embedding �

j :u

<--+

x 1 = { ( w, z)

E C2

:

1w1 < 1}

with j ( U ) = YI : { (w , z) E XI : w2 - z2 = 0}. We choose an c with 0 < c < 1 and put P := { ( w ,z) E XI : lwl < c} and Y{ : = Y1 - P . We define =

Y2

:= Y - r 1 (YI n P)

and

Y;

: = r 1 (Y ) n Y2 = j- I (YI - P) I

The manifold Y£ consists of two disk shells y0 Yoo

I 2 j- ( { (w , z) E C : c < lwl < 1 and z = w } ) , 2 r i ( {(w, z) E C : c < lwl < 1 and z = -w}).

On Y1 we have the holomorphic function WI : = pr I I y1 1 which can serve as local (and even global) coordinate. The set Y2 is the complement of the union of two neighborhoods of 0 and of oo in JID I and therefore isomorphic to a disk shell in @* = JIDI - {0, oo }. We have a global coordinate w2 on Y2 , and a holomorphic function w 1 = j(w2) on Y£ with

Now we put x2 = y2 X c and use in x2 the product coordinates (w2 , Z2) · We choose suitable small open neighborhoods WI = WI (YI ) c X I and W2 = W2 (Y2) c X2 , and define the gluing map F : (X 1 - P ) n W1 -+ (X2 I Y; ) n W2 by

I ( (w ) , z - w ) f{ (w2 1 z2 ) = F (w ' z) = I u - (w) , z + w )

near z = W , near z = -W .

So we obtain by this gluing procedure a complex manifold X around Y. The singular curve Y is a hypersurface in X . The covariant normal bundle N* = NX. (Y) is given by the adjunction formula NX. (Y) = [-Y] IY ,

where [ - Yjs the line bundle associated to the divisor - Y in X . The normal bundle N = Nx (Y) is trivial over Y2 . Therefore, the holomorphic function z2 can be regarded as a holomorphic function on N that is linear on the fibers. This defines a holomorphic cross section in N* over Y2 that is given by the holomorphic function s 2 ( w2 ) - 1 with respect to the local tri vialization over Y2 . It extends to a global meromorphic section s over Y. In the local trivialization over YI , s is given by the meromorphic function

2. Subelliptic Estimates

363

+w) on z = w, 8 !(w , z) = 221 = { 1/(z 1/(z - w) on z = - w, since the transition function gi2 for N* is given by *91 2 = z2 Z-2 w 2 Now we lift everything to IP'1 . The lifted section s of n* ( N*) has poles of first order in 0 and and no other poles and no zeros. So n* (N*) has Chern class - 2. To construct a bundle with positive Chern class, we have to alter X2 . For this Y - Y and two small disks D' we take a point D around such that D is still in Y - Y 1 C Y2 . Let S be the disk shell D - v'. We glue ( Y- v') x
p E

cc

1

p

:

X2

X2 N0 _k 1(Y) IP'1 dimc H (1P' , n*(N_k (Y))) -

X. +1 -

We replace our old by a new and get a new manifold Now the has Chern class 3. It lifting of the covariant normal bundle to follows by Riemann-Roch that =3 = 4. By -+ Y the fibers over 0 and oo are identified. This gives an the projection additional condition for the existence of global holomorphic cross sections in the covariant normal bundle of Y C So we have 3 linearly independent holomorphic cross sections in

IP'1

X. 8 1 , 82 , 83 N*. One can see by computation that 8 1 l y1 may be assumed to be given by (z - w)(z + w ) . The function lz2 -3 2w2 1 2 is an extension of 8 1 81 to X1 . 1f we add the function l (z - w)2(z +w) 1 , we obtain an extension that is strictly plurisubharmonic outside of Yi . We can also extend the other 8 iiY1 to X 1 and the 8 iiY2 to X2 , form the 8 i si , sum up, and glue together after a small change in a neighborhood U(Y) C C X to obtain an extension h of 8 1 8 1 + 82 82 + 83 83 in U. The extension vanishes on Y and is positive outside. It is strictly plurisubharmonic in a neighborhood U' of Y. Then for very small > 0, the tube n = { h < } c c U' will have a smooth boundary. It is strongly pseudoconvex. We have dim H 1 ( Y,O) = 1. The cohomology class can be given by a cocy c

c

cle with coefficients in
364

VII. Boundary Behavior

Exercises 1.

Show that the Sobolev space H_ 8 may be identified with the dual of H, . Show that the Sobolev norm 11 - . - l i s comes from a scalar product ( . . . . . ) and prove the following generalized Schwarz inequality: If f , g E Ho and f E Hs for some s > 0, then ( !, 9 ) s < II fil s · ll 9 ll s · It is a consequence of the famous "Rellich lemma" that the following holds: If s > t, then for any c > 0 there is a neighborhood V (0) such that II J i l t < ell f i l s for all f with compact support in V . Use this to prove the same estimate for the tangential Sobolev norm. Assume that the Neumann problem is solvable for (p, q)-forms. Let a form r.p E L�, q with ar.p = 0 be given that is orthogonal to Jt'P ,q . Prove that 'ljJ := a N r.p is the only solution cf the equation fhjJ = r.p that is orthogonal to every 5-closed form. Let n c c en be a strongly pseudoconvex domain. Prove that for every positive number C there is a smooth plurisubharmonic function f2 on n with 0 < f2 < 1 such that Lev( g) (z, t) > C l l t l l 2 for every z E a n . Take Example 2 and prove that that the derivative d( r.p l y ) is different from 0 and that this form is not harmonic. Take Example 1 and prove that £1,0 has infinite dimension. The following problem requires some knowledge about sheaves and com plex spaces. Take integers r < s such that r, s are free of common divisors. Consider at 0 E e the holomorphic functions that are given by conver gent power series in wr and w 8 • Then replace the ring of holomorphic functions in 0 E IP\ by these holomorphic functions and repeat the pro cedure of Example 2. Since now the "structure sheaf' 0 of JID 1 is smaller, a new curve Y is obtained. Construct an X around Y such that there is a 2-dimensional strongly pseudoconvex tube n around Y. Compute the first cohomology of Y. . ,

2.

3.

"

-

-*

4. 5.

6.

7.

3.

Nebenhullen

General Domains . Assume that n c en is a general domain. We take the open kernel of the intersection of all domains of holomorphy n :::) n and its connected component N(f!) that contains R . (More precisely, we work in the category of unbranched domains over en as in the case of the construction of the hull of holomorphy). �

Definition.

The domain N(f!) is called the Nebenhiille of !1.

We can see very easily:

The Nebenhulle is a dom a in of holomorphy. The Nebenhulle contains the enve lope of holomorphy of !1 .

-

3. Nebenhiillen

365

If n cc e n is a domain of holomorphy with (smooth) real-analytic boundary, then an does not contain any germ of an analytic set of positive dimension. Moreover, one can show that N(n) n. That implies that one can approxi mate n by Stein neighborhoods. See [DiFo77] for details.

=

2 : l w l l zl <

{(w, z)

E
A Domain with Nontrivial Nebenhulle. We define a real-analytic function {J :
o(w, z) = l z t exp( i ln(ww)) l 2 - 1

and put

:= {(w, z) E
n

.

-

�

w

3 . 1 Propos ition. PROOF:

lf

an

is

smooth.

(Wo, zo) E on where d[! = 0, then oo (wo, zo) = 0. oz ln( l wol 2 )) and hence o(w0, z0) -1. But such a

there were a point

z0 = -

exp( i This gives point is not on the boundary.

•

3 . 2 Propos ition. The boundary of n is pseudoconvex. It is not strongly pseudoconvex exactly along M

= {(w,z) : z = 0}.

PROOF: The complex dimension of the complex tangent space at every +a at a point point of an is 1. A vector .;

= a ofow + bojoz ofow + bojoz

366

VII. Boundary Behavior

of an lies in the complex tangent space if and only if ( ao/ow + b ojoz) [g] So the vector space of complex tangents is spanned by the vector �

=

=

0.

_ og !}_ + og !!__ · oz ow

ow az

The function Q is the potential of a Hermitian form H. The boundary of n is pseudoconvex (strongly pseudoconvex) if and only if H > 0 (respectively H > 0) on this vector. By a calculation we get

It vanishes only for z

=

0.

We take any neighborhood U of n that is a domain of holomorphy. Then U contains
Bounded Domain s. The domain n of the preceding paragraph lies over @* and is not bounded. But one can obtain a pseudoconvex bounded domain by making n smaller. One takes an T > 31f, leaves n unchanged over the line segment [ 1 , r] , and deforms the circles n n (
-

The new domain n is a "semitube": The translations in the imaginary wdirection act on it. Hence, the Nebenhulle N(n) is also a semitube. So it can be written as the set S (M) { (w, Z) : (Re(w), Z) E M } , where M is the smallest open set in { {U,Z) : U E JR, z E
1 . S (M) � n. 2. S ( M) is ps�doconvex. 3 . lR x {o} e M.

4.

Boundary Behavior of Biholomorphic Maps

3 67

We have F (S (M) ) = { (w, z) : (l w l , z) E M } , where M is the image of M under the mapping ( u , z) H ( exp ( u ) , z ) . The domain S (M) is pseudoconvex if and only if F ( S ( M) ) is pseudoconvex, since F is locally biholomorphic with respect to w. So the Nebenhulle N (n ) is the set {w E e : l zl < 2}, just as N (n) is { w E @* : l z l < 2}. Exercises

1. Let n C C e 2 be a domain of holomorphy. 1f there is an analytic plane

E such that n n E has isolated boundary points, then n has a nontrivial Nebenhulle. 2. Consider domains n c 0 such � that K c n(t) c n for It - to I < c. (a) Construct a continuous family of domains n(t) such that the family of the envelopes of holomorphy H ( n(t)) is not continuous. (b) Define "upper semicontinuity" for families of domains. Find condi tions such that the upper semicontinuity of H (rl(t )) follows from the continuity of n(t ) . =

-

4.

Boun d ar y Beh avior of Bih olomor p h ic M ap s

The O ne-Dimensional C a se. Assume that n, n' c e are bounded domainS With 1&'00-SmOOth bOUndary and that f : n -+ n' iS a bih0l0m0rphiC mapping. The following theorem is well known:

4 . 1 Theorem. The m ap f can be unique ly extended to an invertible '1&'00 � -

-

m ap f : n -+ n '.

In particular, if an, an' are real-analytic, then the extension is likewise real analytic.

The Theor y of Henkin and Vor moor . Assume now that n is a domain in e n . A holomorhic map F : n -+ eN is called Holder continuous of index c > 0 if for all points w , z E n the estimate IIF ( w ) - F ( z ) l l < M . ll w - z l l € holds with a constant M > 0. Here we state a special case of a theorem that was independently proved by Henkin and by Vormoor (see [He73] and [Vo73] ) .

368

VII.

Boundary Behavior

4.2 Theorem. Assume that !1 1 , !1 2 E
-

-

The PROOF of this theorem is very technical. The main ideas can be found in the book of Diederich/Lieb ( [ DL81 ]) . An important tool is the notion of a pseudodifferential metric. Since these metrics are interesting themselves , we give the definition for general complex manifolds: Assume that X is an n-dimensional complex manifold. Definition. A pseudodifferential metric on X is an upper semicontinuous map Fx :

T(X ) --+ JR6 with the property

for all vectors � E T(X) and all a E
:

:

:=

=

The Kobayashi metric is the pseudodifferential metric dx : T(X) --+ JR6 defined by dx (�) inf { l a l l : :3 f : D --+ X holomorphic with f(O) Xo , !'(0) ao . The following statement is a very simple consequence of the definition. Definition. :=

-

=

=

4.3 Proposition. Assume that F : X --+ Y is a holomorphic map of com plex manifolds. Then dy (F* (O) < dx (�) .

So the Kobayashi metric is invariant under biholomorphic mappings. In cer tain cases it is completely degenerate. For example, the Kobayashi metric on en is identically 0.

4. Boundary Behavior of Biholomorphic Maps

369

For the Kobayashi metric in the unit disk we obtain d o ( �)

=

� � � , for z E D and � E Tz (D) "'
For a bounded domain n

cc

.

en one shows that

11�11 dist (z, 8!1) '

where � E Tz (!1 ) "' <en , z E !1 , and a E a. dn (O dist ( z, 8!1) 1 / 2

·

We consider again the continuation of biholomorphic maps F !1 1 -+ !1 2 to the boundary. But we assume now that the boundaries are real-analytically smooth. In this case we can drop the a..;;sumpti on of strong pseudoconvexity. Real-Analytic Boundaries. :

4.4 Theorem. If F : !11 -+ !1 2 is a biholomorphic map of pseudoconvex do mains with smooth real-analytic boundaries, then F extends to a (uniformly) � Holder continuous map F : !1 1 -+ !1 2 . --

-

This was proved by Diederich and Fornress ( see [DiFo78]) . Fefferman's Result .

theorem ( see [Fe74] ) .

In 1974 Charles Fefferman proved the following

4.5 Theorem. Assume that !11 , !1 2 c c en are strongly pseudoconvex do mains with 1&'00 -smooth boundaries and that F : !11 -+ � !1 2 is a biholomorphic mapping. Then F extends to a 1&'00 diffeomorphism F : !11 "' !12 . -

-

For his proof Fefferman used the Bergman metric, which is invariant under biholomorphic mappings. His proof was very difficult. There were many sim plifying methods written by authors like S.M. Webster, N. Kerzman, S.R. Bell, E. Ligor;ka, L. Nirenberg, P. Yang. Later, an interesting proof that uses the Bergman projection was discovered by Catlin (see [ Ca84] ) . We wish to indicate some ideas concerning the Bergman kernel function. We will define it in the general case of a strongly pseudoconvex domain n c c X with smooth boundary, where X is an n-dimensional complex manifold. If F is a complex analytic line bundle on X , one can prove as in Chapter V that the cohomology H 1 (!1 , F) is finite. Then, following the ideas of Chapter V, we can construct many holomorphic cross sections in F over n. We take for F the line bundle of holomorphic n-forms X = h( Z1 , . . . , Zn ) dz1 1\ · · · 1\ d zn . The norm ll xll is defined by

370

VII.

Boundary Behavior

ll x ll 2 : =

fn x A * x.

Let Jt' be the space of holomorphic n-forms X on n that are square inte grable (i.e., with ll x ll < oo) . The forms x E Jt' with ll x ll < 1 are uniformly bounded (with respect to their coefficients in local coordinate systems) on every compact set M c n. If A is a compact analytic subset of n of positive dimension, all forms x may vanish there. This will lead to difficulties in defin ing the Bergman metric. The space Jt' is a Hilbert space and a subspace of the Hilbert space L 2 (n) of arbitrary square integrable complex n-forms (with the scalar product ( cp, 7/J ) = fn cp 1\ * 7/J) . Next we define the kernel form. We take an orthonormal basis {Xi i E N} of £. Then :

L Xi (z) xi (w)

= K(z, w) dz1 1\ · · ·

1\

dzn 1\ d'iih 1\ · · · 1\ dwn

i is called the kernel form on n, and the coefficient K is called the Bergman kernel. The theory of K is well known for domains in en . We extend it here to complex manifolds, but we give only an overview and avoid going into detail. It is possible to prove that the kernel form is independent of the choice of the orthonormal basis. The convergence of the defining series is locally uniform, also for all derivatives. The kernel K is real-analytic on n, holomorphic in z, and antiholomorphic in w. As (z , w) approaches an, K(z, w) tends to oo to order n + 1 (i.e., its coefficients with respect to local coordinates). When we pass over to the conjugate complex form, interchange w with z, and multiply by ( - 1 ) n , we get the kernel form back. Moreover, the kernel is invariant under biholomorphic mappings of n onto itself. In local coordinates, but independent of these coordinates, we have sup l f (z) l 2 K(z, z) = x=fdz 1 /\ · · · l\dzn , l l x l l = l

We have the orthogonal projection P : L 2 (n) -+ Jt' with P (a) =

L (a , X i ) · Xi

i n ( 1t (n l ) f 2 d z1 1\ · · · 1\ dzn · -

-

i a (w) 1\ K(z, w) dw1

1\

· · · 1\ dwn .

It is called the Bergman projection. We denote by don , O (n) the set of �00 forms X of type (n, 0) in n for which all derivatives of any order of the coefficient of X vanish on an, and by A00 (n) the set of �00 forms in n that are holomorphic in n.

4.6 Theorem. In the case X = e n , for every regular pseudoconvex (and therefore for every strongly pseudoconvex) domain n c en the Bergman pro

jection P maps .0"0n , o ( n ) into A00 (n ) .

4. Boundary Behavior of Biholomorphic Maps

371

This was proved by Catlin in [Ca84] . The result is known as "n satisfies condition (R) ." The worm domain is an example where condition (R) does not hold. Now we consider another strongly pseudoconvex domain n' c c en with � oo smooth boundary, and a biholomorphic mapping F : n --+ n'. We can transform any form a of type ( n, 0) on n' to a form a o F of the same type on n. We state without complete proof a lemma that is trivial in the case of forms with compact support: 0 -1 4.7 oLemma. For a E Ji10n , (n ) the transform a o F is an element of don ( n ) . M appings .

,

We give a brief idea of the proof. We represent a/\ a as a multiple of the kernel form with a factor c(x) > 0. The function c vanishes on an' to arbitrarily high order. We just have to prove that c o F also has this property in n. To do this we take in n and in n' the metric o(x, y) = SUPt lf(x) - f(Y) I . Here f runs through the holomorphic functions in n (respectively n') with I f ( x) I < 1 (this is the so-called Caratheodory distance) . If x0 is a fixed point of n (respectively n') , then the balls around approach an (respectively an') to first order for r --+ oo ; i.e., the Euclidean distance of the ball to the boundary of the domain behaves like exp( -r) . If Xo E n and Yo = F(xo) E n', then any ball of radius r around x0 is mapped onto a ball of the same radius around YO· This implies the vanishing of c F with all derivatives on an. Xo

0

We need the following �00 version of the Cauchy-Kovalevski theorem:

4.8 Lemma. Assume that h E A 00 ( n' ) . Then there is a form g E Jifn , o (n ) 0 such that ag = 0 on an' and h - t::..g E Jd0n ' (n ' ) . -I

We use the lemma to complete the proof. It follows by partial integration that t::..g is orthogonal to all elements from A00 ( n' ) . We can define an orthonormal basis of £(n') consisting of such functions. So t::..g is orthogonal to £(n'). It follows that hoF

= (PB (h - 6..g )) o F = Pc ((h - t::..g ) o F) E A00 ( !1 ) .

When we take for h the functions z1 1 11' , . . . , zn ln ' , we get the differentiability of F on n. The proof of Fefferman ' s theorem is completed. The Bergman Metric .

The Bergman metric is interesting in its own right. It is a Kahler metric and depends only on the complex structure of the manifold under consideration. Using its Riemannian curvature one can prove interesting theorems that give a deeper insight into the complex structure. The notion is well known for domains in en . Here we generalize it to complex manifolds.

372

VII. Boundary Behavior

We will consider the situation where X is an n-dimensional complex manifold and n CC X a subdomain with strongly pseudoconvex boundary. Then !1 is holomorphically convex. We can construct forms x belonging to £. For that we use the boundary of n. We consider the analytic subset E c n given by E

{ x E n : all x vanish at x or there is a tangent vector � at such that � (b) = 0 for the coefficient b of every x } .

x

It can be proved that E is compact and has positive dimension everywhere. So in the classical case of a domain in en it is empty. This is no longer true in complex manifolds. Therefore, we make the following additional assumption:

( ) The set E is empty. *

Under this assumption we have the Bergman metric in R. In every local coordinate system we just define the coefficients

and the Bergm an metric

The definition is independent of local coordinates, and we obtain a Kahler metric in n that is invariant by biholomorphic transformations of R. If there are no compact analytic subsets of positive dimension in n, then n is a Stein manifold. In general, n may contain compact analytic subsets of positive dimension. For example, we may take for X a complex analytic line bundle on JID 1 with Chern class c < - 3. Then a strongly pseudoconvex tube n around the zero section exists, and there are enough forms x on R. So in this case as well we have the Bergman metric. Near to the boundary the following estimates hold If � is a tangent vector in Tz (fl) , z E !1, 11 �11 is the Euclidean norm, and dist(z, 8!1) is the Euclidean distance (everything with respect to local coordinates), then there are positive constants a, b such that

4 . 9 Pr op osition .

a.

11�11 II � II . b < < ll ll � dist(z, on) " dist (z, 8!1) 1 / 2

The distance (with respect to the Bergman metric) between afixed point zo E n and a point z (near to the boundary) is approximately vn + 1 · (- ln(dist(z, 8!1))).

4.

Boundary Behavior of Biholomorphic Maps

373

xer cises 1. Prove the properties ofthe Bergman kernel. Compute the Bergman kernel

for the unit ball. 2. Assume that n c en is a bounded domain. Then n carries the Caratheo dory metric. This is a positive pseudodifferential metric on n and is defined in the following way: If � is a vector at a point z0 E n, we take all holomorphic functions f on n with I f (z ) I < 1 in n and measure the Euclidean length of the image !* (�) . Then we take the supremum over all these f . Prove that the Kobayashi length is at least as big as the Carathkodory length. 3. Consider the bicylinder pn (0, 2) = { (w, z) : lw l < 1 , 1 zl < 1 ) c
!1 = {z E
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[BoStr99] [Bou66] [Br54] [BrMo78] [CS53] [Ca83] [Ca84] [ChoKo52]

[DL8 1 ] [DiFo77] [DiFo77b] [DiFo77c]

W. B arth, C. Peters, A. Van de Ven: Compact Complex Surfaces. Erg. Math. 3 .4, Springer, Berlin- Heidelberg (1984). H. Behnke, K. Stein: Entwicklungen analytischer Funktionen auf R iem annschen Flachen. Math. Ann. 1 20, 430-461 (1948). H. Behnke, P. Thullen: Theo rie der Funktionen mehrerer komp lexer Veranderlichen, 2nd ed., Erg. Math. 51, Springer, Berlin Heidelberg-New York ( 1970). E. Bishop: Mappings of partially analytic spaces. Am. J . Math. 8 3 , 209-242 (1 961). L. Bieberbach: Beispiel zweier ganzer Funktionen zweier kom p lexer Variablen, welche eine schlichte vo lumentreue Abbildung des R4 auf einen Teil seiner selbst vermitteln. Sitzungsb. der preuB. Akad. d. Wissensch. (1 933). H.P. Boas, E.J. Straube: Global Regularity of the a-Neumann Problem: A Survey of the L 2 -So bolev Theory. Several Complex Variables, MSRI Publications 3 7 , 79- 111 (1999). N. Bourbaki: Elements of Mathematics, General Topology, part 1. Hermann, Paris (1 �66) . .. H. Bremermann: Uber die A quivalenz der pseudokonvexen Ge biete und der Holomorphiegebiete im R aume von n komplexen Veranderlichen. Math. Ann. 1 2 8, 63-91 (1954). n L. Brenton, J . Morrow: Compactifications of e . Trans. Am. Math. Soc. 246, 139- 153 (1978) . H. Cartan: Seminaire E.N.S. 1953/54, exposes XVI and XVII. D. Catlin: Necessary conditions fo r subellipticity of the a Ne umann problem. Ann. Math. 1 1 7, 147- 171 ( 1983). D. Catlin: Global regularity of the a-Neumann problem. Symp. Pure Appl. Math . 4 1 , 46-49 ( 1 984). W.L. Chow, K. Kodaira: On analytic surfaces with two independent meromorphicfunctions. Proc. Nat. Acad. Sci. USA 3 8 , 3 1 9325 (1 952). K. Diederich, I . Lieb: Konvexitat in der komplexen Analysis. Birkhauser Basel 1981, 1- 150 (DMV Seminar 2 ) . K. Diederich, J.E. Fornress: Pseudoconvex Domains: A n Examp le with Nontrivial Nebenhulle. Math. Ann. 225, 275-292 ( 1977). K. Diederich, J.E. Fornress: Pseudoconvex domains. Existence of Ste in neighborhoods. Duke Math. J. 44 , 641-662 (1977) . K. Diederich, J.E. Fornress: Pseudoconvex domains. Bounded strictly p lurisubharmonic exhaustion/unctions. Invent. Math. 3 9, 129-141 (1977).

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I n d ex of Not a t ion natural numbers, integers, etc . =

=

=

w u

(0)

:

{X E JR X > 0} e { 0} -

imaginary unit A V* V' F(V)

real dual space of a real (or complex) vector space complex dual space of a complex vector space C-valued real linear forms on a complex vector space

Mp,q(k) Mn (k) GLn (C) A B ( or A · B)

k-valued matrices with p rows and q columns k-valued square matrices of order n general linear group product of matrices

z zt (z l w) m ll z l l , dist (z , w ) \ z l w) lzl

vector z = ( z 1 , . . . , Zn ) and transposed vector standard Euclidean scalar product in lRm Euclidean norm and distance standard Hermitian scalar product sup-norm (or maximum-norm)

D r ( a) D

Br (z) pn (z, r) P� (z) pn T : en --+ "f/ Pz Tn (z , r) Tz

open disk (in q around a with radius r unit disk D 1 (0) open ball with radius r around z polydisk in e n around z with polyradius r polydisk with polyradius r ( r, . . . , r ) unit polydisk P� (O) natural projection onto absolute value space polydisk pn (0, T (z)) distinguished boundary (torus) distinguished boundary Tn ( o , T(z) )

u cc v

U lies relatively compact in V

Ct (z)

Cauchy integral of f

V'f ( z) V'f ( z) V,J(z) V'yf (z)

holomorphic gradient ( fz , ( z ) , . . . Jzn (z)) antiholomorphic gradient (fz, (z), . . . , fzn (z))

'

=

= =

:

( fx, (z) , . . . ,fxn (z) ) ( fy, (z) , · . . , fYn (z) )

T

--+

e at z E P"

382

Index of Notation

Dv J (z) DJ (z) (of)z (w ) (of)z( w ) (df)z Jr (z) JIR,r (z) 6"(G) O(G) t

=

(z , w )

Tz a(O)

N(JI , . . . , Jq )

A (or Reg( A )) Sing( A) rkz (fr , . . . , fq ) (P" , H) (P, H) �

�

bSt

higher partial derivative total real derivative of f at z = fz, (z)wr + . . · + fzn (z)wn = !z1 (z)wr + . . · + fzn (z)wn (of)z + (of)z complex (or holomorphic) Jacobian matrix real Jacobian matrix space of smooth functions on G space of holomorphic functions on G =

tangent vector at z with direction w tangent space at z tangent vector (a(O) , a' (O)) common zero set of fr , . . . , fq set of regular points of A set of singular points of A rank of the Jacobian J(/1 , . . . , /q ) Euclidean Hartogs figure general Harto gs figure boundary of an analytic disk St

=

4?t (D)

connected component of A1 containing boundary distance (for a domain G) Lev (f) ( z, w ) Hz (oG) Hess( f ) (z, w )

z

Levi form I:: v /-L fzvz, (z)wvw ,_. ,

complex (or holomorphic) tangent space of oG Hessian 'E v, ,_. fxvx, (z)wvw,_. holomorphically convex hull of K in G

61 -< 52

H,� (Q) , H( g )

a

qz] Bt Hn

m

I * ' JX JD [u] fz

the Riemann domain r;;1 is contained in r;;2 9 - h u 11 of g and envelope of holomorphy set of accessible boundary points of G ring of formal power series = { ! E qz] : ll f l l t < oo } ring of convergent power series maximal ideal in H, set of nonzero element, set of units in an ideal I set of monic polynomials with coefficients in I germ of the holomorphic function f at z

fudex of Notation w

(u , ) z

D (w ) .6-w , Dw

pseudopolynomial

dimz ( A ) , dim ( A )

local and global dimension of an analytic set

Riemann sphere
algebraic derivative of w discriminant and discriminant set of

w

tangent space of a manifold tangential map

X Xz Y Kx , T(X)

fiber product of manifolds canonical bundle and tangent bundle on X

r( u, v ) = O(U, v ) 6" ( U, V)

space of holomorphic sections in a bundle V space of smooth sections in V

V EB W, V Q9 W V' p k r v , v;w Nx (Y) Ox , O'X

Whitney sum and tensor product of bundles complex dual bundle of V, tensor power of F pullback and quotient bundle normal bundle of Y in X trivial bundles X x
C' ('Pt' , V) Z (%' , V) Bi (%', V) Hi (%' , V) Hi (X, V )

Cech cochains with values in V Cech cocycles Cech coboundaries Cech cohomology group with values in V cohomology group with values in V singular homology and cohomology of X

Ox

ring of germs of holomorphic functions polar set of a meromorphic function set of meromorphic functions on X divisor of a meromorphic function line bundle associated to the divisor Z Picard group and set of all divisors

'

'

Hq (X) , H q (X)

P(m) div(m) [Z] Pic(X) , �(X)

yn pn

=

C"jr

St(k, n ) , Gk , n pl Gk , n --+ pN 0(1), 0( - 1 ) :

n-dimensional complex torus complex projective space Stiefel manifold and Grassmannian Plucker embedding hyperplane bundle and tautological bundle Osgood space

pi

x . . . x

pi

( m times)

383

384

Index of Notation

c(h) = c(L)

Chern class of a cocycle h (of a line bundle L)

1\ r F(X) n� .'Zfr(U) JZtM (U) cp /\ '
bundle of r-dimensional differential forms bundle of holomorphic p-forms on X r-forms forms of type (p ,q) on U wedge product total, holomorphic and antiholomorphic differential (such that ddc = 2 i aa ) = i (a - a) =

(1/(27r i)) fc ! (() /(( - z ) d( 1\ d(

Dolbeault group of type ( p ,q) on X rth de Rham group of X WH

PU(n t 1)

dV * 4?

(cp ' '
fundamental form of the Hermitian metric H group of projective unitary transformations volume element Hodge star operator (JZtr (X) --+ JZt2n -r (X)) inner product Jx cp 1\ *$, with * '
J, rJ , rJ Lcp A � D

Jf'r(X) £M (X) Pr (X) , Pp,q (X)

adjoint operators of d, a, and a = ( 1/2)cp 1\ w, with Kahler form adjoint operator of L

w

real Laplacian (do + od) complex Laplacian ( arJ + rJa) space of hannonic r-forms on X space of hannonic (p, q)-forms Betti numbers zero section of the bundle F generalized Siegel upper halfplane Hilbert space closure of dp,q (!1) domain of the operator T Neumann operator N : L�, q (f!) --+ dom(D)

Index cf Notation

rapidly decreasing functions Sobolev space of index s tangential Sobolev norm

N (n)

Nebenhiille of !1

Fx K(z, w) p : £ 2 (!1) A00 (!1)

pseudo-differential metric on X Bergman kernel Bergman projection

--+

£

=

.0"n,o (n) n o (n)

385

I n d ex 1-complete, 267 1-convex, 267 Abel's lemma, 12 abelian variety, 351 absolute space, 7 accessible boundary point, 101 acyclic covering, 185 adjoint operator, 330 adjunction formula, 215 affine algebraic, 217 affine algebraic manifold, 224 affine algebraic set, 224 affine part, 210 algebraic derivative, 125 analytic disk, 47, 102 analytic polyhedron, 85 analytic set, 36, 160 analytic tangent, 275 analytically dependent, 320 antiholomorphic form, 303 antiholomorphic gradient, 28 Ascoli - lemma of, 23 atlas, 155 automorphism, 226 ball, 5 Banach algebra, 105 Bergman kernel, 370 Bergman metric, 372 Bergman projection, 370 Bertini - theorem of, 225 Betti number, 190 biholomorphic, 33, 159 blowup, 237 boundary conditions, 357 boundary distance, 54, 58 boundary function, 64 boundary operator, 189 bracket, 332 branch locus, 227 branch point, 128 branched covering, 128, 227 branched domain, 229 bubble method, 283 canonical bundle, 174

Caratheodory distance, 371 Cart an-Thullen - theorem of, 77 Cauchy integral, 17 Cauchy-Riemann equations, 29 Cech cohomology, 185 center - of a modification, 236 chain rule, 32 Chern class, 256 Chow - theorem of, 217 closed map, 226 closure - of en , 245 - of a manifold, 244 coboundary, 184, 187 coboundary operator, 183 cochain, 183, 187 cocycle, 1 83, 187 codimension, 151 cohomologous, 174 cohomology group, 184, 187 commutator, 332 compact exhaustion, 163 compact map, see completely continuous compactification - one-point, 153 compatibility condition, 173 compatible coordinate systems, 154 complete hull, 24 completely continuous, 285 completely singular, 49 complex 1-forrn, 261 complex covariant r-tensor, 261 complex differentiable, 14 complex gradient, 15 complex manifold, 155 complex n-space, 1 complex structure, 1 , 155 complexification, 2 cone, 216 conical set, 216 conjugation, 2 connected, 6, 88 continuity of roots, 1 34 continuity principle, 47 continuity theorem, 43

388

Index

contractible space, 189 convergence - compact, 1 2 - domain of, 1 3 convergent series - of numbers, 10 convex, 8, 66, 74 - strictly, 67 convex hull, 73 coordinate system, 154 Cousin-! distribution, 253 Cousin-11 distribution, 254 covariant tangent vector, 261 critical locus, 227 cross section, 176 cuboid, 276 - distinguished, 276 current, 328 - harmonic, 336 8-Neumann conditions, 357 de Rham group, 3 1 1 de Rham isomophism, 312 de Rham sequence, 310 decomposition theorem - of Hodge- Kodaira, 339 defining function, 64 degree - of a hypersurface, 220 derivation, 33, 164 derivative - directional, 16 - of a holomorphic map, 30 - partial, 15 - real, 26 derivatives - Wirtinger, 28 diagonal, 162 differential, 260, 300 differential form, 297, 300 dimension, 161 local, 150 of a submanifold, 39 of an analytic set, 142 of an irreducible analytic set, 141 Dirichlet' s principle, 52 discrete map, 228 discriminant, 127 discriminant set, 128 distinguished boundary, 6 divisor, 198 of a meromorphic function, 198 - of meromorphic section, 201 a

Dolbeault group, 311 Dolbeault sequence, 310 Dolbeault' s lemma, 309 Dolbeault' s theorem, 312 domain, 6 domain of existence, 97 domain of holomorphy, 49, 99 - weak, 49 dual bundle, 178 duality theorem of Poincare, 337 - of Serre- Kodaira, 338 effective divisor, see positive divisor effective group operation, 172 elementary symmetric polynomial, 126 elliptic operator, 337 embedded-analytic - irreducible component, 135 - set, 135 embedding, 214 embedding dimension, 152 embedding theorem, 145, 258 energy form, 359 envelope of holomorphy, 87, 97 equivalence - of bundles, 174 - of systems of transition functions, 174 Euclidean coordinates, 322 Euclidean domain, 125 Euler sequence, 223 exact sequence, 223 exceptional set, 236 exhaustion function, 58 exterior algebra, 298 exterior derivative, 301 factorial monoid, 125 faithful, see effective group operation Fefferman - theorem of, 369 fiber bundle, 173 fiber bundle isomorphism, 174 fiber metric, 341 fiber product, 171 filter basis, 100 finite map, 227 finite module, 121 fixed point, 172 form of type (p, q), 298 frame, 177, 313 Frechet space, 285 Frechet topology, 285

Index free group operation, 172 Fubini metric, 319 fundamental form, 315 Gauss' s lemma, 1 1 7 general linear group, 172 genus, 232 - of a Riemann surface, 340 geodesic coordinates, 316 geometric series, 11 germ, 123, 192 globally generated, 176 gluing, 170 - of fiber bundles, 173 of manifolds, 170 Godeaux surface, 225 good topological space, 189 graph, 161 - of a meromorphic function, 242 Grassmannian, 212, 225 greatest common divisor, 1 17 Griffiths negative, 341 Griffiths positive, 342 harmonic form, 335, 356 harmonic function, 52 Hartogs - theorem of, 307 Hartogs convex, 48, 99 Hartogs figure - Euclidean, 25 - general, 43, 99 Hartogs triangle, 365 Hausdorff space, 153 Hensel' s lemma, 120 Hermitian form, 3, 261, 321 Hermitian metric, 314 Hessian, 67 Hn , 108 Hodge manifold, 346 Hodge metric, 346 Hodge star operator, 323 Holder continuous, 367 holomorphic form, 303 holomorphic function, 13, 96, 257 - on a complex manifold, 156 - on an analytic set, 134 holomorphic gradient, 28 holomorphic map, 30, 158 holomorphically convex, 75, 100, 251 holomorphically convex hull, 75 holomorphically separable, 251 holomorphically spreadable, 251

homogeneous, 9 homogeneous coordinates, 209 Hopf bundle, 238 Hopf manifold, 208, 225 Hopf's a-process, 237 hyperbolic manifold, 368 hyperelliptic involution, 234 hyperelliptic surface, 232 hyperplane at infinity, 210 hyperplane bundle, 220, 221 hyperquadric, 219 hypersurface, 127, 220 - analytic, 160 ideal, 108 identity theorem for analytic sets, 142 for holomorphic functions, 22 for meromorphic functions, 197 for power series, 21 on manifolds, 156 immersion, 168 implicit function theorem, 34 incidence set, 237 indeterminacy - set of, 243 inner product - Euclidean, 4 - Hermitian, 3 integral class, 346 integral domain, 1 1 3 inverse mapping theorem, 33 irreducible analytic set, 141 irreducible component, 131 irreducible element, 1 17 isomorphism of complex manifolds, 159 - of vector bundles, 177 Jacobian - complex, 30 - real, 31 Kahler manifold, 316 Kahler metric, 316 kernel form, 370 Kobayashi metric, 368 Kodaira' s embedding theorem, 344 Krull topology, 1 10 Kugelsatz, 44, 307 Laplacian, 334 Leray covering, see acyclic covering Levi condition, 66

389

390

Index

Levi convex, 66 - strictly, 66 Levi form, 57, 263 Levi's extension theorem, 197 Levi's problem, 5 1 Lie group, 1 7 1 lifted bundle, 1 80 lifting - of a differential form, 304 limit - of a filter, 1 0 1 line bundle, 175 - associated to a hypersurface, 200 line element, 314 linear subspace, 218 Liouville - theorem of, 22 local ®-algebra, 192 local homeomorphism, 8 8 local parametrization, 39, 41 local potential, 318 local uniformization, 231 locally compact, 153 locally finite covering, 153 logarithmically convex, 8 3 maximal ideal, 108 maximum principle, 22 - for plurisubharmonic functions, 264 - for subharmonic functions, 55 mean value property, 52 meromorphic function, 196 - on a Riemann surface, 158 meromorphic map, 243 meromorphic section, 201 minimal defining function, 194 modification, 236 - generalized, 236 module, 120 Moishezon manifold, 320, 353 monic polynomial, 1 1 3 monoidal transformation, 242 monomial, 9 Montel - theorem of, 23 Nebenhulle, 364 negative line bundle, 287 neighborhood filter, 100 Neil parabola, 1 34 Neumann operator, 359 Neumann problem, 357 noetherian, 1 2 1

normal bundle, 1 8 2 normal exhaustion, 78 normalized polynomial, 1 1 3 normally convergent, 1 1 nowhere dense. 37 Oka - theorem of, 100 Oka's principle, 253 open covering, 153 open map, 88 orbit space, 205 order - of a power series, 1 1 9 Osgood - theorem of, 1 9 Osgood space, 243 paracompact, 153 parameter system, 151 partially differentiable, 16 partition of unity, 1 63 pathwise connected, 8 8 period matrix, 348 Picard group, 187 Plucker embedding, 218 pluriharmonic function, 318 plurisubharmonic, 56, 261, 263 - strictly, 59 Poincare map, 301 point of indeterminacy, 196 polar set, 196 pole, 196 polydisk, 5 polynomial, 9 polynomially convex, 87 positive definite, 261 positive divisor, 198 posilive form, 303 positive line bundle, 288 power series - convergent, 108 - formal, 1 1 , 105 power sum, 1 34 prime element, 1 17 primitive polynomial, 1 17 principal divisor, 254 principal ideal domain, 1 17 projective algebraic - manifold, 217 - set, 217 projective space, 208 projective unitary transformations, 3 1 8 proper map, 35, 226

Index

properly discontinuous, 205 pseudoconvex, 60, 1 03 - strongly, 73 pseudoconvex manifold, 267 pseudodifferential metric, 368 pseudopolynomial, 124 pullback, see lifted bundle pure-dimensional, 143 quadratic transformation, 239 quotient bundle, 1 8 1 quotient field, 1 17 quotient topology, 203 rational function, 223 refinement, 153 refinement map, 153 region, 6 regular closure, 245 regular domain, 360 regular function, 224 regular point, 39, 140, 1 6 1 Reinhardt domain, 7 - complete, 8 , 85 - over en' 95 - proper, 8 Remmert-Stein - extension theorem, 150 removable boundary point, 102 removable singularity, 196 resolution of singularities, 240 Riemann domain - branched, 229 - with distinguished point, 89 Riemann extension theorem - first, 38 - second, 1 5 1 Riemann surface - abstract, 157 - concrete, 230 - of Vz, 8 8 Ritt' s lemma, 1 5 1 Riickert basis theorem, 122 Runge domain, 87 saturated set, 203 scalar product, 3 schlicht domain, 90 Schwartz - theorem of, 285 second axiom of countability, 153 section, see cross section - in a fiber bundle, 186 Segre map, 225

self-intersection number, 241 seminegative line bundle, 352 seminorm, 284 semipositive line bundle, 352 Serre duality, 233 Serre problem, 259 shear, 109 Siegel upper halfplane, 352 singular cochain, 189 singular cohomology group, 189 singular homology group, 189 singular locus, 147 singular point, 39 singular q-chain, 188 singular q-simplex, 188 smooth boundary, 64 smoothing lemma, 59 Sobolev norm - tangential, 358 Sobolev space, 358 standard simplex, 1 8 8 star, 283 Stein manifold, 251 Stiefel manifold, 212 Stokes' s theorem, 305 strict transform, 240 strictly pseudoconvex manifold, 267 strongly pseudoconvex, 267 structure group, 173 Stiitzflache, 275 subbundle, 1 80 subelliptic, 358 subharmonic function, 53 submanifold, 4 1 , 161 submersion, 168 symbol - of a differential operator, 336 symmetric polynomial, 126 tangent bundle, 176 - of projective space, 222 tangent space, 32, 165 - holomorphic, 66 tangent vector, 32, 165 tangential map, 167 tautological bundle, 238 tensor power, 178 tensor product, 179, 261 Theorem A, 252 Theorem B , 253 topological map, 4 1 topological quotient, 203 toric closure, 246

391

n

E

3

3

1

•

n

<

1

2

Index

392

toric variety, 246

torsion point, 230

torus, 207, 225, 348

transformation group, 172 transition functions, 1 7 3

transversal, 1 7 1

trivial vector bundle, 177

trivialization, 173

- of

a

vector bundle, 175

tube, 288

tube domain, 87 unbranched point, 1 3 3

Veronese map, 224 volume element . 323 weakly hol omor phi c , 1 6 wedge product , 29 7

Weierstrass

- theorem of 19 ' Weierstrass condition, 1 1 3 Weierstrass division formula, Weierstrass formula, 1 1 0

Weierstrass function, 226

Weierstrass polynomial, 1 14

uniformization, 8 8 , 162

Weierstrass preparation theor

unique factorization domain, 1 1 7

Whitney sum, 1 7 7

union of Riemann domains, 94

116 worm domain, 366

vector bundle, 175

vector bundle c h art, 175

vector bundle homomorphism, 177

vector field, 176

z1- regular, 109 Zariski tangential space, 152 Zariski topology, 1 4 1

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