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Graduate Texts in Mathematics 213 Editorial Board

S. Axler F.W. Gehring K.A. Ribet

Graduate Texts in Mathematics TAKEtrrIlZARiNG. Introduction to 34 SPITZER. Principles of Random Walle Axiomatic Set Theory. 2nd ed. 2nded.

2 OXTOBY. Measure and Category. 2nd ed. 35 ALExANDERlWERMER. Several Complex 3 ScHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.

2nded. 36 KEu.sy/NAMIOKA et al. Linear 4 Hn.TONISTAMMBACH. A Course in Topological Spaces.

Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERTIFRrrzsam. Several Complex

Mathematician. 2nd ed. Variables. 6 HUGlIBSIPtPER. Projective Planes. 39 ARVESON. An Invitation to c*-Algebras. 7 SERRE. A Course in Arithmetic. 40 KEMENY/SNEUiKNAPP. Denumerable 8 TAKEtrrIlZARiNG. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APosroL. Modular Functions and

and Representation Theory. Dirichlet Series in Number Theory. IO COHEN. A Course in Simple Homotopy 2nded.

Theory. 42 SERRE. Linear Representations of Finite 11 CONWAY. Functions of One Complex Groups.

Variable I. 2nd ed. 43 GIU.MANlJERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSONlFtJu.EJt. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry.

of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GoLUBITSKy/GUIU.EMIN. Stable Mappings 46 LotiVE. Probability Theory D. 4th ed.

and Their Singularities. 47 MOISE. Geometric Topology in IS BERBERIAN. Lectures in Functional Dimensions 2 and 3.

Analysis and Operator Theory. 48 SAOfs/WU. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLAtT. Random Processes. 2nd ed. 49 GRUENBERGlWEIR. Linear Geometry. 18 HALMos. Measure Theory. 2nd ed. 19 HALMos. A Hilbert Space Problem Book. 50 EDWARDS. Fermat's Last Theorem.

2nd ed. 51 KuNGENBERG. A Course in Differential 20 HUSEMOU.ER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNESIMAcK. An Algebraic Introduction 53 MANJN. A Course in Mathematical Logic.

to Mathematical Logic. 54 GRAVERIWATKINS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HOLMES. Geometric Functional Analysis 55 BRoWNIPEARCY. Introduction to Operator

and Its Applications. Theory I: Elements of Functional Analysis. 25 HEwrrr/SmOMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An

Analysis. Introduction. 26 MANES. Algebraic Theories. 57 CRoWElllFox. Introduction to Knot 27 KElLEy. General Topology. Theory. 28 ZARIsKIlSAMUEL. Commutative Algebra. 58 KOBUTZ. p-adic Numbers. p-adic

Vol.l. Analysis. and Zeta-Functions. 2nd ed. 29 ZARIsKIlSAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.

V01.ll. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed.

Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra D. Theory.

Linear Algebra. 62 KARGAPOwvIMERuJAKOV. Fundamentals 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups.

m. Theory of Fields and Galois Theory. 63 BOu.DBAS. Graph Theory. 33 HIRsoi. Differential Topology.

(continued after index)

Klaus Fritzsche Hans Grauert

From Holomorphic Functions to Complex Manifolds

With 27 Illustrations

Springer

Klaus Fritzsche Hans Grauert Bergische Universitiit Wuppertal GauBstra6e 20 0-42119 Wuppertal

Mathematisches Institut Georg-August-Universitiit G6ttingen Bunsenstra6e 3-5

Gennany Klaus.Fritzsche@math.uni-wuppertal.de

0-37073 G6ttingen

Editorial Board S. Axler Mathematics Department San Francisco State

University San Francisco, CA 94132 USA axler@sfsu.edu

Gennany

F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA fgehring@math.lsa.

umich.edu

K.A. Ribet Mathematics Department University of California,

Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu

Mathematics Subject Classification (2000): 32-01, 32Axx, 32005, 32Bxx, 32Qxx, 32E35

Library of Congress Cataloging-in-Publication Data Fritzsche. Klaus.

From holomorphic functions to complex manifolds 1 Klaus Fritzsche, Hans Grauert. p. cm. - (Graduate texts in mathematics; 213)

Includes bibliographical references and indexes. ISBN 978-1-4419-2983-9 ISBN 978-1-4684-9273-6 (eBook) DOl 10.1007/978-1-4684-9273-6

I. Complex manifolds. 2. Holomorphic functions. I. Grauert. Hans. 1930- n. Title. III. Series. QA331.7 .F75 2002 515'.98---4c21 2001057673

© 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 in connection with any form of information storage and retrieval, electronic adaptation, computer soft­ware, 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.

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SPIN 10857970

Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

This reprint has been authorized by Springer-Verlag (Berlin/Heidelberg/New York) for sale in the Mainland China only and not for export therefrom.

Preface

The aim of this book is to give an understandable introduction to the the­ory 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 co­cycles 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 sub­sets of the space en. Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi­mum 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: 2, there are pairs of open subsets H c Peen 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 of 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 en.

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 of 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 un­branched coverings of en, 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 divi-

vi Preface

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 ti­tle Seveml 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 a methods are required. As an application Levi's problem is solved. In particular, we show that every pseudoconvex domain in en 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, higher­dimensional Dolbeault and de Rham cohomology, and Kahler metrics. Using results of the previous sections we show that every compact complex mani­fold 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 opera­tors, 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 an­alysis 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, Fornress, Catlin, and Fefferman (a-Neumann, subeUip­tic estimates), and sketch an application of harmonic forms to pseudoconvex domains containing nontrivial compact analytic subsets. The Kobayashi met­ric 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.

Preface 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 sug­gestions 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

Preface

I 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 Hartogs 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 .

v

1 1 1 3 5 6 7 9 9 9

11 14 14 15 16 17 17 19 22 23 23 25 26 26 28 29 30 30 32 32 33 36 36 38 39 41

x Contents

II Domains of Holomorphy 43 l. The Continuity Theorem 43

General Hartogs Figures 43 Removable Singularities . 45 The Continuity Principle 47 Hartogs Convexity . . . . 48 Domains of Holomorphy 49

2. Plurisubharmonic Functions 52 Subharmonic Functions . 52 The Maximum Principle 55 Differentiable Subharmonic Functions 55 Plurisubharmonic Functions 56 The Levi Form . . . . 57 Exhaustion Functions 58

3. Pseudoconvexity . . . . . 60 Pseudoconvexity . . . 60 The Boundary Distance 60 Properties of Pseudoconvex Domains 63

4. Levi Convex Boundaries 64 Boundary Functions 64 The Levi Condition 66 Affine Convexity . . 66 A Theorem of Levi. 69

5. Holomorphic Convexity 73 Affine Convexity . . 73 Holomorphic Convexity 75 The Cartan-Thullen Theorem 76

6. Singular Functions. . . . . . . . . 78 Normal Exhaustions ...... 78 Unbounded Holomorphic Functions 79 Sequences ......... 80

7. Examples and Applications .... 82 Domains of Holomorphy 82 Complete Reinhardt Domains 83 Analytic Polyhedra .. 85

8. Riemann Domains over en . . . 87 Riemann Domains . . . . . . 87 Union of Riemann Domains . 91

9. The Envelope of Holomorphy. . 96 Holomorphy on Riemann Domains . 96 Envelopes of Holomorphy 97 Pseudoconvexity . 99 Boundary Points 100 Analytic Disks .. 102

III Analytic Sets l. The Algebra of Power Series

The Banach Algebra Bt .

Expansion with Respect to Zl

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 H n .

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 . .

Contents Xl

105 105 105 106 107 108 109 110 110 113 114 115 116 116 117 119 119 120 123 123 124 125 125 126 126 127 130 130 132 135 135 137 138 140 141 141 143 143 144 145 147 147 150

xii Contents

IV Complex Manifolds 1. The Complex Structure .

Complex Coordinates Holomorphic FUnctions Riemann Surfaces . . . Holomorphic Mappings Cartesian Products "

153 153 153 156 157 158 159

Analytic Subsets . . . . 160 Differentiable Functions. 162 Tangent Vectors . . . . . 164 The Complex Structure on the Space of Derivations 166 The Induced Mapping. . . . 167 Immersions and Submersions 168 Gluing ............ 170

2. Complex Fiber Bundles . . . . . 171 Lie Groups and 'Transformation Groups 171 Fiber Bundles . . . . . . 173 Equivalence. . . . . . . . 174 Complex Vector Bundles 175 Standard Constructions . 177 Lifting of Bundles . . ... 180 Subbundles and Quotients 180

3. Cohomology....... 182 Cohomology Groups . 182 Refinements ... 184 Acyclic Coverings .. 185 Generalizations. . . . 186 The Singular Cohomology. 188

4. Meromorphic FUnctions and Divisors 192 The Ring of Germs .. 192 Analytic Hypersurfaces 193 Meromorphic Functions 196 Divisors . . . . . . . . . 198 Associated Line Bundles 200 Meromorphic Sections. . 201

5. Quotients and Submanifolds 203 Topological Quotients . . 203 Analytic Decompositions 204 Properly Discontinuously Acting Groups 205 Complex Tori ......... 206 Hopf Manifolds. . . . . . . . . 207 The Complex Projective Space 208 Meromorphic FUnctions . 210 Grassmannian Manifolds . . . 211

v

Submanifolds and Normal Bundles. Projective Algebraic Manifolds Projective Hypersurfaces The Euler Sequence . . . Rational FUnctions. . . .

6. Branched Riemann Domains Branched Analytic Coverings Branched Domains. . . . . . Torsion Points . . . . . . . . Concrete Riemann Surfaces . Hyperelliptic Riemann Surfaces

7. Modifications and Toric Closures Proper Modifications .. Blowing Up ........ . The Tautological Bundle . Quadratic Transformations Monoidal Transformations Meromorphic Maps Toric Closures . . . . . . .

Stein Theory 1. Stein Manifolds

Introduction ..... . Fundamental Theorems Cousin-I Distributions. Cousin-II 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 ..

3. Pseudoconvexity......... Pseudoconvex Complex Manifolds Examples ..... Analytic Tangents . . .

4. Cuboids ......... . Distinguished Cuboids. Vanishing of Cohomology Vanishing on the Embedded Manifolds

Contents xiii

214 216 219 222 223 226 226 228 229 230 231 235 235 237 237 239 241 242 244

251 251 251 '252 253 254 255 257 257 258 259 260 260 261 262 263 264 266 266 267 274 276 276 277 278

xiv Contents

Cuboids in a Complex Manifold 278 Enlarging U' .. 280 Approximation .. 281

5. Special Coverings .. 282 Cuboid Coverings 282 The Bubble Method 283 Frechet Spaces . . . 284 Finiteness of Cohomology . 286 Holomorphic Convexity . . 286 Negative Line Bundles. . . 287 Bundles over Stein Manifolds . 288

6. The Levi Problem'. . . . . . . . . 289 Enlarging: The Idea of the Proof . 289 Enlarging: The First Step . . . 290 Enlarging: The Whole Process 292 Solution of the Levi Problem 293 The Compact Case ...... 295

VI Kahler Manifolds 297 1. Differential Forms .... 297

The Exterior Algebra 297 Forms of Type (p, q) . 298 Bundles of Differential Forms . 300

2. Dolbeault Theory . . . . . . . . . 303 Integra;tion of Differential Forms 303

. The Inhomogeneous Cauchy Formula 305 The a-Equation in One Variable 306 A Theorem of Hartogs . 307 Dolbeault's Lemma 308 Dolbeault Groups 310

3. Kahler Metrics ..... 314 Hermitian metrics . 314 The Fundamental Form 315 Geodesic Coordinates . 316 Local Potentials .... 317 Pluriharmonic Functions 318 The Fubini Metric 318 Deformations . . . . . 320

4. The Inner Product . .. 322 The Volume Element 322 The Star Operator . . 323 The Effect on (p, q)-Forms 324 The Global Inner Product 327 Currents ...... 328

5. Hodge Decomposition ..... 329

Adjoint Operators . The Kiihlerian Case Bracket Relations The Laplacian . Harmonic Forms Consequences. .

6. Hodge Manifolds .. Negative Line Bundles. Special Holomorphic Cross Sections Projective Embeddings Hodge Metrics .

7. Applications ....... . Period Relations . . . . The Siegel Upper Halfplane . Semi positive Line Bundles Moishezon Manifolds

VII Boundary Behavior 1. Strongly Pseudoconvex Manifolds

The Hilbert Space . . Operators ...... . Boundary Conditions

2. Subelliptic Estimates .. Sobolev Spaces .... The Neumann Operator . Real-Analytic Boundaries . Examples .....

3. Nebenhiillen.......... General Domains ..... A Domain with Nontrivial Nebenhiille . Bounded Domains . . . . . . . . . . . . Domains in ((:2 . . . . . . . . . . . . . .

4. Boundary Behavior of Biholomorphic Maps.

References

The One-Dimensional Case ..... The Theory of Henkin and Vormoor Real-Analytic Boundaries Fefferman's Result . . Mappings ...... . The Bergman Metric

Index of Notation

Index

Contents xv

329 331 332 334 335 338 341 341 342 344 345 348 348 352 352 353

355 355 355 355 357 357 357 359 360 360 364 364 365 366 366 367 367 367 369 369 371 371

375

381

387