Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory 9783110838350, 9783110041507

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de Gruyter Studies in Mathematics 3 Editors: Heinz Bauer · Peter Gabriel

Ludger Kaup · Burchard Kaup

Holomorphic Functions of Several Variables An Introduction to the Fundamental Theory With the Assistance of Gottfried Barthel Translated by Michael Bridgland

w DE

G

Walter de Gruyter Berlin-New York 1983

Authors Dr. Ludger Kaup Professor of Mathematics Universität Konstanz Dr. Burchard Kaup Professor of Mathematics Universität Fribourg

Unseren Eltern gewidmet

Library of Congress Cataloging in Publication Data

Kaup, Ludger, 1939Holomorphic functions of several variables. (De Gruyter studies in mathematics ; 3) Bibliography: p. Includes index. 1. Holomorphic functions. I. Kaup, Burchard, 1940. II. Barthel, Gottfried. III. Title. IV. Series. QA331.K374 1983 515.9 83-10120 ISBN 3-11-004150-2

CIP-Kurztitelaufiiahme der Deutschen Bibliothek

Kaup, Ludger:

Holomorphic functions of several variables : an introd. to the fundamental theory / Ludger Kaup ; Burchard Kaup. With the assistance of Gottfried Barthel. Transl. by Michael Bridgland. - Berlin ; New York : de Gruyter, 1983. (De Gruyter studies in mathematics ; 3) ISBN 3-11-004150-2 NE: Kaup, Burchard: ; GT

© Copyright 1983 by Walter de Gruyter & Co., Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means - nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hübler, Berlin. Typesetting and Printing: Tutte Druckerei GmbH, Salzweg-Passau. Binding: Lüderitz & Bauer, Berlin.

Foreword

,

In the preface to his pioneering book, Die Idee der Riemannschen Fläche [Wl], Hermann Weyl wrote "Erst Klein1* hat ja jene freiere Auffassung der Riemannschen Fläche recht zur Geltung gebracht, welche ihre Verbindung mit der komplexen Ebene als eine über der Ebene sich ausbreitende Überlagerungsfläche aufhebt, und hatte dadurch den Grundgedanken Riemanns erst seine volle Wirkungskraft gegeben— Ich teilte seine Überzeugung, daß die Riemannsche Fläche nicht bloß ein Mittel zur Veranschaulichung der Vieldeutigkeit analytischer Funktionen ist, sondern ein unentbehrlicher sachlicher Bestandteil der Theorie; nicht etwas, was nachträglich mehr oder minder künstlich aus den Funktionen herausdestilliert wird, sondern ihr prius, der Mutterboden, auf dem die Funktionen erst wachsen und gedeihen können."2) Likewise in the theory of holomorphic functions of several complex variables, the investigation of holomorphic functions and their natural domains of existence is rooted in an abstract version of a Riemann surface, called a "complex space". Two phenomena appearing only in the multidimensional theory are particularly striking: i) For n > 1, a domain in C" is not in general a domain of holomorphy; that is, it need not be the maximal domain of definition of a holomorphic function. Complements of finite point-sets in C" fall into that category. ii) In the generalization of concrete Riemann surfaces to higher dimensions, called "Riemann domains", ramification points need not possess local uniformizations, as can be seen even in such a simple example as the "origin" in the Riemann domain of \/z1z2 · Such points are called singularities. Those two phenomena profoundly influenced the development of (multidimensional) complex analysis; they also determine the content and organization of our book: i) In the study of domains of holomorphy in C" and their function theory, it soon became evident that the concept of "analytic convexity" is of fundamental importance. In attempting to free oneself, in the sense indicated by Weyl, of the concrete realization of domains of holomorphy as subsets of C", one is led naturally to the concept of " Felix Klein (1849-1925). Klein had been the first to develop the freer conception of a Riemann surface, in which the surface is no longer a covering of the complex plane; thereby he endowed Riemann's basic ideas with their full power... .1 shared his conviction that Riemann surfaces are not merely a device for visualizing the manyvaluedness of analytic functions, but rather an indispensable essential component of the theory; not a supplement, more or less artificially distilled from the functions, but their native land, the only soil in which the functions grow and thrive. [Wl]2

2)

VI

Foreword

Stein spaces. Due to their rich function-theoretic structure, such spaces also are called "holomorphically complete"; they play a leading role in complex analysis. Both Chapter 1 in the first part of the book and the entire third part are devoted to the topics just outlined. ii) Even at its singular points, a Riemann domain can be described locally as an "analytic set", i.e., the solution set of a system of holomorphic equations. Also the systematic study of complex manifolds depends heavily on analytic sets, for example, in many inductive arguments. Consequently, function theory on analytic sets is indispensible for an understanding of Riemann domains. A further attempt to obtain a "freer conception" ä la Weyl, this time to describe analytic sets without reference to an embedding in a complex number space, leads to the construction of the complex spaces mentioned previously. That explains why they have become the central object of investigation in complex analysis - or, as some authors say in analogy to algebraic geometry, in complex analytic geometry. A systematic treatment of complex spaces entails the analysis of punctual, local, and global properties. With its investigation of power series algebras and their homomorphic images, Chapter 2 can be viewed as a central portion of the punctual theory. The " Weierstrass Preparation Theorem" occupies such a key position in that investigation that Chapter 2 might well have been entitled, "Variations on the Weierstrass Preparation Theorem". In Chapter 3, we prepare for the transition to the local and global theories of complex spaces by developing a more general formal framework in the concept of a "ringed space", which is a topological space with a distinguished "structure sheaf". We then develop a geometric intuition through the presentation of numerous examples of manifolds and reduced complex spaces. In Chapter 4, the local theory is the main topic. The concept of coherent sheaves plays a decisive role in the step from "punctual" to "local"; the central results, aside from the actual Coherence Theorems, are the Representation Theorem for Prime Germs 46.1 and the Local Characterization Theorem for Finite Morphisms 45.4. The global theory is particularly well developed for two classes of complex spaces that can be characterized in terms of topological or function-theoretic "completeness", namely, compact spaces and Stein spaces. In this book, the emphasis in the global theory is on Stein spaces, since their function theory yields fundamental results for the local investigation of complex spaces as well. For the step from "local" to "global", we apply cohomology theory; the central result is Theorem B: "Stein spaces have trivial analytic cohomology". In a supplement, Chapter 7, we treat a class of complex spaces whose function theory closely resembles that of manifolds, namely, the "normal spaces", characterized by the validity of the Riemann Removable Singularity Theorems. In the Normalization Theorem, we show how to modify an arbitrary reduced complex space so that it becomes a normal space (that may be viewed as a first step toward "regularizing" it into a manifold). A much deeper result is Hironaka's Theorem on the resolution of singularities [Hr], whose proof lies well beyond the scope of this book; however, we do present the tools for the resolution of singularities of complex curves and surfaces: normalization and quadratic transformation.

Foreword

VII

The coherence of the structure sheaves of complex spaces leads to an interplay between algebra and geometry that we find especially charming, and we have used it as a methodical leitmotiv for Chapter 4. The algebraic objects "analytic algebras" (which appear punctually as stalks of structure sheaves) correspond to the local geometric objects "germs of complex spaces". As a result, geometric statements have algebraic proofs (which are frequently easier and more transparent), and algebraic statements can be interpreted geometrically. In order to exploit this "antiequivalence" to the fullest, it is necessary to drop the restrictive condition "reduced" from the definition of complex spaces in Chapter 3. Consequently, stalks of structure sheaves of complex spaces, which are simply analytic algebras, may contain nilpotent elements. We thereby have an appropriate tool for the treatment of solution sets of systems of holomorphic equations with "multiplicities". For example, consider the zero of the function z m ; its multiplicity is reflected in the analytic algebra C {z} / (z™), rather than in the corresponding reduced algebra C {z} / (z). That more general approach does not complicate the proofs of the essential results; on the contrary, it is advantageous - for example, for the proof of Cartan's Coherence Theorem. There is a similar antiequivalence, which we discuss to a limited extent in connection with the Character Theorem (§ 57), between Stein spaces on the geometric side and their algebras of global holomorphic functions on the algebraic side. This introduction to complex analysis is intended both as a textbook and as a guide for independent study. In general, we neither discuss the historical development of the theory, nor mention the discoverers of propositions or proofs. However, those familiar with the subject cannot fail to notice the tremendous influence on our presentation, both direct and indirect, of the Münster school that grew up around Heinrich Behnke (1898-1979), particularly through the ideas of Grauert and Remmert. The limited scope of an introductory textbook permits the presentation of only a small selection of topics from a rich and living branch of mathematics; the resulting omissions are all the more apparent if the discipline under consideration distinguishes itself through numerous connections to other subjects, as is the case with complex analysis and, say, commutative algebra, differential geometry, algebraic geometry, functional analysis, the theory of partial differential equations, and algebraic topology. The reader who wishes to pursue the subject in greater depth should consult the relevant monographs (such as [Fi] and the appendices in [BeTh] for an overview, [GrRe] and [Ab] for the punctual theory, [GrRe]2 and [BaSt] for the global theory, [Ho] for ^-theory, etc.). Since this book is an introduction to the foundations of the theory, we generally do not treat subjects that have not already appeared in one form or another in textbooks (two major exceptions are the investigation of quotient structures and part of the discussion of normalization). We have given considerable attention to the (generally simple) exercises, which serve to test the reader's understanding of the material, to vary or complete simple proofs, and sometimes to augment the text. In general,

VIII

Foreword

an exercise whose conclusion is needed later in the book is provided with hints, if necessary, and an indication of the pkce at which it is applied for the first time. We use small print to indicate both material from other disciplines and supplementary material that may be left out during a first reading of the text without sacrificing rigor. Aside from sheaf theory and cohomology theory (for which we summarize the necessary concepts and results in such a way as to motivate their application), we assume very little prior knowledge of the reader: some elements of one-dimensional complex analysis, and some basic results from differential calculus, algebra, and functional analysis. Even for those topics, we frequently give explicit references. We are indebted to many colleagues for valuable suggestions and detailed comments on the text; in particular, we should like to mention Milos Dostal, George Elenczwajg, Gerd Fischer, Wilhelm Kaup, Leopolde Nachbin, and Reinhold Remmert. We are pleased to acknowledge the work of Karl-Heinz Fieseier and Ernst-Ulrich Kolle, who read the German manuscript with great care, and corrected many inaccuracies. We are also grateful to our students for their contributions, in the form of questions and comments, to the betterment of the text. Mrs. Gisela Schroff patiently and carefully typed the various drafts of both the German and English manuscripts, for which we are particularly grateful. Most especially, our thanks go to Michael Bridgland, who, under circumstances that were not always of the best, has rendered the German manuscript into English with so much care. Finally, we would like to thank Heinz Bauer for suggesting to us that we write this book. The publishers have earned our gratitude both for their patience and understanding during the many delays that accompanied the writing of the book, and for their friendly cooperation during the printing. We also thank everyone else, colleagues and institutions, who have contributed either directly or indirectly to the completion of the book, not least of all our families. Konstanz/Fribourg, Summer 1983

Ludger Kaup Burchard Kaup

Contents

Interdependence of chapters Courses and Seminars: Guide to the Essentials of Specific Topics

XIV XV

Part One: Function Theory on Domains in C

Chapter 0 Elementary Properties of Holomorphic Functions §1 §2 §3 §4

Definition of Holomorphic Functions 0 (X) as a Topological Algebra Holomorphic Mappings Cauchy's Integral Formula

1 2 4 7 10

Cauchy's Estimate, Abel's Lemma, Taylor Series, Liouville's Theorem

§ 4A §5 §6 §7

Supplement: Cartan's Uniqueness Theorem and Automorphisms of Bounded Domains Montel's Theorem The Identity Theorem, the Maximum Principle, and Runge's Theorem The Riemann Removable Singularity Theorems

16 17 19 21

Analytic Sets, Codimension, Kugelsatz

§8

The Implicite Mapping Theorem

26

Complex Functional Determinant, Inverse Mapping Theorem, Rank Theorem, Submanifolds

Chapter 1 Regions of Holomorphy § 11 § 12 § 12A § 12B

31

Domains of Convergence of Power Series and Reinhardt Domains .. 32 Hartogs's Figure, Hartogs's Kontinuitätssatz Regions of Holomorphy and Holomorphic Convexity 36 Holomorphically Extendible at a Point, Thullen's Lemma Supplement: Further Extension Theorems for Holomorphic Functions 43 Supplement: The Edge-of-the-Wedge Theorem 46

X

§ 13 § 14

Contents

Plurisubharmonic Functions Levi Form Pseudoconvex Domains

49 56

Part Two: Function Theory on Analytic Sets Chapter 2 The Weierstrass Preparation Theorem and its Applications

64

§ 21

65

§ 22

§ 23

§23A

§ 24

Power Series Algebras : the Norm || ||r, Homomorphisms of Power Series Algebras The Weierstrass Theorems Distinguished Power Series, Weierstrass Preparation Theorem, Weierstrass Division Formula, Shearing Algebraic Properties of „6% and „^ Noetherian Rings, Factorial Rings, Normal Rings, Hensel's Lemma, Representation of „00[T]/(P) Supplement: Finite Ring-extensions, Normality Dedekind's Lemma, Nakayama's Lemma, Krull's Intersection Theorem, Equation of Integral Dependence Analytic Algebras

71

80

86

89

Finite Ring-homomorphism, Noether's Normalization Theorem, Finite Extensions of Analytic Algebras

Chapter 3 Complex Manifolds and the Elementary Theory of Complex Spaces . 92 § 30

Bringing in the Sheaves

§ 31

Ringed Spaces and Local Models of Complex Spaces 97 Reduction, Subspaces and Ideal Sheaves, Comorphisms, Restriction Lemma, Nilpotent Elements Complex Manifolds and Reduced Complex Spaces 106

§ 32

93

Projective Algebraic Varieties, Set of Singular Points, Tangent Space

§ 32A § 32B

Supplement: Submersions and Immersions Supplement: Examples of Complex Manifolds

114 117

Lie Groups, Manifolds as Quotient Spaces, Torus, Grassmann Manifold, Transformation Groups, Quadratic Transformation

§ 33 § 33A § 33B

Zeros of Polynomials Branched Coverings, Finite Holomorphic Mappings Supplement: Resultants and Discriminants Supplement: Proper Mappings and Equivalence Relations

126 131 133

Contents

Chapter 4 Complex Spaces

XI

135

§ 41

Coherent Sheaves 136 Finite Type, Sheaf of Relations § 42 The Coherence of „(9 145 § 43 Complex Spaces 148 § 44 Germs of Complex Spaces and Analytic Algebras 154 Decomposition of Germs into Irreducible Components § 45 Discrete and Finite Holomorphic Mappings 161 Finite Coherence Theorem, Local Characterization of Finite Morphisms, Embedding, Open Lemma, Complex Spaces over X § 45 A Supplement: Image Sheaves 170 §45B Supplement: The Analytic Spectrum 172 § 46 The Representation Theorem for Prime Germs 173 §46A Supplement: Injective Holomorphic Mappings between Manifolds of the Same Dimension 177 § 46B Supplement: Universal Denominators for Prime Germs 178 Thin Subsets, Weakly Holomorphic Functions § 47 Hubert's Nullstellensatz and Cartan's Coherence Theorem 180 § 48 Dimension Theory 183 Active Lemma § 49 Set of Singular Points and Decomposition into Irreducible Components 191 Identity Theorem, Semicontinuity of Dimension and of Fiber Dimension §49A Supplement: Fiber Products and Quotients 198 Kernel and Cokernel of a Pair of Morphisms, Analytic Equivalence Relations, Group Actions, Orbit Spaces, Weighted Projective Spaces, Stein's Factorization Theorem

Part Three: Function Theory on Stein Spaces Chapter 5 Applications of Theorem B

213

§ 50

Introductory Remarks on Cohomology Resolutions, Abstract de Rham Theorem, Cech Cohomology, Leray's Theorem

214

§ 50A § 51 §51A

Supplement: Automorphic Functions Stein Spaces Supplement: Countable Topology in Complex Spaces

220 223 227

XII

§ 52

Contents

Theorem B and B-Spaces

230

Existence Theorem for Global Holomorphic Functions, Exactness of the Analytic Section-Functor, Local Coordinates by Global Holomorphic Functions, Theorem A, Hartogs's Kugelsatz

§ 53

The Additive Cousin Problem

238

Meromorphic Functions §53A

Supplement: Meromorphic Functions on Reduced Complex Spaces .. 241 Normal Analytic Algebras, Indeterminate Points, Remarks on the Meromorphic Function Field of a Compact Complex Space

§ 54

The Multiplicative Cousin Problem Divisors,-Exponential Sequence, Chern Class, Locally Free Sheaves of Rank 1, Logarithm

244

§ 54A § 54B

Supplement: The Poincare Problem Supplement: Holomorphic Line Bundles

249 251

Locally Free Sheaves, Oka's Principle, Line Bundles on (Pj

§ 55

Coherent Analytic Sheaves as Frechet Sheaves

255

§ 56

Canonical Topology, Privileged Neighborhoods The Exhaustion Theorem Runge Pairs, Globally Generated Analytic Submodules

263

§ 57

The Character Theorem and Holomorphic Hulls Spectrum, Maximal Ideals in Global Function Spaces, Holomorphic Hull, a Stein Quotient Space for a Holomorphically Convex Space

267

§ 58

The Holomorphic Version of de Rham's Theorem

272

Holomorphic Version of Poincare's Lemma

§ 58A

Supplement: The Grassmann Algebra and Differential Forms

275

Vector Fields, Tangent Spaces

Chapter 6 Proof of Theorem B

277

§ 61

278

Dolbeault's Lemma Poincare Lemma (5-Version), de Rham's Theorem (^-Version), Polynomial Polyhedra, the Structure Sheaf of a Polydisk is Acyclic

§ 62

Theorem B for Strictly Pseudoconvex Domains

285

Finiteness Theorem of Cartan-Serre, Grauert's Solution of Levi's Problem

§ 63

Characterization of Stein Spaces

293

Weakly Holomorphically Convex Spaces, Runge Pairs, Characterization of Domains of Holomorphy in C"

§63A § 63 B

Supplement: Levi's Problem for Pseudoconvex Domains Supplement: Weakly Holomorphically Convex Spaces are Holomorphically Convex

298 301

Contents

XIII

Supplement: Chapter 7 Normal Complex Spaces

302

§ 71

302

§ 72

Normalization Normal Complex Spaces, Riemann Removable Singularity Theorem, Universal Denominators Maximal Complex Structure Characterization of Biholomorphic Mappings, Maximalization, Normality of Quotient Spaces

310

§ 73

Finite Mappings on Stein Spaces

313

§ 74

A Criterium for Normality R-Sequence, Riemann Continuation Theorem for Cohomology Classes, Koszul Complex, Local Cohomology

314

List of Examples Bibliography Glossary of Notations Index

323 325 329 337

Interdependence of chapters

0 Elementary Theory

2 Weierstrass Theory

4 3 Reduced Spaces 1 Domains of Holomorphy 4 Complex Spaces

6 Proof of Theorem B

5 Applications of Theorem Β 52.5 Existence of Global Functions

61.8 fc is acyclic

7 Normal Spaces

Courses and Seminars: Guide to the Essentials of Specific Topics

Here we propose ways of using the book for various courses or seminars: 1) Function theory on domains in C": §§ 1-14. 2) Complex manifolds: §§ 1, 3, 4, 6.1-6.6, 8; definition of manifolds with E. 32a; § 32 without the results on singular spaces; § 32 A, B. 3) Analytic algebras and their dimension theory: §§21-24; 45.6 proved as in [GrRe II. 2 Satz 2] ; all of the algebraic results of § 48, up to and including 48. 1 10/g, with proofs; possibly E.48e, (32. 11.1)-32.13, 45.7, and 45.8, as well; for a geometric interpretation of algebraic statements, we recommend the definition of germs of complex spaces and their morphisms as geometric realizations of analytic algebras and their homomorphisms, to be followed with 44.7-44.14 and 45.1-45.14, possibly without the proof of 45.2. 4) The theory of complex spaces can be appended to 3): central themes: Coherence Theorems (Oka, Cartan, Finite), Representation Theorem for Prime Germs, decomposition into irreducible components - §31, examples from §32, §33, and the remainder from Chapter 4 (without the supplement). 5) Domains of holomorphy, either in C" or in Stein manifolds X: the goal is either 63.7 or 63.2: Chapter 1; §§30, 41, 42; specialization to domains in C" (resp., manifolds) of §§ 50-53, § 55, 56.2, and Chapter 6. (The Finite Coherence Theorem in the proof of 63.3 can be avoided for manifolds X that admit local coordinates by global functions: then, for each K = K = K a 17 «re AT, there exists a φ e Hoi (X, Cm) such that

and such that φ \ν is an injective mapping that induces on every stalk a surjective homomorphism; hence, φ: W-.= φ~1(Ρ)η U— *P is a (closed) embedding, i.e., W is a B-space and Runge in X, thus in every analytic polyhedron in X that includes W.) 6) Solution of Levi's problem: Chapter 1, §§ 61, 62, 63 A (prerequisites: Approximation Theorem, sheaf theory, cohomology theory). 7) Quotients of complex spaces: 32 B.3 and its applications in §32 B, §49 A (prerequisites: ringed spaces, coherent sheaves, Finite Coherence Theorem). 8) Normalization: §§45 B, 46 B, 71, 72, possibly 74. 9) Finiteness Theorem ofCartan-Serre: §§ 30, 41, 42, 50, 51, 52, and 55; 56.2; §§ 61, 62. 10) Finite Mappings: 33.1, §§45, 45 B, 46.

Part One: Function Theory on Domains in C"

We begin our exposition of complex analysis by adapting familiar propositions and methods of proof from the one-dimensional theory to the multidimensional case in Chapter 0. In order to make the analogy between the two cases particularly transparent, we introduce holomorphic functions as continuous partially holomorphic functions; we then show that approach to be equivalent to the other usual definitions. In Chapter 1, we work out basic differences between the two theories, the existence of which is discernible in Chapter 0 only in connection with the Kugelsatz. Thus, we show that, contrary to the impression left by Chapter 0, the multidimensional theory does not consist of mere adaptations, but rather that completely new methods must be developed.

Chapter 0: Elementary Properties of Holomorphic Functions In analogy to the historical development of multidimensional function theory, we begin with the basic concepts and results that carry over from the one-dimensional theory relatively easily. The set &(X) of holomorphic functions on an open set1' X oc If the topology of the (locally compact) space Τ is countable4', then Τ can be written as a 00

0

countable union Τ = (J 7J with compacta T j c T j + l S } (seeSl A.2). It follows that #(Γ), provided J=l

with the family of seminorms || · ||r., is still complete, since every Cauchy sequence in #(Γ) (i.e., Cauchy sequence with respect to every || · ||r.) has a unique limit on each TJ. If ||/||r. = 0 for every j, then / = 0. A family of seminorms (Sj)jeJ on a vector space V determines a unique topology, for which the family of sets of the form

W(v, ε; Sj) = {w e V; s}(v - w) < ε} is a subbasis. A vector space with a Hausdorff topology is called a Frechet space if its topology is determined by an at most countable family of seminorms in the manner described above, and if it is complete. For (locally compact) Τ with a countable topology, ^(T) is thus a Frechet space. E. 2a. All families (7}),-eJ with TJoz τ and (J 7) = Τ determine the same topology on J (for 4.3). That topology is called the topology of compact convergence. Every || · \\A is submultiplicative; i. e.,

E. 2b. Find polynomials f,ge C[z] and a Κ αζ C such that

< 11/11*11*11*. If a Frechet space topology is defined on an algebra V by means of submultiplicative seminorms, then V is called a Frechet algebra. In particular, %>(Τ) is a Frechet algebra if Τ has a countable topology. 3)

A o= Γ means that Λ is a relatively compact subset of Γ; i.e., its closure, A, is compact. A countable topology is one that satisfies the second axiom of countability (i.e., there exists a countable basis of open sets). 5) f denotes the interior of T. 4)

6

Function Theory on Domains in C"

2.1 Proposition. For X cr C", the subalgebra G(X) of /+ £, are continuous. Since F, /(-»/+ A, is a homeomorphism for each A e F. Many local topological statements are translation-invariant; such statements need only be investigated at 0 e V. E. 2d. Show that #(Γ) is a locally convex vector space; i.e., every fe^(T) has arbitrarily small convex neighborhoods U. A convex neighborhood U is one such that for each pair of functions f,geU, the segment joining them, {a/+ (1 - a)g; 0 g a g 1}, is included in U. E. 2e. If a Hausdorff topology is given for a vector space V by means of seminorms (Sj)JeW, then σο

ίφ, w) != £ 2"J" min (1, ί,·(ι? - w)) J=l

is a metric that generates the topology of V (hints: i) i/ is translation-invariant: d(h + v,h + w) = d(v,w) = d(v - w, 0), VA e F; ii)K— »|β^ 0 , PH-» e™·—- ; # e IR} is the set of those automorphisms of P1 which send α to 0 az — 1

[Co VI. 2. 5]. Specifically, Aut(P1) is a group with three real parameters. 3.11 Proposition. Iff:B"—>P"

is biholomorphic, then η = 1.

Proof. Since P" is homogeneous, we may assume that /(O) = 0. Moreover, by 3.9 and the Chain Rule 8.2, we may assume that / is linear, and thus defined on all of C". In particular, f(dB") = dP". If Π ;>2, then the boundary dP" includes the real segment {(1, i,0,...,0)eC"; -l g t ^ 1}; since f ~ v is linear, the boundary dB" =/~ 1 (9P") must include a real segment as well. \\ · The identity mapping on a region X cr C" is holomorphic, and the composition of holomorphic mappings is holomorphic. That can be stated concisely: the regions in complex number spaces and their holomorphic mappings form a category. The sets Hol( \ Y) are called the sets ofmorphisms of the category. Similarly, the Frechet algebras with the sets of morphisms .E', F)~ {φ : Ε—> F; φ is a continuous algebra-homomorphism} form a category. E. 3b. For /e Ηο1(Χ, Υ), g e Hol (Υ, Ζ), verify that Ο (id*)0 = i-C?(Ar), ft->@(f)-=f°, between the category of regions and the category of Frechet algebras is called a (contravariant) functor, and i) and ii) are the functor properties. E. 3c. Determine φ° for the following «". For fixed (z l 5 ...,z n ) and F(Cn)-=f(z ,,..., zn _,,(„),

J If Cn is fixed and z 1 ,...,z n _ 1 are variable, then it follows from the induction hypothesis that

The assertion follows by substitution.

·

4.2 Remarks, i) As the nature of the proof indicates, 4.1 holds for a product D1x ... χ D„ in place of P if the one-dimensional version of Cauchy's Formula holds for each D . 7) 8)

For locally analytic subsets of C", we shall introduce a more general notion in 32.3. "Near" a subset A of a topological space means „in some neighborhood of A".

12

Function Theory on Domains in C"

ii) To ensure that f\r = h in 4.1, it suffices that / be continuous on Ρ (α; ρ) and holomorphic on Ρ (α; ρ) [Os I §45]. For a multi-index ν — (v l 5 ..., vJelNT, we use the following notation: v|:= Σ vj; ν!:=ν 1 !·...·ν Ι Ι !;ζ*.= ζϊ.·...·ζ;;»; v + l,= ( V l + 1,..., vn + 1) . 7=1

If /: A'—> C is a |v|-fold partially differentiable function, then Dvf denotes the function dz\i...dzvn* 4.3 Corollary. For each multi-index v, Dv is a continuous endomorphism of (9(X). For each polydisk Ρ = Ρ" (α; ρ)α= Χ with distinguished boundary T, and for each holomorphic function f on X, the following statements hold:

i)

DvJ/(z) = —— f - ν +. 1, άζ fe J for ^ ' " (ζ-ζ)

ii)

| Dvf(a) \£\\.f\\T

zeP.

(Cauchy's Estimate) .

iii) Fix we Ρ and m e INI; //^/or every ze Ρ, Χ includes the "closed complex segment" {(1 - λ)η + λζ; AeP^l)}, then

up| Σ supi L· ~~τ" -M^n·4 ~ w> ι = iiy \\χ· |v| = m v·

The following fact is an immediate consequence of the continuity of D v :

4.4 Weierstrass's Theorem. If (/j) ώ a sequence in &(X) that converges to f, then (£>v/;.) converges in &(X) to Dvf for each v e IM". · Proof of 4.3. The continuous function /(£) / (ζ — z)1 has continuous first partial derivatives with respect to z and z, so differentiation can be performed under the integral in 4.1. Thus, the integral in i) is partially holomorphic and continuous; for v — (v l 5 ..., vn), we have that Dv = Dv> ··· Dv", so i) follows by induction on v (with a suitably chosen order), and we conclude that D v is a vector space endomorphism n

on &(X). Assertion ii) follows from the fact that \άζ = Υ[ 2πρ Λ = (2π)"ρ1, since T fc = we know that *

Elementary Properties of Holomorphic Functions

13

iii) Fix weP, and define a polynomial Pm by means of the assignment

ρ

^}··= Σ ^£>ν/(νΦν-

Now fix z e P; by assumption, then, V-.= [λ e C; (1 - λ)\ν + λζ e X} is an open neighborhood of P1 (1) in C. The function g : V— * C, λ Η->·/(Λζ + (l — A)iv), is holomorphic; by the Chain Rule, ^(0)= Σ ^D'f(w)(z-wr

= mlPm(z-w).

The one-dimensional Cauchy Estimate for g, and the inequality \Pm(z-w)\ = -i-\^jr (0)| g sup m< αλ U| = 1 imply the validity of assertion iii). The continuity of D v remains to be shown; to that end, we apply the following simple fact: 4.5 Lemma. For a linear mapping q>:E—>F of topological vector spaces whose topologies are given by families of seminorms p( and q^ respectively, if9} there exist for each q} an r e IR and a Pi such thatqj°v/||j»(ii;r) for each pair a,r such that P(a; 2r) en X. Then, by ii), we see that

for each b E P(a; r), and it follows that

With the help of 4.3, we want to show that a function is holomorphic iff it can be represented locally by its Taylor series (Weierstrass's definition of holomorphic functions). Before doing that, we need to clarify the concept of convergence of an infinite series of holomorphic functions, by which one could understand, say, pointwise convergence or convergence in 0(X). The following result is an easy consequence of [Di V.3]: 4.6 Remark. Consider a family of functions (/ν)νείκιη with fve&(X). If, for some order of summation, £ |/v| converges in ^ ( X ) , then,for any order of summation, velN"

i)

Σ /v(z) converges absolutely for each ζ e X , vetM"

91

The converse is also easy to show; see [Bou EVT II.5.6 Prop. 9].

14

Function Theory on Domains in C"

"V

Σ l/v I converges in ^(X), and velM"

Hi)

Σ Λ

converges in & (Χ)

velM"

to a limit that is independent of the order of summation.

·

As indicated by the next lemma, pointwise convergence and convergence in are equivalent for a power series on a polydisk P (however, see E.2f). 4.7 Abel's Lemma. If

Σ avzv is a power series and weC*", then the following velM"

statements hold10): i) If the set {avwv; velM"} c C is bounded, then Σ\ανζν converges in #(PB(w)) and Σανζν converges in &(P"(w)). ii) Otherwise, the series Σ\ανζν\ and Σανζν diverge for every zeC" suchthat vveP(z). Proof, i) For z e P(w), we have that ^-^- < 1 for every j; thus, by E. 4a, for each ε > 0, there exists a k E IM such that

z-

Ι Σ *,ζΊ^ Σ K* v l = Σ Ι«ν*ΤΙ — I g c - ^ i - i g c - e , |ν|

fc

|v| i f t

Ιν| a*

w

w

where C ··— sup | av wv \. v

ii) If the series Σανζν were convergent, then {avzv; veiM"} would be bounded; by i), that would imply the boundedness of {avwv; v e l M n } U . · E. 4a. Geometric series. The series Σ Ι Ζ Ί converges in < ^(P n (l)), and Σ veN"

ζ =

veN"

*

Π

j = l 1 ~ zj

in ΛΓ

For a fixed e and sufficiently large JV, the second summand is smaller than ε; for a fixed Ν and sufficiently large β^μ0(ε), the first summand is also smaller than ε: for each velN", we have that (,· αν)μ^η^>·αν, due to the fact that J/1:E^F between two topological vector spaces is called compact if there is a neighborhood U of zero in E such that φ (U) is relatively compact in F. 5.4 Corollary. If i : X —* Υ is the inclusion mapping between two regions in C", and if X is relatively compact in Y, then i°:&(Y)— > (9(X) is a compact mapping. Proof. The set W(Q, 1; || · \\x) = {/e 0(7); \\f\\ x < 1} is a neighborhood of zero in G(Y), since X is compact by assumption. The set ^(W^O, 1; || · \\x)) is bounded in G(X); thus, by Montel's Theorem, it is relatively compact. · E. 5 a. In #(IR), the set {smjx;j e Ν } of real analytic functions is closed and bounded, but not compact.

Elementary Properties of Holomorphic Functions

19

§ 6 The Identity Theorem, the Maximum Principle, and Runge's Theorem A holomorphic function is completely determined on a domain by its values in a neighborhood of a single point : 6. 1 Identity Theorem. Let G be a domain, and suppose that f and g are in &(G). If there exists a nonempty open subset W of G such that f\w= g\w, thenf = g. Proof. Suppose that a e W. Since partial derivatives are formed locally, we see that Dvf(a) = Dv(f\w}(a) = Dv(g\w)(d) = Dvg(a) for every veiN"; the theorem follows from E. 4d. · 6.2 Corollary. 0(X) is an integral domain (i.e., a commutative ring with a unit element and no zero-divisors) iff X is a domain. Proof. If X includes, say, the two components Xl + X2, then the characteristic functions ^ of Xj are holomorphic on X; moreover, χ\χ2 — 0> so ®(X) nas zerodivisors. If X is connected, and if f,gE(9(X) with /V 0, then, by virtue of continuity, there exists an open set W in X on which / has no zeros. Thus, if fg = 0, it follows that g\w = 0; by the Identity Theorem, g = 0. · E. 6a. If a compact subset K of a domain G contains interior points, then || · ||K is a norm on 0«?).

For the next theorem, we need the following terminology: a mapping is called open if it sends open sets to open sets; closed mappings are defined analogously. 6.3 Theorem. If f e & ( X ) is nonconstant on each connected component of X, then f is an open mapping. Proof. For a e X, choose a B (a; r) 0 such that G n [Ρ" ~~ 1 (ε) χ Ρ 1 (r)] is connected. Suppose that there exists a sequence of points (ja) = (, ',jiJ,,);etN converging to 0 such that each [,.α'} χ Pl(jdn;r) is included in G. Show that there exists an open neighborhood U of Ο χ Ρ 1 (r) such that the restriction-mapping i° : Θ (G u 17) —> G (G) is an isomorphism. E. 7d. If φ: Χι —> X2 is a holomorphic mapping, then its graph, Γ(φ)··={(ζ,φ(:))ε Χίχ Χ2}, is an analytic set in Xr χ X2. Factor φ over a holomorphic bijection \\ι: Χι —>Γ(φ). (For 32 Β. 8; in the language of 8.8, Γ(φ) is a submanifold; ψ is actually biholomorphic (see §32).) E. 7e. Laurent series. For domains //ccC"" 1 and R— {zeC 1 ; a < \z\ < b] cc C 1 , show «.' that every function f e ( 9 ( R χ //) has a unique representation of the form / = £ gjz' with j=-x

gj e Θ (Η) such that the series converges absolutely in # (R χ Η). (Hint: With the help of the one-

26

Function Theory on Domains in C"

dimensional Laurent series, construct a representation of the form 1

f f(w>u) A Μ —ζ

l

c f(w,u)

,

§ 8 The Implicit Mapping Theorem In the real differential calculus, linear approximation is one of the fundamental techniques. With its help, the investigation of nonlinear mappings is led back to the methods and results of linear algebra. In our discussion of the adaptation of that method to complex calculus, Υ is a region in Cm and / = (/Ί, ..., fm) : X— >Y is a holomorphic mapping. 8.1 Definition. For fe Hoi (X, Y), let -^- denote the mapping dz X -> Hom(C", C m ), a ^

- (a) =

- (a) iijim IS k gn

The matrix -^— (a) is called the holomorphic functional matrix (or Jacobian matrix) off at a. dz Note that, for such an /, the (a priori IR-linear) derivative daf: C" —»· Cm is Clinear, as it satisfies the equation dj= Σ ^Wzj j=l

OZj

(see E. l a v)); hence, — (a) is the matrix that describes daf with respect to the dz canonical bases of C" and Cm. Thus, the usual computational rules for derivatives in real analysis remain valid for holomorphic functional matrices; in particular, we have the following fact: 8.2 Chain Rule. If g e Uo\(Y, Z), then

dz

=

_0

\dw

.

_

J dz

For n = m, the determinant

exists; it is called the complex functional determinant (or Jacobian determinant) of / at a. With that notation, we have the following fact: P /· -^- (a) is an isomorphism f(X) has an inverse mapping g:f(X)—>X with continuous real derivatives. In order to prove that g is holomorphic, we verify the Cauchy-Riemann differential equation dz —^ — 0 (see E. la). The identity mapping idc« =f°g is holomorphic, so the Chain dw

Rule E. l a implies that

d(f°g)

(df

Λ

dg.

dw •

fl

f*.

since — ° g is invertible, we have that —— = 0 . dz dw E. 8 a. Let the mapping /: X —> C" have partial derivatives with respect to the real variables Xj and yj, and let jj = gj + ihj denote the decomposition of the component f} of / into real and imainar parts. imaginary arts. For

and 8zk/

!(z1,z1z2 + ζ^,ζ^ζ^ — z 2 + 2z 2 z 3 ), of 6.9, Jv(z)= l - 2 z 3 ; hence, for 0 < ε < 1 / 2 , φ is locally invertible on P — P 3 (l + ε, Ι +ε,ε). If φ:Ρ—κρ(Ρ) were not biholomorphic for small ε, then there would exist two convergent sequences (jO) and (jb) in Ρ with }a =£ jb, 1ίηι/ζ3 = 0 = lim^ 3 , and φ(ρ) = φ(β). It is easy to see that the two sequences would have the same limit, so φ would not be bijective near a. 8.5 Theorem. For X a: C", if f: X —>·€" is a holomorphic injection, then f(X) is open in C", and f is a biholomorphic mapping from X to f(X). Theorem 8.5 is equivalent to the following statement: If /e Hoi (X, C") is injective, then Jf(z) Φ 0 for every z e X. For, by 8.3, if that statement holds, then f, being injective, maps X biholomorphically onto f(X); the reverse implication follows from 8.3. We forego the proof of 8.5 at this point, because a general version of the theorem will be proved in 46 A. 1.

28

Function Theory on Domains in C"

E. 8c. Give an example showing that 8.5 does not hold for real analytic functions. Let /: Χ χ Y—> C" be holomorphic. Then, for fixed a e X, the mapping f(a,-):Y-+C>,wi-*f(a,w), is also holomorphic. 8.6 Implicit Mapping Theorem. Suppose thatXxY(rC"x Cm, and that f:Xx Y^> Cm Λ Γ

is holomorphic; let a point ( a , w 0 ) e N ( f )

such that rank ~-(a,w0) — m be given. ow Then there exists an open neighborhood U x W of (a, w0) such that the (a priori setvalued) assignment u^{weW;f(u,w) = 0} determines a holomorphic mapping g: U —> W. Proof. After shrinking X and Υ if necessary, we see by 8.3 that the holomorphic mapping

induces a biholomorphic mapping F:XxY—>F with an inverse mapping H = (pr„, HY). We may assume that V includes .Υ χ 0; then the mapping h ··= HY\ x x 0 is holomorphic, and we conclude that /*/ * /y» j f \ f(z, w)\ = f\Q iff w = h(z).

m^—

As in the real case, those holomorphic functions whose functional matrices have constant rank admit a canonical representation: 8.7 Rank Theorem. Let f : X —» Cm be a holomorphic mapping for which ·— has oz constant rank r near a. Then there exist open neighborhoods U of a and V of b = /(a), polydisks P" e= C" and Pm c Cm, each centered at 0, and biholomorphic mappings φ:Ρ"—> U and φ-.V —*Pm with /m)( z ) = /(*) = /°&(&)

g

g

of homomorphisms of topological algebras. By 3.4, each of the mappings is continuous; by 11.4, σ is bijective. The construction by means of the Cauchy integral, as in the proof of 7.3 (or 55.8 ii)), shows that σ is open. Thus σ is a homeomorphism, and the same must hold for ρ. Now we prove 11.6: for an fe 0(X], 11.7 implies the existence of a unique extension f€&(P) of f\f). By the Identity Theorem, / and / coincide on the domain Pr\X, so / determines a unique extension of / to X u P. It follows immediately from 11.7 that the restriction-mapping is open. · E. lie. i) Show that C χ Ο has a fundamental system of neighborhoods that are logarithmically convex complete Reinhardt domains in C 2 (hint: in IR 2 a 0 > the set (IR^o) χ Ο has a fundamental system of neighborhoods determined by decreasing step functions), ii) Generalize i) to C" χ Ο in C" χ Cm (for E. 63 e).

§ 12 Regions of Holomorphy and Holomorphic Convexity In section 11, we investigated certain Reinhardt domains to see whether every holomorphic function on G could be extended to a domain that properly includes G. In this section, we consider the analogous question for arbitrary regions in C"; our discussion is couched in the following terms: 12.1 Definition. A function fe (9 (X) is called holomorphically extendible

(z — a)v,

a point aeX) to a poly disk Ρ (α; ρ) if its Taylor series, £ -— velN"

V

(from

'

converges on P(a; ρ); it is called holomorphically extendible at a point yeC"\X if, for some aeX, the point y lies in a poly disk Ρ (α; ρ) to which f is holomorphically extendible. The fact that a function fe&(X) is holomorphically extendible from aeX to Ρ = Ρ (α; ρ) does not necessarily imply that / is holomorphically extendible to XuP, for the extension can depend on the given point a (that is the case, for example, with /= |Ae0(C*\[R