Graduate Texts in Mathematics

Transcription

1 Graduate Texts in Mathematics 38 Editorial Board F. W. Gehring P. R. Halmos Managing Editor c. C. Moore

2 H. Grauert K. Fritzsche Several Complex Variables Springer-Verlag New York Heidelberg Berlin

3 H. Grauert K. Fritzsche Mathematischen Institut der Universitiit B unsenstrasse 3-5 Mathematischen Institut der Universitat Bunsenstrasse Gottingen 34 Gottingen Federal Republic of Germany Federal Republic of Germany Editorial Board P. R. Halmos Managing Editor University of California Department of Mathematics Santa Barbara, California F. W. Gehring University of Michigan Department of Mathematics Ann Arbor, Michigan C. C. Moore University of California at Berkeley Department of Mathematics Berkeley, California AMS Subject Classifications: 32-01, 32A05, 32A07, 32AIO, 32A20, 32BIO, 32CIO, 32C35, 32D05, 32DlO, 32ElO Library of Congress Cataloging in Publication Data Grauert, Hans, Several complex variables. (Graduate texts in mathematics; 38) Translation of Einftihrung in die Funktionentheorie mehrerer Veranderlicher. Bibliography: p. 201 Includes index. 1. Functions of several complex variables. I. Fritzsche, Klaus, joint author. II. Title. III. Series. QA331.G69 515' All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag by Springer-Verlag Inc. Softcover reprint of the hardcover 1st edition 1976 ISBN-13: e-isbn-13: DOl: /

4 Preface The present book grew out of introductory lectures on the theory offunctions of several variables. Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains. It turns out that, in l:ontrast to the theory of a single variable, for n ~ 2 there exist domains G, G c en with G c G and G "# G such that each function holomorphic in G has a continuation on G. Domains G for which such a G does not exist are called domains of holomorphy. In Chapter 2 we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem). We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued. The third chapter presents the Weierstrass formula and the Weierstrass preparation theorem with applications to the ring of convergent power series. It is shown that this ring is a factorization, a Noetherian, and a Hensel ring. Furthermore we indicate how the obtained algebraic theorems can be applied to the local investigation of analytic sets. One achieves deep results in this connection by using sheaf theory, the basic concepts of which are discussed in the fourth chapter. In Chapter V we introduce complex manifolds and give several examples. We also examine the different closures of en and the effects of modifications on complex manifolds. Cohomology theory with values in analytic sheaves connects sheaf theory v

5 Preface with the theory of functions on complex manifolds. It is treated and applied in Chapter VI in order to express the main results for domains ofholomorphy and Stein manifolds (for example, the solvability of the Cousin problems). The seventh chapter is entirely devoted to the analysis of real differentiability in complex notation, partial differentiation with respect to z, z, and complex functional matrices, topics already mentioned in the first chapter. We define tangential vectors, differential forms, and the operators d, d', d". The theorems of Dolbeault and de Rham yield the connection with cohomology theory. The authors develop the theory in full detail and with the help of numerous figures. They refer to the literature for theorems whose proofs exceed the scope of the book. Presupposed are only a basic knowledge of differential and integral calculus and the theory of functions of one variable, as well as a few elements from vector analysis, algebra, and general topology. The book I is written as an introduction and should be of interest to the specialist and the nonspecialist alike. Gottingen, Spring 1976 H. Grauert K. Fritzsche vi

6 Contents Chapter I Holomorphic Functions 1 Power Series 2 Complex Differentiable Functions 3 The Cauchy Integral 4 Identity Theorems 5 Expansion in Reinhardt Domains 6 Real and Complex Differentiability 7 Holomorphic Mappings Chapter II Domains of Holomorphy 1 The Continuity Theorem 2 Pseudo convexity 3 Holomorphic Convexity 4 The Thullen Theorem 5 Holomorphically Convex Domains 6 Examples 7 Riemann Domains over en 8 Holomorphic Hulls Chapter III The Weierstrass Preparation Theorem 1 The Algebra of Power Series. 2 The Weierstrass Formula vii

7 Contents 3 Convergent Power Series 4 Prime Factorization 5 Further Consequences (Hensel Rings, Noetherian Rings) 6 Analytic Sets Chapter IV Sheaf Theory 1 Sheaves of Sets 2 Sheaves with Algebraic Structure 3 Analytic Sheaf Morphisms 4 Coherent Sheaves Chapter V Complex Manifolds Complex Ringed Spaces 2 Function Theory on Complex Manifolds 3 Examples of Complex Manifolds 4 Closures of en Chapter VI Cohomology Theory 1 Flabby Cohomology 2 The Cech Cohomology 3 Double Complexes 4 The Cohomology Sequence 5 Main Theorem on Stein Manifolds Chapter VII Real Methods 1 Tangential Vectors 2 Differential Forms on Complex Manifolds 3 Cauchy Integrals 4 Dolbeault's Lemma 5 Fine Sheaves (Theorems of Dolbeault and de Rham) List of symbols Bibliography Index Vlll