From Holomorphic Functions to Complex Manifolds / Edition 1

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Overview

This book is an introduction to the theory of complex manifolds. The authors' 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 involving sheaves, coherence, and higher-dimensional cohomology have been completely 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. Each chapter is complemented by a variety of examples and exercises. The only prerequisite needed to read this book is a knowledge of real analysis and some basic facts from algebra, topology, and the theory of one complex variable. The book can be used as a first introduction to several complex variables as well as a reference for the expert.
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Editorial Reviews

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From the reviews:

MATHEMATICAL REVIEWS

"This new book is a valuable addition to the literature."

K. Fritzsche and H. Grauert

From Holomorphic Functions to Complex Manifolds

"A valuable addition to the literature."—MATHEMATICAL REVIEW

"The book is a nice introduction to the theory of complex manifolds. The authors’ intention is to introduce the reader in a simple way to the most important branches and methods in the theory of several complex variables. … The book is written in a very readable way; it is a nice introduction into the topic." (EMS, March 2004)

"About 25 years ago, the same couple of authors published the forerunner of this work with the title Several Complex Variables … . The experience of forty years of active teaching besides the well-known research career resulted in an admirably well readable simple clean and polished style. … I find this book of extraordinary importance and I recommend it to all students, teachers and researchers in mathematics and even in physics as well." (László L. Stachó, Acta Scientarum Mathematicarum, Vol. 69, 2003)

"This book gives an easily understandable introduction to the theory of complex manifolds. It is self-contained … and leads to deep results such as the Remmert-Stein theorem for analytic sets, finiteness theorems for spaces of cross sections in holomorphic vector bundles, and the solution if the Levi problem, using only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles." (F. Haslinger, Monatshefte für Mathematik, Vol. 142 (3), 2004)

"The book is an essentially extended and modified version of the classical monograph 'Several complex variables' by the same authors. … The monograph is strongly recommended to everybody interested in modern complex analysis, both for students and researchers." (Marek Jarnicki, Zentralblatt MATH, Vol. 1005, 2003)

"The authors state that this book ‘grew out of’ their earlier graduate textbook [Several complex variables, Translated from the German, Springer, New York, 1976; MR 54 # 3004]. The book should not, however, be thought of as merely a second edition. … Where the two books do overlap in content, the exposition in the new volume has been largely rewritten. This new book is a valuable addition to the literature." (Harold P. Boas, Mathematical Reviews, 2003 g)

"This book is an introduction to the theory of complex manifolds. The authors’ 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. … The book can be used as a first introduction to several complex variables as well as a reference for the expert." (L’ENSEIGNEMENT MATHEMATHIQUE, Vol. 48 (3-4), 2002)

"Due to its interior unity and its many-sided applicability, Complex Analysis became an absolutely essential part of today’s Mathematics. … It is a merit of the authors that their book is an introduction into holomorphic functions of several complex variables which is easily understandable. … K. Fritzsche’s and H. Grauert’s book will give a fresh impetus not only to mathematicians who are interested in holomorphic functions in several complex variables but also to those who deal with generalized multi-regular functions." (W. Tutschke, ZAA, Vol. 22 (1), 2003)

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Product Details

  • ISBN-13: 9780387953953
  • Publisher: Springer New York
  • Publication date: 4/1/2002
  • Series: Graduate Texts in Mathematics Series , #213
  • Edition description: 2002
  • Edition number: 1
  • Pages: 398
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.00 (d)

Table of Contents

Preface
I Holomorphic Functions 1
1 Complex Geometry 1
2 Power Series 9
3 Complex Differentiable Functions 14
4 The Cauchy Integral 17
5 The Hartogs Figure 23
6 The Cauchy-Riemann Equations 26
7 Holomorphic Maps 30
8 Analytic Sets 36
II Domains of Holomorphy 43
1 The Continuity Theorem 43
2 Plurisubharmonic Functions 52
3 Pseudoconvexity 60
4 Levi Convex Boundaries 64
5 Holomorphic Convexity 73
6 Singular Functions 78
7 Examples and Applications 82
8 Riemann Domains over C[superscript n] 87
9 The Envelope of Holomorphy 96
III Analytic Sets 105
1 The Algebra of Power Series 105
2 The Preparation Theorem 110
3 Prime Factorization 116
4 Branched Coverings 123
5 Irreducible Components 135
6 Regular and Singular Points 143
IV Complex Manifolds 153
1 The Complex Structure 153
2 Complex Fiber Bundles 171
3 Cohomology 182
4 Meromorphic Functions and Divisors 192
5 Quotients and Submanifolds 203
6 Branched Riemann Domain 226
7 Modification and Toric Closures 235
V Stein Theory 251
1 Stein Manifolds 251
2 The Levi Form 260
3 Pseudoconvexity 266
4 Cuboids 276
5 Special Coverings 282
6 The Levi Problem 289
VI Kahler Manifolds 297
1 Differential Algebra 297
2 Dolbeault Theory 303
3 Kahler Metrics 314
4 The Inner Product 322
5 Hodge Decomposition 329
6 Hodge Manifolds 341
7 Applications 348
VII Boundary Behavior 355
1 Strongly Pseudoconvex Manifolds 355
2 Subelliptic Estimates 357
3 Nebenhullen 364
4 Boundary Behavior of Biholomorphic Maps 367
References 375
Index of Notation 381
Index 387
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