From Holomorphic Functions to Complex Manifolds - Fritzsche, Klaus

From Holomorphic Functions to Complex Manifolds

Klaus Fritzsche

Yayınevi: Springer

Yayın tarihi: 04/2002

ISBN: 9780387953953

Yazar : Hans Grauert

Ciltli | İngilizce | 392 Sayfa |

Tür: Matematik-İstatistik

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This book is an introduction to the theory of complex manifolds. The authors's 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.

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

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