From Holomorphic Functions to Complex Manifolds / Edition 1

by Klaus Fritzsche, Hans Grauert

Overview

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 co­ cycles 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 sub­ sets of the space en. Many of the well-known properties of holomorphic functions of one variable, such as the Cauchy integral formula or the maxi­ mum 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.

Product Details

ISBN-13:	9780387953953
Publisher:	Springer New York
Publication date:	04/12/2002
Series:	Graduate Texts in Mathematics , #213
Edition description:	2002
Pages:	397
Product dimensions:	6.14(w) x 9.21(h) x 0.04(d)

Table of Contents

I Holomorphic Functions.- 1. Complex Geometry.- 2. Power Series.- 3. Complex Differentiable Functions.- 4. The Cauchy Integral.- 5. The Hartogs Figure.- 6. The Cauchy-Riemann Equations.- 7. Holomorphic Maps.- 8. Analytic Sets.- II Domains of Holomorphy.- 1. The Continuity Theorem.- 2. Plurisubharmonic Functions.- 3. Pseudoconvexity.- 4. Levi Convex Boundaries.- 5. Holomorphic Convexity.- 7. Examples and Applications.- 8. Riemann Domains over Cn.- 9. The Envelope of Holomorphy.- III Analytic Sets.- 1. The Algebra of Power Series.- 2. The Preparation Theorem.- 3. Prime Factorization.- 4. Branched Coverings.- 5. Irreducible Components.- 6. Regular and Singular Points.- IV Complex Manifolds.- 1. The Complex Structure.- 2. Complex Fiber Bundles.- 3. Cohomology.- 4. Meromorphie Functions and Divisors.- 5. Quotients and Submanifolds.- 6. Branched Riemann Domains.- 7. Modifications and Toric Closures.- V Stein Theory.- 1. Stein Manifolds.- 2. The Levi Form.- 3. Pseudoconvexity.- 4. Cuboids.- 5. Special Coverings.- 6. The Levi Problem.- VI Kahler Manifolds.- 1. Differential Forms.- 2. Dolbeault Theory.- 3. Kähler Metrics.- 4. The Inner Product.- 5. Hodge Decomposition.- 6. Hodge Manifolds.- 7. Applications.- VII Boundary Behavior.- 1. Strongly Pseudoconvex Manifolds.- 2. Subelliptic Estimates.- 3. Nebenhüllen.- 4. Boundary Behavior of Biholomorphic Maps.- References.- Index of Notation.

Editorial Reviews

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 toall 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)

From the Publisher