Analytic functions of several complex variables / Robert C. Gunning, Hugo Rossi.

Format:

Book

Author/Creator:

Gunning, Robert C. (Robert Clifford), 1931- author.

Contributor:

Rossi, Hugo.

Series:

AMS Chelsea Publishing Series

AMS Chelsea Publishing, v. 368

Language:

English

Subjects (All):

Functions of several complex variables.

Functional analysis.

Physical Description:

1 online resource (338 pages) : illustrations

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : AMS Chelsea Publishing, 2015.

Summary:

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincaré and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods.The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces.Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.Further information about this book, including updates, can be found at the following URL: www.ams.org/publications/authors/books/postpub/chel-368.

Contents:

Front Cover

Preface to the AMS/Chelsea Edition

PREFACE

CONTENTS

CHAPTER I: HOLOMORPHIC FUNCTIONS

A. The Elementary Properties of Holomorphic Functions

B. Holomorphic Mappings and Complex Manifolds

C. Removable Singularities

D. The Calculus of Differential Forms

E. The Cousin Theorem

F. Polynomial Approximations

G. Envelopes of Holomorphy

H. Some Applications to Uniform Algebras

Notes

CHAPTER II: LOCAL RINGS OF HOLOMORPHIC FUNCTIONS

A. The Elementary Properties of the Local Rings

B. The Weierstrass Theorems

C. Modules over the Local Rings

D. The Extended Weierstrass Division Theorem

E. Germs of Varieties

CHAPTER III: VARIETIES

A. The Nullstellensatz for Prime Ideals, and Local Parameterization

B. Analytic Covers

C. Dimension

CHAPTER IV: ANALYTIC SHEAVES

A. The Elementary Properties of Sheaves

B. Sheaves of Modules

C. Analytic Sheaves on Subdomains of C n

D. Analytic Sheaves of Subvarieties of C n

CHAPTER V: ANALYTIC SPACES

A. Definitions and Examples

B. Holomorphic Functions on an Analytic Space

C. The Proper Mapping Theorem

D. Nowhere Degenerate Maps

CHAPTER VI: COHOMOLOGY THEORY

A. Soft Sheaves and Fine Sheaves

B. The Axioms of Sheaf Cohomology

C. The Theorem of Dolbeault on Cohomology

D. Leray's Theorem on Cohomology

E. Cartan's Lemma

F. Amalgamation of Syzygies

CHAPTER VII: STEIN SPACES, GEOMETRIC THEORY

A. Approximation Theorems

B. Special Analytic Polyhedra

C. The Imbedding Theorem

D. Uses of Special Analytic Polyhedra

CHAPTER VIII: STEIN SPACES, SHEAF THEORY

A. Frechet Sheaves

B. Meromorphic Functions

C. Locally Free Sheaves

CHAPTER IX: PSEUDOCONVEXITY

A. The Complex Hessian

B. Grauert's Solution of Levi's Problem.

C. Plurisubharmonic Functions

D. Oka's Pseudoconvexity Theorem

E. Kodaira's Theorem on Projective Varieties

APPENDIX A: PARTITIONS OF UNITY

APPENDIX B: THE THEOREM OF SCHWARTZ ONFRECHET SPACES

REFERENCES

BIBLIOGRAPHY

INDEX

Back Cover.

Notes:

Originally published: Englewood Cliffs, N.J. : Prentice-Hall, 1965, in series: Prentice-Hall series in modern analysis.

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1576-3

OCLC:

982020365