Analytic Function Theory of Several Variables
Elements of Oka’s Coherence

Authors: Junjiro Noguchi



Is an easily readable and enjoyable text on the classical analytic function theory of several complex variables for new graduate students in mathematics

Includes complete proofs of Oka's Three Coherence Theorems, Oka–Cartan's Fundamental Theorem, and Oka's Theorem on Levi's problem for Riemann domains

Can easily be used for courses and lectures with self-contained treatments and a number of simplifications of classical proofs

The purpose of this book is to present the classical analytic function theory of several variables as a standard subject in a course of mathematics after learning the elementary materials (sets, general topology, algebra, one complex variable). This includes the essential parts of Grauert–Remmert's two volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a lowering of the level for novice graduate students (here, Grauert's direct image theorem is limited to the case of finite maps).The core of the theory is "Oka's Coherence", found and proved by Kiyoshi Oka. It is indispensable, not only in the study of complex analysis and complex geometry, but also in a large area of modern mathematics. In this book, just after an introductory chapter on holomorphic functions (Chap. 1), we prove Oka's First Coherence Theorem for holomorphic functions in Chap. 2. This defines a unique character of the book compared with other books on this subject, in which the notion of coherence appears much later.The present book, consisting of nine chapters, gives complete treatments of the following items: Coherence of sheaves of holomorphic functions (Chap. 2); Oka–Cartan's Fundamental Theorem (Chap. 4); Coherence of ideal sheaves of complex analytic subsets (Chap. 6); Coherence of the normalization sheaves of complex spaces (Chap. 6); Grauert's Finiteness Theorem (Chaps. 7, 8); Oka's Theorem for Riemann domains (Chap. 8). The theories of sheaf cohomology and domains of holomorphy are also presented (Chaps. 3, 5). Chapter 6 deals with the theory of complex analytic subsets. Chapter 8 is devoted to the applications of formerly obtained results, proving Cartan–Serre's Theorem and Kodaira's Embedding Theorem. In Chap. 9, we discuss the historical development of "Coherence".It is difficult to find a book at this level that treats all of the above subjects in a completely self-contained manner. In the present volume, a number of classical proofs are improved and simplified, so that the contents are easily accessible for beginning graduate students.

Table of contents

Front Matter
Pages i-xviii

Holomorphic Functions
Pages 1-32

Oka’s First Coherence Theorem
Pages 33-63

Sheaf Cohomology
Pages 65-110

Holomorphically Convex Domains and the Oka–Cartan Fundamental Theorem
Pages 111-153

Domains of Holomorphy
Pages 155-201

Analytic Sets and Complex Spaces
Pages 203-279

Pseudoconvex Domains and Oka’s Theorem
Pages 281-341

Cohomology of Coherent Sheaves and Kodaira’s Embedding Theorem
Pages 343-366

On Coherence
Pages 367-373

Erratum to: Analytic Function Theory of Several Variables
Pages E1-E1

Back Matter
Pages 375-397

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