Noguchi, Junjiro, "Analytic Function Theory of Several Variables: Elements of Okas Coherence"
English | 2016 | ISBN-10: 9811002894 | 397 pages | pdf | 4.5 MB
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 Okas Three Coherence Theorems, Oka-Cartans Fundamental Theorem, and Okas Theorem on Levis 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-Remmerts two volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a lowering of the level for novice graduate students (here, Grauerts direct image theorem is limited to the case of finite maps).The core of the theory is "Okas 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 Okas 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-Cartans Fundamental Theorem (Chap. 4); Coherence of ideal sheaves of complex analytic subsets (Chap. 6); Coherence of the normalization sheaves of complex spaces (Chap. 6); Grauerts Finiteness Theorem (Chaps. 7, 8); Okas 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-Serres Theorem and Kodairas 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.