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Analytic Function Theory of Several Variables - Elements of Oka's Coherence (Hardcover, 1st ed. 2016)
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Analytic Function Theory of Several Variables - Elements of Oka's Coherence (Hardcover, 1st ed. 2016)
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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.
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