Overview
- 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
- Includes supplementary material: sn.pub/extras
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Table of contents (11 chapters)
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About this book
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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Bibliographic Information
Book Title: Analytic Function Theory of Several Variables
Book Subtitle: Elements of Oka’s Coherence
Authors: Junjiro Noguchi
DOI: https://doi.org/10.1007/978-981-10-0291-5
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media Singapore 2016
Hardcover ISBN: 978-981-10-0289-2Published: 07 October 2016
Softcover ISBN: 978-981-10-9124-7Published: 15 June 2018
eBook ISBN: 978-981-10-0291-5Published: 16 August 2016
Edition Number: 1
Number of Pages: XVI, 397
Number of Illustrations: 116 b/w illustrations
Topics: Several Complex Variables and Analytic Spaces, Category Theory, Homological Algebra, Algebraic Geometry