Analytic Function Theory of Several Variables

Elements of Oka’s Coherence

Authors: Noguchi, Junjiro

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  • 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
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Softcover $89.99
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About this Textbook

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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Reviews

“The book is extremely readable and yet detailed and clear. Its novel viewpoint on Oka’s theory, essentially inverting the order in which the main theorems are proved, will certainly benefit beginning graduate students who are interested in several complex variables. This book is destined to become a classic on the topic of coherence in complex analysis.” (Valentino Tosatti, zbMATH 1380.32001, 2018)

Table of contents (10 chapters)

Table of contents (10 chapters)

Buy this book

eBook $69.99
price for USA in USD
  • ISBN 978-981-10-0291-5
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $89.99
price for USA in USD
  • ISBN 978-981-10-0289-2
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
Softcover $89.99
price for USA in USD
  • ISBN 978-981-10-9124-7
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Analytic Function Theory of Several Variables
Book Subtitle
Elements of Oka’s Coherence
Authors
Copyright
2016
Publisher
Springer Singapore
Copyright Holder
Springer Science+Business Media Singapore
eBook ISBN
978-981-10-0291-5
DOI
10.1007/978-981-10-0291-5
Hardcover ISBN
978-981-10-0289-2
Softcover ISBN
978-981-10-9124-7
Edition Number
1
Number of Pages
XVIII, 397
Number of Illustrations
1 b/w illustrations, 27 illustrations in colour
Topics