Noguchi J. Analytic Function Theory of Several Variables: Elements of Oka’s Coherence

Springer Science+Business Media, Singapore, 2016. — 404 p. — ISBN10: 9811002894Is 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 the present volume, a number of classical proofs are improved and simplified, so that the contents are easily accessible for beginning graduate students.Holomorphic Functions
Oka’s First Coherence Theorem
Sheaf Cohomology
Holomorphically Convex Domains and the Oka–Cartan Fundamental Theorem
Domains of Holomorphy
Analytic Sets and Complex Spaces
Pseudoconvex Domains and Oka’s Theorem
Cohomology of Coherent Sheaves and Kodaira’s Embedding Theorem
On Coherence