``Kiyoshi Oka, at the beginning of his research, regarded the collection of problems which he encountered in the study of domains of holomorphy as large mountains which separate today and tomorrow. Thus, he believed that there could be no essential progress in analysis without climbing over these mountains ... this book is a worthwhile initial step for the reader in order to understand the mathematical world which was created by Kiyoshi Oka.'' --from the Preface This book explains results in the theory of functions of several complex variables which were mostly established from the late nineteenth century through the middle of the twentieth century. In the work, the author introduces the mathematical world created by his advisor, Kiyoshi Oka. In this volume, Oka's work is divided into two parts. The first is the study of analytic functions in univalent domains in ${\mathbf C}^n$. Here Oka proved that three concepts are equivalent: domains of holomorphy, holomorphically convex domains, and pseudoconvex domains; and moreover that the Poincare problem, the Cousin problems, and the Runge problem, when stated properly, can be solved in domains of holomorphy satisfying the appropriate conditions. The second part of Oka's work established a method for the study of analytic functions defined in a ramified domain over ${\mathbf C}^n$ in which the branch points are considered as interior points of the domain. Here analytic functions in an analytic space are treated, which is a slight generalization of a ramified domain over ${\mathbf C}^n$. In writing the book, the author's goal was to bring to readers a real understanding of Oka's original papers. This volume is an English translation of the original Japanese edition, published by the University of Tokyo Press (Japan). It would make a suitable course text for advanced graduate level introductions to several complex variables.
Author(s): Toshio Nishino
Series: TMM193
Publisher: AMS
Year: 2001
Language: English
Pages: 382
Front Cover......Page 1
Title......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 10
Preface to the English Edition......Page 14
Part 1. Fundamental Theory......Page 16
1.1. Complex Euclidean Space......Page 18
1.2. Analytic Functions......Page 23
1.3. Holomorphic Functions......Page 27
1.4. Separate Analyticity Theorem......Page 38
1.5. Domains of Holomorphy......Page 42
2.1. Implicit Functions......Page 52
2.2. Analytic Sets (Local)......Page 59
2.3. Weierstrass Condition......Page 71
2.4. Analytic Sets (Global)......Page 75
2.5. Projections of Analytic Sets in Projective Space......Page 79
3.1. Meromorphic Functions......Page 88
3.2. Cousin Problems in Polydisks......Page 92
3.3. Cousin I Problem in Polynomially Convex Domains......Page 95
3.4. Cousin I Problem in Domains of Holomorphy......Page 100
3.5. Cousin H Problem......Page 106
3.6. Runge Problem......Page 113
4.1. Pseudoconvex Domains......Page 120
4.2. Pseudoconvex Domains with Smooth Boundary......Page 131
4.3. Boundary Problem......Page 139
4.4. Pseudoconcave Sets......Page 146
4.5. Analytic Derived Sets......Page 152
5.1. Holomorphic Mappings of Elementary Domains......Page 162
5.2. Holomorphic Mappings of C^n......Page 167
Part 2. Theory of Analytic Spaces......Page 180
6.1. Ramified Domains......Page 182
6.2. Fundamental Theorem for Locally Ramified Domains......Page 194
6.3. Appendix 1......Page 211
6.4. Appendix 2......Page 215
7.1. Holomorphic Functions on Analytic Sets......Page 224
7.2. Universal Denominators......Page 230
7.3. 0-Modules......Page 235
7.4. Combination Theorems......Page 246
7.5. Local Finiteness Theorem......Page 266
8.1. Analytic Spaces......Page 282
8.2. Analytic Polyhedra......Page 285
8.3. Stein Spaces......Page 295
8.4. Quantitative Estimates......Page 306
8.5. Representation of a Stein space......Page 316
8.6. Appendix......Page 331
9.1. Normal Pseudoconvex Spaces......Page 336
9.2. Linking Problem......Page 343
9.3. Principal Theorem......Page 354
9.4. Unramified Domains Over C'......Page 359
Bibliography......Page 374
Index......Page 378
Back Cover......Page 382
