Several Complex Variables and the Geometry of Real Hypersurfaces

✍ Scribed by John P. D’Angelo

• Publisher: CRC Press
• Year: 1993
• Tongue: English
• Leaves: 288
• Series: Studies in Advanced Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

Several Complex Variables and the Geometry of Real Hypersurfaces covers a wide range of information from basic facts about holomorphic functions of several complex variables through deep results such as subelliptic estimates for the ?-Neumann problem on pseudoconvex domains with a real analytic boundary. The book focuses on describing the geometry of a real hypersurface in a complex vector space by understanding its relationship with ambient complex analytic varieties. You will learn how to decide whether a real hypersurface contains complex varieties, how closely such varieties can contact the hypersurface, and why it's important. The book concludes with two sets of problems: routine problems and difficult problems (many of which are unsolved). Principal prerequisites for using this book include a thorough understanding of advanced calculus and standard knowledge of complex analysis in one variable. Several Complex Variables and the Geometry of Real Hypersurfaces will be a useful text for advanced graduate students and professionals working in complex analysis.

✦ Table of Contents

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 10
1.1 Preliminaries ......Page 16
1.2 Holomorphic mappings ......Page 26
1.3 Further applications ......Page 40
1.4 Basic analytic geometry ......Page 46
2.1 Finite analytic mappings ......Page 57
2.2 Intersection numbers ......Page 75
2.3 The order of contact of an ideal ......Page 86
2.4 Higher order invariants ......Page 100
3.1 CR geometry......Page 103
3.2 Algebraic real hypersurfaces and complex varieties......Page 115
3.3 Real analytic subvarieties ......Page 123
4.1 Orders of contact ......Page 138
4.2 Local bounds ......Page 150
4.3 Other finite type conditions ......Page 156
4.4 Conclusions ......Page 163
5.1 Rational proper mappings ......Page 166
5.2 Invariance under fixed-point-free finite unitary groups ......Page 186
5.3 Boundary behavior......Page 203
6.1 Introduction to the problem ......Page 209
6.2 Existence and regularity results on the \bar\partial and \bar\partial-Neumann problems......Page 213
6.3 Subellipticity ......Page 214
6.4 Subelliptic multipliers ......Page 221
6.5 Varieties of positive holomorphic dimension ......Page 240
7.1 The Bergman projection......Page 257
7.2 Boundary invariants and CR mappings ......Page 264
Problems ......Page 269
Index of Notation......Page 276
Bibliography......Page 280
Index ......Page 286
Back Cover......Page 288