Holomorphic Functions of Several Variables

Name: Holomorphic Functions of Several Variables | An Introduction to the Fundamental Theory
Brand: De Gruyter
Price: 139.95 EUR
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An Introduction to the Fundamental Theory

Ludger Kaup Burchard Kaup(Autor*in)

De Gruyter (Verlag)

1. Auflage

Erschienen am 9. Mai 2011

XIII, 350 Seiten

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978-3-11-083835-0 (ISBN)

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Inhalt

Intro
Interdependence of chapters
Courses and Seminars: Guide to the Essentials of Specific Topics
Part One: Function Theory on Domains in Cn
Chapter 0. Elementary Properties of Holomorphic Functions
§ 1 Definition of Holomorphic Function
§ 2 O (X) as a Topological Algebra
§ 3 Holomorphic Mappings
§ 4 Cauchy's Integral Formula
§ 4A Supplement: Cartan's Uniqueness Theorem and Automorphisms of Bounded Domains
§ 5 Montel's Theorem
§ 6 The Identity Theorem, the Maximum Principle, and Runge's Theorem
§ 7 The Riemann Removable Singularity Theorems
§ 8 The Implicite Mapping Theorem
Chapter 1 Regions of Holomorphy
§ 11 Domains of Convergence of Power Series and Reinhardt Domains
§ 12 Regions of Holomorphy and Holomorphic Convexit
§ 12A Supplement: Further Extension Theorems for Holomorphic Functions
§ 12B Supplement: The Edge-of-the-Wedge Theorem
§ 13 Plurisubharmonic Functions
§ 14 Pseudoconvex Domains
Part Two: Function Theory on Analytic Sets
Chapter 2 The Weierstrass Preparation Theorem and its Applications
§ 21 Power Series Algebras
§ 22 The Weierstrass Theorems
§ 23 Algebraic Properties of nO0 and
§23A Supplement: Finite Ring-extensions, Normality
§ 24 Analytic Algebras
Chapter 3 Complex Manifolds and the Elementary Theory of Complex Spaces
§ 30 Bringing in the Sheaves
§ 31 Ringed Spaces and Local Models of Complex Spaces
§ 32 Complex Manifolds and Reduced Complex Spaces
§32A Supplement: Submersions and Immersions
§32B Supplement: Examples of Complex Manifolds
§ 33 Zeros of Polynomials
§33A Supplement: Resultants and Discriminants
§33B Supplement: Proper Mappings and Equivalence Relations
Chapter 4 Complex Spaces
§ 41 Coherent Sheaves
§ 42 The Coherence of nO
§ 43 Complex Spaces
§ 44 Germs of Complex Spaces and Analytic Algebras
§ 45 Discrete and Finite Holomorphic Mappings
§ 45A Supplement: Image Sheaves
§ 45B Supplement: The Analytic Spectrum
§ 46 The Representation Theorem for Prime Germs
§ 46A Supplement: Injective Holomorphic Mappings between Manifolds of the Same Dimension
§ 46B Supplement: Universal Denominators for Prime Germs
§ 47 Hilbert's Nullstellensatz and Cartan's Coherence Theorem
§ 48 Dimension Theory
§ 49 Set of Singular Points and Decomposition into Irreducible Components
§ 49A Supplement: Fiber Products and Quotients
Part Three: Function Theory on Stein Spaces
Chapter 5 Applications of Theorem B
§ 50 Introductory Remarks on Cohomology
§ 50A Supplement: Automorphic Functions
§ 51 Stein Spaces
§ 51A Supplement: Countable Topology in Complex Spaces
§ 52 Theorem B and B-Spaces
§ 53 The Additive Cousin Problem
§ 53A Supplement: Meromorphic Functions on Reduced Complex Spaces
§ 54 The Multiplicative Cousin Problem
§ 54 A Supplement: The Poincaré Problem
§ 54B Supplement: Holomorphic Line Bundles
§ 55 Coherent Analytic Sheaves as Fréchet Sheaves
§ 56 The Exhaustion Theorem
§ 57 The Character Theorem and Holomorphic Hulls
§ 58 The Holomorphic Version of de Rham's Theorem
§ 58A Supplement: The Grassmann Algebra and Differential Forms
Chapter 6 Proof of Theorem B
§ 61 Dolbeault's Lemma
§ 62 Theorem B for Strictly Pseudoconvex Domains
§ 63 Characterization of Stein Spaces
§ 63A Supplement: Levi's Problem for Pseudoconvex Domains
§ 63B Supplement: Weakly Holomorphically Convex Spaces are Holomorphically Convex
Supplement: Chapter 7 Normal Complex Spaces
§ 71 Normalization
§ 72 Maximal Complex Structure
§ 73 Finite Mappings on Stein Spaces
§ 74 A Criterium for Normality
List of Examples
Bibliography
Glossary of Notations
Index

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