Holomorphic functions of several variables: an introduction to the fundamental theory
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Berlin [u.a.]
Walter de Gruyter
1983
|
| Series: | De Gruyter Studies in Mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 349 Seiten graph. Darst. |
| ISBN: | 3110041502 |
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MARC
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| 100 | 1 | |a Kaup, Ludger |d 1939- |e Verfasser |0 (DE-588)106035959 |4 aut | |
| 245 | 1 | 0 | |a Holomorphic functions of several variables |b an introduction to the fundamental theory |c Ludger Kaup, Burchard Kaup. With the assistance of Gottfried Barthel. Translated by Michael Bridgland |
| 264 | 1 | |a Berlin [u.a.] |b Walter de Gruyter |c 1983 | |
| 300 | |a XIII, 349 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
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| adam_text | Contents
Interdependence of chapters XIV
Courses and Seminars: Guide to the Essentials of Specific Topics XV
Part One: Function Theory on Domains in C
Chapter 0 Elementary Properties of Holomorphic Functions 1
§ 1 Definition of Holomorphic Functions 2
§ 2 0 (X) as a Topological Algebra 4
§ 3 Holomorphic Mappings 7
§ 4 Cauchy s Integral Formula 10
Cauchy s Estimate, Abel s Lemma, Taylor Series, Liouville s Theorem
§4A Supplement: Cartan s Uniqueness Theorem and Automorphisms of
Bounded Domains 16
§ 5 Montel s Theorem 17
§ 6 The Identity Theorem, the Maximum Principle, and Runge s
Theorem 19
§ 7 The Riemann Removable Singularity Theorems 21
Analytic Sets, Codimension, Kugelsatz
§ 8 The Implicite Mapping Theorem 26
Complex Functional Determinant, Inverse Mapping Theorem, Rank
Theorem, Submanifolds
Chapter 1 Regions of Holomorphy 31
§ 11 Domains of Convergence of Power Series and Reinhardt Domains .. 32
Hartogs s Figure, Hartogs s Kontinuitatssatz
§ 12 Regions of Holomorphy and Holomorphic Convexity 36
Holomorphically Extendible at a Point, Thullen s Lemma
§ 12A Supplement: Further Extension Theorems for Holomorphic
Functions 43
§ 12B Supplement: The Edge of the Wedge Theorem 46
X Contents
§ 13 Plurisubharmonic Functions 49
Levi Form
§ 14 Pseudoconvex Domains 56
Part Two: Function Theory on Analytic Sets
Chapter 2 The Weierstrass Preparation Theorem and its Applications 64
§ 21 Power Series Algebras : 65
the Norm || ||r, Homomorphisms of Power Series Algebras
§ 22 The Weierstrass Theorems 71
Distinguished Power Series, Weierstrass Preparation Theorem, Weierstrass
Division Formula, Shearing
§ 23 Algebraic Properties of n(90 and „ 80
Noetherian Rings, Factorial Rings, Normal Rings, Hensel s Lemma,
Representation of nCr0[T]/(P)
§ 23 A Supplement: Finite Ring extensions, Normality 86
Dedekind s Lemma, Nakayama s Lemma, Krull s Intersection Theorem,
Equation of Integral Dependence
§ 24 Analytic Algebras 89
Finite Ring homomorphism, Noether s Normalization Theorem, Finite
Extensions of Analytic Algebras
Chapter 3 Complex Manifolds and the Elementary Theory of Complex Spaces . 92
§ 30 Bringing in the Sheaves 93
§ 31 Ringed Spaces and Local Models of Complex Spaces 97
Reduction, Subspaces and Ideal Sheaves, Comorphisms, Restriction Lemma,
Nilpotent Elements
§ 32 Complex Manifolds and Reduced Complex Spaces 106
Projective Algebraic Varieties, Set of Singular Points, Tangent Space
§ 32A Supplement: Submersions and Immersions 114
§ 32B Supplement: Examples of Complex Manifolds 117
Lie Groups, Manifolds as Quotient Spaces, Torus, Grassmann Manifold,
Transformation Groups, Quadratic Transformation
§ 33 Zeros of Polynomials 126
Branched Coverings, Finite Holomorphic Mappings
§ 33A Supplement: Resultants and Discriminants 131
§ 33B Supplement: Proper Mappings and Equivalence Relations 133
Contents XI
Chapter 4 Complex Spaces 135
§ 41 Coherent Sheaves 136
Finite Type, Sheaf of Relations
§ 42 The Coherence of nC 145
§ 43 Complex Spaces 148
§ 44 Germs of Complex Spaces and Analytic Algebras 154
Decomposition of Germs into Irreducible Components
§ 45 Discrete and Finite Holomorphic Mappings 161
Finite Coherence Theorem, Local Characterization of Finite Morphisms,
Embedding, Open Lemma, Complex Spaces over X
§ 45 A Supplement: Image Sheaves 170
§ 45 B Supplement: The Analytic Spectrum 172
§ 46 The Representation Theorem for Prime Germs 173
§46A Supplement: Injective Holomorphic Mappings between Manifolds of
the Same Dimension 177
§ 46B Supplement: Universal Denominators for Prime Germs 178
Thin Subsets, Weakly Holomorphic Functions
§ 47 Hilbert s Nullstellensatz and Cartan s Coherence Theorem 180
§ 48 Dimension Theory 183
Active Lemma
§ 49 Set of Singular Points and Decomposition into Irreducible
Components 191
Identity Theorem, Semicontinuity of Dimension and of Fiber Dimension
§ 49A Supplement: Fiber Products and Quotients 198
Kernel and Cokernel of a Pair of Morphisms, Analytic Equivalence Relations,
Group Actions, Orbit Spaces, Weighted Projective Spaces, Stein s Factorization
Theorem
Part Three: Function Theory on Stein Spaces
Chapter 5 Applications of Theorem B 213
§ 50 Introductory Remarks on Cohomology 214
Resolutions, Abstract de Rham Theorem, Cech Cohomology, Leray s
Theorem
§50A Supplement: Automorphic Functions 220
§ 51 Stein Spaces 223
§ 51A Supplement: Countable Topology in Complex Spaces 227
XII Contents
§ 52 Theorem B and B Spaces 230
Existence Theorem for Global Holomorphic Functions, Exactness of the
Analytic Section Functor, Local Coordinates by Global Holomorphic Functions,
Theorem A, Hartogs s Kugelsatz
§ 53 The Additive Cousin Problem 238
Meromorphic Functions
§ 53 A Supplement: Meromorphic Functions on Reduced Complex Spaces .. 241
Normal Analytic Algebras, Indeterminate Points, Remarks on the
Meromorphic Function Field of a Compact Complex Space
§ 54 The Multiplicative Cousin Problem 244
Divisors, Exponential Sequence, Chern Class, Locally Free Sheaves of
Rank 1, Logarithm
§ 54A Supplement: The Poincare Problem 249
§ 54B Supplement: Holomorphic Line Bundles 251
Locally Free Sheaves, Oka s Principle, Line Bundles on Pt
§ 55 Coherent Analytic Sheaves as Frechet Sheaves 255
Canonical Topology, Privileged Neighborhoods
§ 56 The Exhaustion Theorem 263
Runge Pairs, Globally Generated Analytic Submodules
§ 57 The Character Theorem and Holomorphic Hulls 267
Spectrum, Maximal Ideals in Global Function Spaces, Holomorphic Hull,
a Stein Quotient Space for a Holomorphically Convex Space
§ 58 The Holomorphic Version of de Rham s Theorem 272
Holomorphic Version of Poincare s Lemma
§58A Supplement: The Grassmann Algebra and Differential Forms 275
Vector Fields, Tangent Spaces
Chapter 6 Proof of Theorem B 277
§ 61 Dolbeault s Lemma 278
Poincare Lemma (5 Version), de Rham s Theorem (^ Version), Polynomial
Polyhedra, the Structure Sheaf of a Polydisk is Acyclic
§ 62 Theorem B for Strictly Pseudoconvex Domains 285
Finiteness Theorem of Cartan Serre, Grauert s Solution of Levi s Problem
§ 63 Characterization of Stein Spaces 293
Weakly Holomorphically Convex Spaces, Runge Pairs, Characterization
of Domains of Holomorphy in C
§ 63A Supplement: Levi s Problem for Pseudoconvex Domains 298
§ 63 B Supplement: Weakly Holomorphically Convex Spaces are
Holomorphically Convex 301
Contents XIII
Supplement: Chapter 7 Normal Complex Spaces 302
§ 71 Normalization 302
Normal Complex Spaces, Riemann Removable Singularity Theorem,
Universal Denominators
§ 72 Maximal Complex Structure 310
Characterization of Biholomorphic Mappings, Maximalization, Normality
of Quotient Spaces
§ 73 Finite Mappings on Stein Spaces 313
§ 74 A Criterium for Normality 314
R Sequence, Riemann Continuation Theorem for Cohomology Classes,
Koszul Complex, Local Cohomology
List of Examples 323
Bibliography 325
Glossary of Notations 329
Index 337
|
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| dewey-raw | 515.9 |
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| institution | BVB |
| isbn | 3110041502 |
| language | English German |
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| spelling | Kaup, Ludger 1939- Verfasser (DE-588)106035959 aut Holomorphic functions of several variables an introduction to the fundamental theory Ludger Kaup, Burchard Kaup. With the assistance of Gottfried Barthel. Translated by Michael Bridgland Berlin [u.a.] Walter de Gruyter 1983 XIII, 349 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics 3 Fonctions holomorphes Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd rswk-swf Mehrere Variable (DE-588)4277015-4 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Holomorphe Funktion (DE-588)4025645-5 s Mehrere Variable (DE-588)4277015-4 s DE-604 Funktionentheorie (DE-588)4018935-1 s Kaup, Burchard 1940- Verfasser (DE-588)172181429 aut De Gruyter Studies in Mathematics 3 (DE-604)BV000005407 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001207162&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kaup, Ludger 1939- Kaup, Burchard 1940- Holomorphic functions of several variables an introduction to the fundamental theory De Gruyter Studies in Mathematics Fonctions holomorphes Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd Mehrere Variable (DE-588)4277015-4 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4025645-5 (DE-588)4277015-4 (DE-588)4018935-1 |
| title | Holomorphic functions of several variables an introduction to the fundamental theory |
| title_auth | Holomorphic functions of several variables an introduction to the fundamental theory |
| title_exact_search | Holomorphic functions of several variables an introduction to the fundamental theory |
| title_full | Holomorphic functions of several variables an introduction to the fundamental theory Ludger Kaup, Burchard Kaup. With the assistance of Gottfried Barthel. Translated by Michael Bridgland |
| title_fullStr | Holomorphic functions of several variables an introduction to the fundamental theory Ludger Kaup, Burchard Kaup. With the assistance of Gottfried Barthel. Translated by Michael Bridgland |
| title_full_unstemmed | Holomorphic functions of several variables an introduction to the fundamental theory Ludger Kaup, Burchard Kaup. With the assistance of Gottfried Barthel. Translated by Michael Bridgland |
| title_short | Holomorphic functions of several variables |
| title_sort | holomorphic functions of several variables an introduction to the fundamental theory |
| title_sub | an introduction to the fundamental theory |
| topic | Fonctions holomorphes Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd Mehrere Variable (DE-588)4277015-4 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Fonctions holomorphes Holomorphic functions Holomorphe Funktion Mehrere Variable Funktionentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001207162&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
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