From holomorphic functions to complex manifolds:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
2002
|
| Series: | Graduate Texts in Mathematics
213 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 392 Seiten graph. Darst. |
| ISBN: | 0387953957 9780387953953 |
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MARC
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| 245 | 1 | 0 | |a From holomorphic functions to complex manifolds |c Klaus Fritzsche, Hans Grauert |
| 264 | 1 | |a New York, NY [u.a.] |b Springer |c 2002 | |
| 300 | |a XV, 392 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Variétés complexes | |
| 650 | 4 | |a Complex manifolds | |
| 650 | 4 | |a Holomorphic functions | |
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Record in the Search Index
| _version_ | 1804129121021722624 |
|---|---|
| adam_text | Contents
Preface v
I Holomorphic Functions 1
1. Complex Geometry 1
Real and Complex Structures 1
Hermitian Forms and Inner Products 3
Balls and Polydisks 5
Connectedness 6
Reinhardt Domains 7
2. Power Series 9
Polynomials 9
Convergence 9
Power Series 11
3. Complex Differentiable Functions 14
The Complex Gradient 14
Weakly Holomorphic Functions 15
Holomorphic Functions 16
4. The Cauchy Integral 17
The Integral Formula 17
Holomorphy of the Derivatives 19
The Identity Theorem 22
5. The Hartogs Figure 23
Expansion in Reinhardt Domains 23
Hartogs Figures 25
6. The Cauchy Riemann Equations 26
Real Differentiable Functions 26
Wirtinger s Calculus 28
The Cauchy Riemann Equations 29
7. Holomorphic Maps 30
The Jacobian 30
Chain Rules 32
Tangent Vectors 32
The Inverse Mapping 33
8. Analytic Sets 36
Analytic Subsets 36
Bounded Holomorphic Functions 38
Regular Points 39
Injective Holomorphic Mappings 41
x Contents
II Domains of Holomorphy 43
1. The Continuity Theorem 43
General Hartogs Figures 43
Removable Singularities 45
The Continuity Principle 47
Hartogs Convexity 48
Domains of Holomorphy 49
2. Plurisubharmonic Functions 52
Subharmonic Functions 52
The Maximum Principle 55
Differentiable Subharmonic Functions 55
Plurisubharmonic Functions 56
The Levi Form 57
Exhaustion Functions 58
3. Pseudoconvexity 60
Pseudoconvexity 60
The Boundary Distance 60
Properties of Pseudoconvex Domains 63
4. Levi Convex Boundaries 64
Boundary Functions 64
The Levi Condition 66
Affine Convexity 66
A Theorem of Levi 69
5. Holomorphic Convexity 73
Affine Convexity 73
Holomorphic Convexity 75
The Cartan Thullen Theorem 76
6. Singular Functions 78
Normal Exhaustions 78
Unbounded Holomorphic Functions 79
Sequences 80
7. Examples and Applications 82
Domains of Holomorphy 82
Complete Reinhardt Domains 83
Analytic Polyhedra 85
8. Riemann Domains over Câ„¢ 87
Riemann Domains 87
Union of Riemann Domains 91
9. The Envelope of Holomorphy 96
Holomorphy on Riemann Domains 96
Envelopes of Holomorphy 97
Pseudoconvexity 99
Boundary Points 100
Analytic Disks 102
Contents xi
III Analytic Sets 105
1. The Algebra of Power Series 105
The Banach Algebra Bt 105
Expansion with Respect to z 106
Convergent Series in Banach Algebras 107
Convergent Power Series 108
Distinguished Directions 109
2. The Preparation Theorem 110
Division with Remainder in Bt HO
The Weierstrass Condition 113
Weierstrass Polynomials 114
Weierstrass Preparation Theorem 115
3. Prime Factorization 116
Unique Factorization 116
Gauss s Lemma 117
Factorization in Hn 119
Hensel s Lemma 119
The Noetherian Property 120
4. Branched Coverings 123
Germs 123
Pseudopolynomials 124
Euclidean Domains 125
The Algebraic Derivative 125
Symmetric Polynomials 126
The Discriminant 126
Hypersurfaces 127
The Unbranched Part 130
Decompositions 130
Projections 132
5. Irreducible Components 135
Embedded Analytic Sets 135
Images of Embedded Analytic Sets 137
Local Decomposition 138
Analyticity 140
The Zariski Topology 141
Global Decompositions 141
6. Regular and Singular Points 143
Compact Analytic Sets 143
Embedding of Analytic Sets 144
Regular Points of an Analytic Set 145
The Singular Locus 147
Extending Analytic Sets 147
The Local Dimension 150
xii Contents
IV Complex Manifolds 153
1. The Complex Structure 153
Complex Coordinates 153
Holomorphic Functions 156
Riemann Surfaces 157
Holomorphic Mappings 158
Cartesian Products 159
Analytic Subsets 160
Differentiable Functions 162
Tangent Vectors 164
The Complex Structure on the Space of Derivations . . . 166
The Induced Mapping 167
Immersions and Submersions 168
Gluing 170
2. Complex Fiber Bundles 171
Lie Groups and Transformation Groups 171
Fiber Bundles 173
Equivalence 174
Complex Vector Bundles 175
Standard Constructions 177
Lifting of Bundles 180
Subbundles and Quotients 180
3. Cohomology 182
Cohomology Groups 182
Refinements 184
Acyclic Coverings 185
Generalizations 186
The Singular Cohomology 188
4. Meromorphic Functions and Divisors 192
The Ring of Germs 192
Analytic Hypersurfaces 193
Meromorphic Functions 196
Divisors 198
Associated Line Bundles 200
Meromorphic Sections 201
5. Quotients and Submanifolds 203
Topological Quotients 203
Analytic Decompositions 204
Properly Discontinuously Acting Groups 205
Complex Tori 206
Hopf Manifolds 207
The Complex Projective Space 208
Meromorphic Functions 210
Grassmannian Manifolds 211
Contents xiii
Submanifolds and Normal Bundles 214
Projective Algebraic Manifolds 216
Projective Hypersurfaces 219
The Euler Sequence 222
Rational Functions 223
6. Branched Riemann Domains 226
Branched Analytic Coverings 226
Branched Domains 228
Torsion Points 229
Concrete Riemann Surfaces 230
Hyperelliptic Riemann Surfaces 231
7. Modifications and Toric Closures 235
Proper Modifications 235
Blowing Up 237
The Tautological Bundle 237
Quadratic Transformations 239
Monoidal Transformations 241
Meromorphic Maps 242
Toric Closures 244
V Stein Theory 251
1. Stein Manifolds 251
Introduction 251
Fundamental Theorems 252
Cousin I Distributions 253
Cousin II Distributions 254
Chern Class and Exponential Sequence 255
Extension from Submanifolds 257
Unbranched Domains of Holomorphy 257
The Embedding Theorem 258
The Serre Problem 259
2. The Levi Form 260
Covariant Tangent Vectors 260
Hermitian Forms 261
Coordinate Transformations 262
Plurisubharmonic Functions 263
The Maximum Principle 264
3. Pseudoconvexity 266
Pseudoconvex Complex Manifolds 266
Examples 267
Analytic Tangents 274
4. Cuboids 276
Distinguished Cuboids 276
Vanishing of Cohomology 277
Vanishing on the Embedded Manifolds 278
xiv Contents
Cuboids in a Complex Manifold 278
Enlarging U 280
Approximation 281
5. Special Coverings 282
Cuboid Coverings 282
The Bubble Method 283
Frechet Spaces 284
Finiteness of Cohomology 286
Holomorphic Convexity 286
Negative Line Bundles 287
Bundles over Stein Manifolds 288
6. The Levi Problem 289
Enlarging: The Idea of the Proof 289
Enlarging: The First Step 290
Enlarging: The Whole Process 292
Solution of the Levi Problem 293
The Compact Case 295
VI Kahler Manifolds 297
1. Differential Forms 297
The Exterior Algebra 297
Forms of Type (p,q) 298
Bundles of Differential Forms 300
2. Dolbeault Theory 303
Integration of Differential Forms 303
The Inhomogeneous Cauchy Formula 305
The 9 Equation in One Variable 306
A Theorem of Hartogs 307
Dolbeault s Lemma 308
Dolbeault Groups 310
3. Kahler Metrics 314
Hermitian metrics 314
The Fundamental Form 315
Geodesic Coordinates 316
Local Potentials 317
Pluriharmonic Functions 318
The Fubini Metric 318
Deformations 320
4. The Inner Product 322
The Volume Element 322
The Star Operator 323
The Effect on {p, ?) Forms 324
The Global Inner Product 327
Currents 328
5. Hodge Decomposition 329
Contents xv
Adjoint Operators 329
The Kahlerian Case 331
Bracket Relations 332
The Laplacian 334
Harmonic Forms 335
Consequences 338
6. Hodge Manifolds 341
Negative Line Bundles 341
Special Holomorphic Cross Sections 342
Projective Embeddings 344
Hodge Metrics 345
7. Applications 348
Period Relations 348
The Siegel Upper Halfplane 352
Semipositive Line Bundles 352
Moishezon Manifolds 353
VII Boundary Behavior 355
1. Strongly Pseudoconvex Manifolds 355
The Hilbert Space 355
Operators 355
Boundary Conditions 357
2. Subelliptic Estimates 357
Sobolev Spaces 357
The Neumann Operator 359
Real Analytic Boundaries 360
Examples 360
3. Nebenhullen 364
General Domains 364
A Domain with Nontrivial Nebenhiille 365
Bounded Domains 366
Domains in C2 366
4. Boundary Behavior of Biholomorphic Maps 367
The One Dimensional Case 367
The Theory of Henkin and Vormoor 367
Real Analytic Boundaries 369
Fefferman s Result 369
Mappings 371
The Bergman Metric 371
References 375
Index of Notation 381
Index 387
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:00:06Z |
| institution | BVB |
| isbn | 0387953957 9780387953953 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009759405 |
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| spelling | Fritzsche, Klaus 1946- Verfasser (DE-588)124716113 aut From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert New York, NY [u.a.] Springer 2002 XV, 392 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate Texts in Mathematics 213 Complexe manifolds gtt Fonctions holomorphes Holomorfe functies gtt Variétés complexes Complex manifolds Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd rswk-swf Komplexe Mannigfaltigkeit (DE-588)4031996-9 gnd rswk-swf Komplexe Mannigfaltigkeit (DE-588)4031996-9 s DE-604 Holomorphe Funktion (DE-588)4025645-5 s Grauert, Hans 1930-2011 Verfasser (DE-588)11921007X aut Graduate Texts in Mathematics 213 (DE-604)BV000000067 213 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009759405&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fritzsche, Klaus 1946- Grauert, Hans 1930-2011 From holomorphic functions to complex manifolds Graduate Texts in Mathematics Complexe manifolds gtt Fonctions holomorphes Holomorfe functies gtt Variétés complexes Complex manifolds Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd Komplexe Mannigfaltigkeit (DE-588)4031996-9 gnd |
| subject_GND | (DE-588)4025645-5 (DE-588)4031996-9 |
| title | From holomorphic functions to complex manifolds |
| title_auth | From holomorphic functions to complex manifolds |
| title_exact_search | From holomorphic functions to complex manifolds |
| title_full | From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert |
| title_fullStr | From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert |
| title_full_unstemmed | From holomorphic functions to complex manifolds Klaus Fritzsche, Hans Grauert |
| title_short | From holomorphic functions to complex manifolds |
| title_sort | from holomorphic functions to complex manifolds |
| topic | Complexe manifolds gtt Fonctions holomorphes Holomorfe functies gtt Variétés complexes Complex manifolds Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd Komplexe Mannigfaltigkeit (DE-588)4031996-9 gnd |
| topic_facet | Complexe manifolds Fonctions holomorphes Holomorfe functies Variétés complexes Complex manifolds Holomorphic functions Holomorphe Funktion Komplexe Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009759405&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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