Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
[2017]
|
| Edition: | Second edition |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
volume 56 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 562 Seiten Illustrationen |
| ISBN: | 9783319610573 9783319869940 |
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Record in the Search Index
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| adam_text | Titel: Stein manifolds and holomorphic mappings
Autor: Forstnerič, Franc
Jahr: 2017
Contents Part I Stein Manifolds 1 Preliminaries.............................. 3 1.1 Complex Manifolds and Holomorphic Maps........... 3 1.2 Examples of Complex Manifolds................. 7 1.3 Subvarieties and Complex Spaces................ 10 1.4 Holomorphic Fibre Bundles................... 13 1.5 Holomorphic Vector Bundles................... 16 1.6 The Tangent Bundle ....................... 21 1.7 The Cotangent Bundle and Differential Forms.......... 24 1.8 Plurisubharmonic Functions and the Levi Form......... 27 1.9 Vector Fields, Flows and Foliations............... 32 1.10 What is the H-Principle?..................... 39 2 Stein Manifolds............................. 45 2.1 Domains of Holomorphy..................... 45 2.2 Stein Manifolds and Stein Spaces................ 49 2.3 Holomorphic Convexity and the Oka-Weil Theorem ...... 50 2.4 Embedding Stein Manifolds into Euclidean Spaces....... 51 2.5 Characterization by Plurisubharmonic Functions . ....... 52 2.6 Cartan-Serre Theorems A B.................. 54 2.7 The 3-Problem.......................... 58 2.8 Cartan-Oka-Weil Theorem with Parameters........... 59 3 Stein Neighborhoods and Approximation .............. 65 3.1 g-Complete Neighborhoods................... 65 3.2 Stein Neighborhoods of Stein Subvarieties............ 70 3.3 Holomorphic Retractions onto Stein Submanifolds....... 73 3.4 A Semiglobal Holomorphic Extension Theorem......... 75 3.5 Approximation on Totally Real Submanifolds.......... 79 3.6 Stein Neighborhoods of Laminated Sets............. 82 3.7 Stein Compacts with Totally Real Handles............ 86 xi
Contents xii 3.8 A Mergelyan Approximation Theorem .............. 88 3.9 Strongly Pseudoconvex Handlebodies.............. 90 3.10 Morse Critical Points of g -Convex Functions.......... 94 3.11 Critical Levels of a g -Convex Function............. 98 3.12 Topological Structure of a Stein Space.............. 102 4 Automorphisms of Complex Euclidean Spaces............ 107 4.1 Shears............................... 107 4.2 Automorphisms of C 2 ...................... 112 4.3 Attracting Basins and Fatou-Bieberbach Domains........ 115 4.4 Random Iterations and the Push-Out Method.......... 123 4.5 Mittag-Leffler Theorem for Entire Maps............. 126 4.6 Tame Discrete Sets in C ..................... 127 4.7 Unavoidable and Rigid Discrete Sets............... 130 4.8 Algorithms for Computing Flows ................ 133 4.9 The Andersén-Lempert Theory.................. 135 4.10 The Density Property....................... 141 4.11 Automorphisms Fixing a Subvariety............... 151 4.12 Moving Polynomially Convex Sets................ 157 4.13 Moving Totally Real Submanifolds ............... 161 4.14 Carleman Approximation by Automorphisms.......... 164 4.15 Automorphisms with Given Jets................. 169 4.16 Mittag-Leffler Theorem for Automorphisms of C” ....... 175 4.17 Interpolation by Fatou-Bieberbach Maps............. 181 4.18 Twisted Holomorphic Embeddings into C ........... 185 4.19 Nonlinearizable Periodic Automorphisms of C ......... 189 4.20 A Non-Runge Fatou-Bieberbach Domain............ 195 4.21 A Long C 2 Without Holomorphic Functions........... 197 Part II Oka Theory 5 Oka Manifolds............................. 207 5.1 A Historical Introduction to the Oka Principle.......... 207 5.2 Cousin Problems and Oka’s Theorem.............. 209 5.3 The Oka-Grauert Principle.................... 212 5.4 What is an Oka Manifold?.................... 215 5.5 Basic Properties of Oka manifolds................ 219 5.6
Examples of Oka Manifolds................... 223 5.7 Cartan Pairs............................ 234 5.8 A Splitting Lemma........................ 235 5.9 Gluing Holomorphic Sprays................... 239 5.10 Noncritical Strongly Pseudoconvex Extensions......... 242 5.11 Proof of Theorem 5.4.4: The Basic Case............. 245 5.12 Proof of Theorem 5.4.4: Stratified Fibre Bundles........ 247 5.13 Proof of Theorem 5.4.4: The Parametric Case.......... 252 5.14 Existence Theorems for Holomorphic Sections......... 256 5.15 Equivalences Between Oka Properties.............. 258
Contents xiii 6 Elliptic Complex Geometry and Oka Theory ............ 263 6.1 Fibre Sprays and Elliptic Submersions.............. 264 6.2 The Oka Principle for Sections of Stratified Subelliptic Submersions........................... 265 6.3 Composed and Iterated Sprays . ................. 267 6.4 Examples of Subelliptic Manifolds and Submersions...... 271 6.5 Lifting Homotopies to Spray Bundles............... 280 6.6 Runge Theorem for Sections of Subelliptic Submersions .... 283 6.7 Gluing Holomorphic Sections on C-Pairs............ 287 6.8 Complexes of Holomorphic Sections............... 290 6.9 C-Strings............................. 293 6.10 Construction of the Initial Holomorphic Complex........ 295 6.11 The Main Modification Lemma................. 297 6.12 Proof of Theorems 6.2.2 and 6.6.6................ 303 6.13 Relative Oka Principle on 1-Convex Manifolds......... 306 6.14 The Oka Principle for Sections of Branched Maps........ 307 6.15 Approximation by Algebraic Maps................ 312 7 Flexibility Properties of Complex Manifolds and Holomorphic Maps 319 7.1 Hierarchy of Holomorphic Flexibility Properties ........ 320 7.2 Stratified Oka Manifolds and Kummer Surfaces......... 325 7.3 Oka Properties of Compact Complex Surfaces.......... 328 7.4 Oka Maps............................. 332 7.5 A Homotopy-Theoretic Viewpoint on Oka Theory ....... 336 7.6 Miscellanea and Open Problems................. 342 Part IH Applications 8 Applications of Oka Theory and Its Methods ............ 353 8.1 Principal Fibre Bundles...................... 353 8.2 The Oka-Grauert Principle for G-Bundles............ 356 8.3 Homomorphisms and Generators of Vector Bundles....... 360 8.4 Generators of Coherent Analytic Sheaves............ 366 8.5 The Number of Equations Defining a Subvariety........ 369 8.6 Elimination of Intersections............. 373 8.7 Holomorphic Vaserstein Problem ................ 375 8.8 Transversality Theorems
for Holomorphic Maps........ 378 8.9 Singularities of Holomorphic Maps............... 386 8.10 Local Sprays of Class A(D) ................... 388 8.11 Stein Neighborhoods of A(D)-Graphs.............. 393 8.12 Oka Principle on Strongly Pseudoconvex Domains....... 398 8.13 Banach Manifolds of Holomorphic Mappings.......... 400 9 Embeddings, Immersions and Submersions............. 403 9.1 The H-Principle for Totally Real Immersions and for Complex Submersions........................... 404 9.2 (Almost) Proper Maps to Euclidean Spaces........... 411
xiv Contents 9.3 Embedding and Immersing Stein Manifolds into Euclidean Spaces of Minimal Dimension.................. 415 9.4 Proof of the Relative Embedding Theorem............ 420 9.5 Weakly Regular Embeddings and Interpolation......... 426 9.6 The Oka Principle for Holomorphic Immersions ........ 429 9.7 A Splitting Lemma for Biholomorphic Maps.......... 431 9.8 The Oka Principle for Proper Holomorphic Maps........ 436 9.9 Exposing Points of Bordered Riemann Surfaces......... 441 9.10 Embedding Bordered Riemann Surfaces in C 2 ......... 446 9.11 Infinitely Connected Complex Curves in C 2 ........... 450 9.12 Approximation of Holomorphic Submersions.......... 457 9.13 Noncritical Holomorphic Functions............... 461 9.14 The Oka Principle for Holomorphic Submersions........ 469 9.15 Closed Holomorphic 1-Forms Without Zeros.......... 470 9.16 Holomorphic Foliations on Stein Manifolds........... 472 10 Topological Methods in Stein Geometry ............... 477 10.1 Real Surfaces in Complex Surfaces ............... 478 10.2 Invariants of Smooth 4-Manifolds................ 482 10.3 Lai Indexes and Index Formulas................. 484 10.4 Cancelling Pairs of Complex Points............... 488 10.5 Applications of the Cancellation Theorem............ 492 10.6 The Adjunction Inequality in Kahler Surfaces.......... 498 10.7 The Adjunction Inequality in Stein Surfaces........... 505 10.8 Well Attached Handles...................... 509 10.9 Stein Structures and the Soft Oka Principle........... 517 10.10 The Case dim R X ^ 4....................... 520 10.11 Exotic Stein Structures on Smooth 4-Manifolds......... 523 References.................................. 533 Index..................................... 557
|
| any_adam_object | 1 |
| author | Forstnerič, Franc 1958- |
| author_GND | (DE-588)1016469241 |
| author_facet | Forstnerič, Franc 1958- |
| author_role | aut |
| author_sort | Forstnerič, Franc 1958- |
| author_variant | f f ff |
| building | Verbundindex |
| bvnumber | BV044513414 |
| classification_rvk | SK 780 |
| ctrlnum | (OCoLC)1005928280 (DE-599)BSZ493702903 |
| dewey-full | 515.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.9 |
| dewey-search | 515.9 |
| dewey-sort | 3515.9 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV044513414 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:54:39Z |
| institution | BVB |
| isbn | 9783319610573 9783319869940 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029913141 |
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| physical | xiv, 562 Seiten Illustrationen |
| publishDate | 2017 |
| publishDateSearch | 2017 |
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| publisher | Springer |
| record_format | marc |
| series | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge |
| spelling | Forstnerič, Franc 1958- Verfasser (DE-588)1016469241 aut Stein manifolds and holomorphic mappings the homotopy principle in complex analysis Franc Forstnerič Second edition Cham, Switzerland Springer [2017] © 2017 xiv, 562 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge volume 56 Holomorphe Abbildung (DE-588)4160471-4 gnd rswk-swf Stein-Mannigfaltigkeit (DE-588)4183070-2 gnd rswk-swf Stein-Mannigfaltigkeit (DE-588)4183070-2 s Holomorphe Abbildung (DE-588)4160471-4 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-61058-0 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge volume 56 (DE-604)BV000899194 56 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029913141&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Forstnerič, Franc 1958- Stein manifolds and holomorphic mappings the homotopy principle in complex analysis Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge Holomorphe Abbildung (DE-588)4160471-4 gnd Stein-Mannigfaltigkeit (DE-588)4183070-2 gnd |
| subject_GND | (DE-588)4160471-4 (DE-588)4183070-2 |
| title | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis |
| title_auth | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis |
| title_exact_search | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis |
| title_full | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis Franc Forstnerič |
| title_fullStr | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis Franc Forstnerič |
| title_full_unstemmed | Stein manifolds and holomorphic mappings the homotopy principle in complex analysis Franc Forstnerič |
| title_short | Stein manifolds and holomorphic mappings |
| title_sort | stein manifolds and holomorphic mappings the homotopy principle in complex analysis |
| title_sub | the homotopy principle in complex analysis |
| topic | Holomorphe Abbildung (DE-588)4160471-4 gnd Stein-Mannigfaltigkeit (DE-588)4183070-2 gnd |
| topic_facet | Holomorphe Abbildung Stein-Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029913141&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT forstnericfranc steinmanifoldsandholomorphicmappingsthehomotopyprincipleincomplexanalysis |