Extension problems in complex and CR-geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Abschlussarbeit Buch |
| Sprache: | English |
| Veröffentlicht: |
Pisa
Ed. della Normale
2008
|
| Schriftenreihe: | Tesi Scuola Normale Superiore
9 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 153 S. graph. Darst. |
| ISBN: | 9788876423383 |
Internformat
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| 100 | 1 | |a Saracco, Alberto |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Extension problems in complex and CR-geometry |c Alberto Saracco |
| 264 | 1 | |a Pisa |b Ed. della Normale |c 2008 | |
| 300 | |a XIV, 153 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804142701276299266 |
|---|---|
| adam_text | Titel: Extension problems in complex and CR-geometry
Autor: Saracco, Alberto
Jahr: 2008
Contents
Introduction ix
Acknowledgments xiii
1 Definitions 1
1.1. Holomorphic functions and tangent spaces........ 1
1.2. Analytic subsets...................... 2
1.3. Hulls of compact sets................... 2
1.4. Harmonic, pluriharmonic and plurisubharmonic functions 2
1.5. CÄ-geometry....................... 3
1.6. The Levi-form and Levi-convexity............ 4
1.7. Representation formulas................. 9
1.7.1. The Cauchy kernel................ 9
1.7.2. The Bergman kernel............... 9
1.7.3. The Bochner-Martinelli kernel.......... 11
1.7.4. The Henkin kernel................ 11
2 Classical extension theorems in one and several complex
variables 13
2.1. Basic theorems...................... 13
2.2. Extension theorems in one complex variable....... 14
2.2.1. Removable singularities for bounded
holomorphic functions.............. 14
2.2.2. Reflection principle................ 15
2.3. Extension theorems in several complex variable..... 15
2.3.1. Extension near small-dimensional sets...... 15
2.3.2. Edge of the wedge theorem............ 17
vi Alberto Saracco
3 Extension of CA-functions up to a Levi-flat boundary
and of holomorphic maps 21
3.1. Global extension for CR-functions............ 21
3.2. Local extension for CR-functions............ 22
3.3. Extension up to a Levi-flat boundary........... 22
3.4. Extension out of holomorphic hulls............ 26
3.5. Extension in Stein manifolds............... 33
3.6. Extension on unbounded domains............ 33
3.7. Extension of holomorphic maps: the reflection principle
in higher dimension.................... 34
3.7.1. Reflection principle in C............. 35
3.7.2. Reflection principle and extension theorems in
C , n 1..................... 37
3.7.3. Extension in the strictly pseudoconvex case ... 38
3.7.4. Holomorphic extension in dimension n — 2 . . . 39
3.7.5. Extension of proper holomorphic maps between
strictly pseudoconvex C -domains........ 40
3.7.6. Algebroid functions ............... 42
3.7.7. Edge of the wedge theorem for the cotangent
bundle....................... 43
3.7.8. Scaling method.................. 44
3.7.9. Non-pseudoconvex case............. 46
3.7.10. Main Theorems.................. 46
3.7.11. Properties of Segre varieties........... 49
3.7.12. Complex structure of the set of Segre varieties . 50
3.7.13. Extending the graph of ƒ............. 51
3.7.14. Conclusion of proof............... 52
3.7.15. Final considerations ............... 53
4 Cohomology vanishing and extension problems for semi
ç-coronae 57
4.1. Introduction........................ 57
4.2. Cohomology and extension of sections.......... 59
4.2.1. Closed q -coronae................. 59
4.2.2. Open ç-coronae.................. 61
4.2.3. Corollaries of the extension theorems....... 64
4.3. Extension of divisors and analytic sets of codimension one 67
5 Cohomology of semi 1-coronae and extension of analytic
subsets 73
5.1. Introduction........................ 73
5.2. Remarks on the proofs of theorems in Chapter 4..... 74
vü Extension Problems ¡n Complex and Cfi-Geometry
5.3. An isomorphism theorem for complete semi 1-coronae . 76
5.3.1. Bump lemma: surjectivity of cohomology .... 77
5.3.2. Approximation.................. 80
5.4. Extension of coherent sheaves and analytic subsets ... 85
5.5. Some generalizations................... 88
5.5.1. Bump lemma for semi ç-coronae ........ 88
5.5.2. Semi ç-coronae in Stein spaces ......... 89
The boundary problem 91
6.1. The boundary problem.................. 91
6.2. The boundary problem for compact curves........ 92
6.2.1. Sketches of the proofs.............. 92
6.2.2. Generalization to several curves......... 97
6.3. The boundary problem for compact manifolds...... 97
6.3.1. The boundary problem in terms of holomorphic
chains....................... 97
6.3.2. The boundary problem in strictly pseudoconvex
domains...................... 100
6.3.3. The boundary problem and the linking number . 100
6.4. The boundary problem in ç-concave domains...... 101
6.5. The boundary problem in CF1.............. 102
6.5.1. The protective hull................ 103
6.5.2. The protective linking number.......... 104
6.5.3. Z-sheeted solutions................ 106
6.6. The boundary problem for non-compact cycles..... 106
6.7. The boundary problem in an arbitrary complex
manifold X........................ 107
Non-compact boundaries of complex analytic varieties 109
7.1. Introduction........................ 109
7.2. The local result...................... Ill
7.3. The global result ..................... 115
7.3.1. M is of dimension at least 5 (m 2)....... 116
7.3.2. M is of dimension 3 (m = 1) .......... 120
7.3.3. M is of dimension 1 (m =0) .......... 122
7.4. Extension to pseudoconvex domains........... 122
7.5. On the Lupacciolu s (*) condition............ 124
Semi-local extension of maximally complex submanifolds 125
8.1. Introduction........................ 125
8.2. Main result........................ 126
8.2.1. M is of dimension at least 5 (m 2) ...... 127
viii Alberto Saracco
8.2.2. M is of dimension 3 (m = 1) .......... 131
8.3. Some remarks....................... 134
8.3.1. Maximality of the solution............ 134
8.3.2. The unbounded case............... 135
8.4. Generalization to analytic sets.............. 136
References 141
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| id | DE-604.BV025524463 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:35:56Z |
| institution | BVB |
| isbn | 9788876423383 |
| language | English |
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| physical | XIV, 153 S. graph. Darst. |
| publishDate | 2008 |
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| publisher | Ed. della Normale |
| record_format | marc |
| series | Tesi Scuola Normale Superiore |
| series2 | Tesi / Scuola Normale Superiore |
| spelling | Saracco, Alberto Verfasser aut Extension problems in complex and CR-geometry Alberto Saracco Pisa Ed. della Normale 2008 XIV, 153 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Tesi / Scuola Normale Superiore 9 Zugl.: Pisa, Univ., Diss., 2008 (DE-588)4113937-9 Hochschulschrift gnd-content Tesi Scuola Normale Superiore 9 (DE-604)BV035447349 9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020132247&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Saracco, Alberto Extension problems in complex and CR-geometry Tesi Scuola Normale Superiore |
| subject_GND | (DE-588)4113937-9 |
| title | Extension problems in complex and CR-geometry |
| title_auth | Extension problems in complex and CR-geometry |
| title_exact_search | Extension problems in complex and CR-geometry |
| title_full | Extension problems in complex and CR-geometry Alberto Saracco |
| title_fullStr | Extension problems in complex and CR-geometry Alberto Saracco |
| title_full_unstemmed | Extension problems in complex and CR-geometry Alberto Saracco |
| title_short | Extension problems in complex and CR-geometry |
| title_sort | extension problems in complex and cr geometry |
| topic_facet | Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020132247&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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