Riemann surfaces by way of complex analytic geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2011
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| Series: | Graduate Studies in Mathematics
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVIII, 236 Seiten |
| ISBN: | 9780821853696 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Riemann surfaces by way of complex analytic geometry
Autor: Varolin, Dror
Jahr: 2011
Contents Preface Chapter 1. Complex Analysis §1.1. Green’s Theorem and the Cauchy-Green Formula § 1.2. Holomorphic functions and Cauchy Formulas §1.3. Power series §1.4. Isolated singularities of holomorphic functions §1.5. The Maximum Principle §1.6. Compactness theorems §1.7. Harmonic functions §1.8. Subharmonic functions §1.9. Exercises Chapter 2. Riemann Surfaces §2.1. Definition of a Riemann surface §2.2. Riemann surfaces as smooth 2-manifolds §2.3. Examples of Riemann surfaces §2.4. Exercises Chapter 3. Functions on Riemann Surfaces §3.1. Holomorphic and meromorphic functions §3.2. Global aspects of meromorphic functions §3.3. Holomorphic maps between Riemann surfaces §3.4. An example: Hyperelliptic surfaces xi 1 1 2 3 4 8 9 11 14 19 21 21 23 25 36 37 37 42 45 54 vii
Contents §3.5. Harmonie and subharmonic functions §3.6. Exercises Chapter 4. Complex Line Bundles §4.1. Complex line bundles §4.2. Holomorphic line bundles §4.3. Two canonically defined holomorphic line bundles §4.4. Holomorphic vector fields on a Riemann surface §4.5. Divisors and line bundles §4.6. Line bundles over §4.7. Holomorphic sections and projective maps §4.8. A finiteness theorem §4.9. Exercises Chapter 5. Complex Differential Forms §5.1. Differential (1, 0)-forms §5.2. T^- 0,1 and (0, l)-forms §5.3. T£ and 1-forms §5.4. A^ 1 and (1, l)-forms §5.5. Exterior algebra and calculus §5.6. Integration of forms §5.7. Residues §5.8. Homotopy and homology §5.9. Poincaré and Dolbeault Lemmas §5.10. Dolbeault cohomology §5.11. Exercises Chapter 6. Calculus on Line Bundles §6.1. Connections on line bundles §6.2. Hermitian metrics and connections §6.3. (1,0)-connections on holomorphic line bundles §6.4. The Chern connection §6.5. Curvature of the Chern connection §6.6. Chern numbers §6.7. Example: The holomorphic line bundle T^’° §6.8. Exercises 57 59 61 61 65 66 70 74 79 81 84 85 87 87 89 89 90 90 92 95 96 98 99 100 101 101 104 105 106 107 109 111 112 Chapter 7. Potential Theory 115
Contents IX §7.1. The Dirichlet Problem and Perron’s Method 115 §7.2. Approximation on open Riemann surfaces 126 §7.3. Exercises 130 Chapter 8. Solving 8 for Smooth Data 133 §8.1. The basic result 133 §8.2. Triviality of holomorphic line bundles 134 §8.3. The Weierstrass Product Theorem 135 §8.4. Meromorphic functions as quotients 135 §8.5. The Mittag-Leffler Problem 136 §8.6. The Poisson Equation on open Riemann surfaces 140 §8.7. Exercises 143 Chapter 9. Harmonic Forms 145 §9.1. The definition and basic properties of harmonic forms 145 §9.2. Harmonic forms and cohomology 149 §9.3. The Hodge decomposition of (?(X) 151 §9.4. Existence of positive line bundles 157 §9.5. Proof of the Dolbeault-Serre isomorphism 161 §9.6. Exercises 161 Chapter 10. Uniformization 165 §10.1. Automorphisms of the complex plane, projective line, and unit disk 165 §10.2. A review of covering spaces 166 §10.3. The Uniformization Theorem 168 § 10.4. Proof of the Uniformization Theorem 174 §10.5. Exercises 175 Chapter 11. Hormander’s Theorem 177 §11.1. Hilbert spaces of sections 177 §11.2. The Basic Identity 180 §11.3. Hormander’s Theorem 183 § 11.4. Proof of the Korn-Lichtenstcin Theorem 184 §11.5. Exercises 195 Chapter 12. Embedding Riemann Surfaces 197 § 12.1. Controlling the derivatives of sections 198 §12.2. Meromorphic sections of line bundles 201
X Contents §12.3. Plenitude of meromorphic functions 202 §12.4. Kodaira’s Embedding Theorem 202 §12.5. Narasimhan’s Embedding Theorem 204 §12.6. Exercises 210 Chapter 13. The Riemann-Roch Theorem 211 §13.1. The Riemann-Roch Theorem 211 §13.2. Some corollaries 217 Chapter 14. Abel’s Theorem 223 §14.1. Indefinite integration of holomorphic forms 223 §14.2. Riemann’s Bilinear Relations 225 §14.3. The Reciprocity Theorem 228 §14.4. Proof of Abel’s Theorem 229 §14.5. A discussion of Jacobi’s Inversion Theorem 231 Bibliography 233 Index 235
|
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| discipline | Mathematik |
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| id | DE-604.BV039535886 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:05:44Z |
| institution | BVB |
| isbn | 9780821853696 |
| language | English |
| lccn | 2011014621 |
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| physical | XVIII, 236 Seiten |
| publishDate | 2011 |
| publishDateSearch | 2011 |
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| publisher | American Mathematical Society |
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| series | Graduate Studies in Mathematics |
| series2 | Graduate Studies in Mathematics |
| spelling | Varolin, Dror 1970- Verfasser (DE-588)1016512597 aut Riemann surfaces by way of complex analytic geometry Dror Varolin Providence, RI American Mathematical Society 2011 XVIII, 236 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate Studies in Mathematics 125 Includes bibliographical references and index Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Komplex-analytische Struktur (DE-588)4340739-0 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 s Komplex-analytische Struktur (DE-588)4340739-0 s DE-604 Graduate Studies in Mathematics 125 (DE-604)BV009739289 125 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024388070&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Varolin, Dror 1970- Riemann surfaces by way of complex analytic geometry Graduate Studies in Mathematics Riemannsche Fläche (DE-588)4049991-1 gnd Komplex-analytische Struktur (DE-588)4340739-0 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4340739-0 |
| title | Riemann surfaces by way of complex analytic geometry |
| title_auth | Riemann surfaces by way of complex analytic geometry |
| title_exact_search | Riemann surfaces by way of complex analytic geometry |
| title_full | Riemann surfaces by way of complex analytic geometry Dror Varolin |
| title_fullStr | Riemann surfaces by way of complex analytic geometry Dror Varolin |
| title_full_unstemmed | Riemann surfaces by way of complex analytic geometry Dror Varolin |
| title_short | Riemann surfaces by way of complex analytic geometry |
| title_sort | riemann surfaces by way of complex analytic geometry |
| topic | Riemannsche Fläche (DE-588)4049991-1 gnd Komplex-analytische Struktur (DE-588)4340739-0 gnd |
| topic_facet | Riemannsche Fläche Komplex-analytische Struktur |
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