Riemann surfaces by way of complex analytic geometry PDF
Preview Riemann surfaces by way of complex analytic geometry
Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society Riemann Surfaces by Way of Complex Analytic Geometry Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 30F10, 30F15, 30F30, 30F45, 30F99, 30G99,31A05, 31A99, 32W05. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-125 Library of Congress Cataloging-in-Publication Data Varolin,Dror,1970– Riemannsurfacesbywayofcomplexanalyticgeometry/DrorVarolin. p.cm. —(Graduatestudiesinmathematics;v.125) Includesbibliographicalreferencesandindex. ISBN978-0-8218-5369-6(alk.paper) 1. Riemann surfaces. 2. Functions of complex variables. 3. Geometry, Analytic. I. Title. II.Series. 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VisittheAMShomepageathttp://www.ams.org/ 10987654321 161514131211 DedicatedtoErin Contents Preface xi Chapter1. ComplexAnalysis 1 §1.1. Green’sTheoremandtheCauchy-GreenFormula 1 §1.2. HolomorphicfunctionsandCauchyFormulas 2 §1.3. Powerseries 3 §1.4. Isolatedsingularitiesofholomorphicfunctions 4 §1.5. TheMaximumPrinciple 8 §1.6. Compactnesstheorems 9 §1.7. Harmonicfunctions 11 §1.8. Subharmonicfunctions 14 §1.9. Exercises 19 Chapter2. RiemannSurfaces 21 §2.1. DefinitionofaRiemannsurface 21 §2.2. Riemannsurfacesassmooth2-manifolds 23 §2.3. ExamplesofRiemannsurfaces 25 §2.4. Exercises 36 Chapter3. FunctionsonRiemannSurfaces 37 §3.1. Holomorphicandmeromorphicfunctions 37 §3.2. Globalaspectsofmeromorphicfunctions 42 §3.3. HolomorphicmapsbetweenRiemannsurfaces 45 §3.4. Anexample: Hyperellipticsurfaces 54 vii viii Contents §3.5. Harmonicandsubharmonicfunctions 57 §3.6. Exercises 59 Chapter4. ComplexLineBundles 61 §4.1. Complexlinebundles 61 §4.2. Holomorphiclinebundles 65 §4.3. Twocanonicallydefinedholomorphiclinebundles 66 §4.4. HolomorphicvectorfieldsonaRiemannsurface 70 §4.5. Divisorsandlinebundles 74 §4.6. LinebundlesoverP 79 n §4.7. Holomorphicsectionsandprojectivemaps 81 §4.8. Afinitenesstheorem 84 §4.9. Exercises 85 Chapter5. ComplexDifferentialForms 87 §5.1. Differential(1,0)-forms 87 §5.2. T∗0,1 and(0,1)-forms 89 X §5.3. T∗ and1-forms 89 X §5.4. Λ1,1 and(1,1)-forms 90 X §5.5. Exterioralgebraandcalculus 90 §5.6. Integrationofforms 92 §5.7. Residues 95 §5.8. Homotopyandhomology 96 §5.9. Poincare´ andDolbeaultLemmas 98 §5.10. Dolbeaultcohomology 99 §5.11. Exercises 100 Chapter6. CalculusonLineBundles 101 §6.1. Connectionsonlinebundles 101 §6.2. Hermitianmetricsandconnections 104 §6.3. (1,0)-connectionsonholomorphiclinebundles 105 §6.4. TheChernconnection 106 §6.5. CurvatureoftheChernconnection 107 §6.6. Chernnumbers 109 §6.7. Example: TheholomorphiclinebundleT1,0 111 X §6.8. Exercises 112 Chapter7. PotentialTheory 115 Contents ix §7.1. TheDirichletProblemandPerron’sMethod 115 §7.2. ApproximationonopenRiemannsurfaces 126 §7.3. Exercises 130 Chapter8. Solving∂¯forSmoothData 133 §8.1. Thebasicresult 133 §8.2. Trivialityofholomorphiclinebundles 134 §8.3. TheWeierstrassProductTheorem 135 §8.4. Meromorphicfunctionsasquotients 135 §8.5. TheMittag-LefflerProblem 136 §8.6. ThePoissonEquationonopenRiemannsurfaces 140 §8.7. Exercises 143 Chapter9. HarmonicForms 145 §9.1. Thedefinitionandbasicpropertiesofharmonicforms 145 §9.2. Harmonicformsandcohomology 149 §9.3. TheHodgedecompositionofE(X) 151 §9.4. Existenceofpositivelinebundles 157 §9.5. ProofoftheDolbeault-Serreisomorphism 161 §9.6. Exercises 161 Chapter10. Uniformization 165 §10.1. Automorphisms of the complex plane, projective line, and unit disk 165 §10.2. Areviewofcoveringspaces 166 §10.3. TheUniformizationTheorem 168 §10.4. ProofoftheUniformizationTheorem 174 §10.5. Exercises 175 Chapter11. Ho¨rmander’sTheorem 177 §11.1. Hilbertspacesofsections 177 §11.2. TheBasicIdentity 180 §11.3. Ho¨rmander’sTheorem 183 §11.4. ProofoftheKorn-LichtensteinTheorem 184 §11.5. Exercises 195 Chapter12. EmbeddingRiemannSurfaces 197 §12.1. Controllingthederivativesofsections 198 §12.2. Meromorphicsectionsoflinebundles 201
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