Riemann surfaces by way of complex analytic geometry

258 Pages·2011·1.15 MB·English

by  Dror Varolin

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.
QA333.V37 2011
515(cid:2).93—dc23
2011014621
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(cid:2)c 2011byDrorVarolin. Allrightsreserved.
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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