Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51) (Mathematical Notes, 51)

228 Pages·2016·2.948 MB·English

by  Paula Tretkoff

Complex Ball Quotients and Line
Arrangements in the Projective Plane

Complex Ball Quotients and Line
Arrangements in the Projective Plane
Paula Tretkoff
WithanappendixbyHans-ChristophImHof
MathematicalNotes51
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:1)c 2016byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
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LibraryofCongressCataloging-in-PublicationData
Tretkoff,Paula,1957–
Complexballquotientsandlinearrangementsintheprojectiveplane/
PaulaTretkoff.
pagescm.–(Mathematicalnotes;51)
Includesbibliographicalreferencesandindex.
ISBN978-0-691-14477-1(pbk.:alk.paper)1.Curves,Elliptic. 2.Geometry,
Algebraic. 3.Projectiveplanes. 4.Unitball. 5.Riemannsurfaces. I.Title.
QA567.2.E44T742016
516.3(cid:1)52–dc23 2015016120
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinMinionPro
Printedonacid-freepaper.∞
press.princeton.edu
TypesetbySRNovaPvtLtd,Bangalore,India
PrintedintheUnitedStatesofAmerica
1 3 5 7 9 10 8 6 4 2
TothememoryofFriedrichHirzebruch

Contents
Preface ix
Introduction 1
1 TopologicalInvariantsandDifferentialGeometry 6
1.1 TopologicalInvariants 7
1.2 FundamentalGroupsandCoveringSpaces 10
1.3 ComplexManifoldsandMetrics 13
1.4 Divisors,LineBundles,theFirstChernClass 16
2 RiemannSurfaces,Coverings,andHypergeometric
Functions 23
2.1 GenusandEulerNumber 23
2.2 MöbiusTransformations 25
2.3 MetricandCurvature 29
2.4 BehavioroftheEulerNumberunderFiniteCovering 33
2.5 FiniteSubgroupsofPSL(2,C) 34
2.6 GaussHypergeometricFunctions 36
2.7 TriangleGroups 41
2.8 TheHypergeometricMonodromyGroup 45
3 ComplexSurfacesandCoverings 47
3.1 CoveringsBranchedoverSubvarietieswithTransverse
Intersections 47
3.2 DivisorClassGroupandCanonicalClass 49
3.3 Proportionality 54
3.4 Signature 59
3.5 BlowingUpPoints 61
4 AlgebraicSurfacesandtheMiyaoka-YauInequality 65
4.1 RoughClassificationofAlgebraicSurfaces 65
4.2 TheMiyaoka-YauInequality,I 70
4.3 TheMiyaoka-YauInequality,II 73
5 LineArrangementsinP (C)andTheirFiniteCovers 85
2
5.1 BlowingUpLineArrangements 87
5.2 Höfer’sFormula 88
viii Contents
5.3 ArrangementsAnnihilating R andHaving
EqualRamificationalongAllLines 92
5.4 Blow-UpofaSingularIntersectionPoint 99
5.5 PossibilitiesfortheAssignedWeights 103
5.6 BlowingDownRationalCurvesand
RemovingEllipticCurves 115
5.7 TablesoftheWeightsGivingProp=0 122
6 ExistenceofBallQuotientsCoveringLineArrangements 126
6.1 ExistenceofFiniteCoversbyBallQuotientsofWeighted
Configurations:TheGeneralCase 128
6.2 RemarksonOrbifoldsandb-Spaces 133
6.3 KX(cid:1)(cid:1) +D(cid:1)(cid:1) forWeightedLineArrangements 135
6.4 ExistenceQuestion 139
6.5 Amplenessof KX(cid:1)(cid:1) +D(cid:1)(cid:1) 140
6.6 Log-TerminalSingularitiesandLCS 145
6.7 ExistenceTheoremforLineArrangements 148
6.8 IsotropySubgroupsoftheCoveringGroup 164
7 AppellHypergeometricFunctions 167
7.1 TheActionof S ontheBlown-UpProjectivePlane 168
5
7.2 AppellHypergeometricFunctions 173
7.3 ArithmeticMonodromyGroups 181
7.4 SomeRemarksabouttheSignature 186
ATorsion-FreeSubgroupsofFiniteIndex 189
A.1 FuchsianGroups 190
A.2 Fenchel’sConjecture 191
A.3 ReductiontoTriangleGroups 192
A.4 TriangleGroups 193
B KummerCoverings 197
Bibliography 205
Index 213
Preface
Thisbookisdevotedtoastudyofquotientsofthecomplex2-ballyieldingfinite
coveringsoftheprojectiveplanebranchedalongcertainlinearrangements.It
is intended to be an introduction for graduate students and for researchers.
Wegiveacompletelistoftheknownweightedlinearrangementsthatcangive
risetosuchballquotients,andthenweprovideajustificationfortheexistence
oftheballquotients.TheMiyaoka-Yauinequalityforsurfacesofgeneraltype,
anditsanalogueforsurfaceswithanorbifoldstructure,playsacentralrole.
The book has its origins in a Nachdiplom course given by F. Hirzebruch
at the ETH Zürich during the Spring of 1996. I (née Cohen) was at that
time Directeur de Recherche au CNRS at the Université de Lille 1, and a
guestofETHZürich.Iattendedthecourse,lecturedonsomeofthematerial
on hypergeometric functions, and made the original set of notes for all
Hirzebruch’s lectures. I also presented related material as a two-semester
graduatecourseatPrincetonUniversityduringtheacademicyear2001/2002
whileIwasVisitingProfessorthere.TheETHNachdiplomcoursenoteswere
subsequentlydevelopedandrefinedbyF.Hirzebruchandmeduringregular
visitstotheMaxPlanckInstituteinBonn.IthankETHZürich,MPIBonn,and
PrincetonUniversityfortheirsupportduringthepreparationofthisbook.
After working on the book together for some years, F. Hirzebruch and
I decided even moreadditional materialseemed desirable and, at that point,
F.Hirzebruchaskedmetocompletethebookundermyownname.Thebook
entitled Geradenkonfigurationen und Algebraische Flächen, Vieweg, 1987, by
G. Barthel, F. Hirzebruch, and T. Höfer, served as a valuable resource and
reference.ThisbookisinGerman,andwehaveminedsomeofitscontentsas
needed.Thepresentbookassumeslessbackgroundthantheearlierbook,and
werevisitinmoredetailseveralofitsimportantsubjects.Wehavealsoadded
materialnotfoundinthatbook.Forexample,wepresentamoreorganizedlist
of the possible weights on line arrangements that yield finite covers that are
ballquotients,andwedevoteawholechapter,ratherthanjustafewpages,to
thequestionoftheexistenceofsuchballquotients.Also,weincludematerial
onhypergeometricfunctions.
The book is dedicated to the memory of F. Hirzebruch. I am grateful
beyondwordsforthetimeandattentionhepaidtothisprojectamidthemany
commitmentsofhisimportantandbusylife.