Lectures on contact 3-manifolds, holomorphic curves and intersection theory

197 Pages·2020·1.551 MB·English

by  Wendl, Chris

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220 LecturesonContact3-Manifolds,HolomorphicCurves
andIntersectionTheory
CAMBRIDGE TRACTS IN MATHEMATICS
GENERALEDITORS
B.BOLLOBA´S,W.FULTON,F.KIRWAN,
P.SARNAK,B.SIMON,B.TOTARO
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ByC.WENDL

Lectures on Contact 3-Manifolds,
Holomorphic Curves
and Intersection Theory
CHRIS WENDL
HumboldtUniversityofBerlin
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Contents
Preface pagevii
Acknowledgments ix
Introduction:Motivation 1
Lecture1 ClosedHolomorphicCurvesinSymplectic4-Manifolds 11
1.1 SomeExamplesofSymplectic4-Manifolds 11
1.2 McDuff’sCharacterizationofSymplecticRuledSurfaces 17
1.3 LocalFoliationsbyHolomorphicSpheres 22
Lecture2 Intersections,RuledSurfaces,andContactBoundaries 26
2.1 PositivityofIntersectionsandtheAdjunctionFormula 26
2.2 ApplicationtoRuledSurfaces 32
2.3 ContactManifolds,SymplecticFillingsandCobordisms 35
2.4 AsymptoticallyCylindricalHolomorphicCurves 38
Lecture3 AsymptoticsofPuncturedHolomorphicCurves 44
3.1 HolomorphicHalf-CylindersasGradient-FlowLines 45
3.2 AsymptoticFormulasforCylindricalEnds 49
3.3 WindingofAsymptoticEigenfunctions 53
3.4 LocalFoliationsandtheNormalChernNumber 55
Lecture4 IntersectionTheoryforPuncturedHolomorphicCurves 62
4.1 StatementoftheMainResults 62
4.2 RelativeIntersectionNumbersandthe∗-Pairing 66
4.3 AdjunctionFormulas,RelativeandAbsolute 71
Lecture5 SymplecticFillingsofPlanarContact3-Manifolds 77
5.1 OpenBooksandLefschetzFibrations 77
5.2 AClassificationTheoremforSymplecticFillings 83
5.3 SketchoftheProof 86
vi Contents
AppendixA PropertiesofPseudoholomorphicCurves 94
A.1 TheClosedCase 94
A.2 CurveswithPunctures 102
AppendixB LocalPositivityofIntersections 108
B.1 RegularityandtheSimilarityPrinciple 108
B.1.1 LinearCauchy–RiemannTypeOperators 109
B.1.2 EllipticRegularity 111
B.1.3 LocalExistenceofHolomorphicSections 120
B.1.4 TheSimilarityPrinciple 123
B.2 TheRepresentationFormula 126
B.2.1 TheGeneralizedTangent-NormalDecomposition 128
B.2.2 ALemmaonNormalPush-Offs 130
B.2.3 LocalCoordinates 134
B.2.4 ConstructingtheNormalPush-Off 137
B.2.5 ConclusionoftheProof 146
B.3 CountingLocalIntersectionsandSingularities 148
AppendixC AQuickSurveyofSiefring’sIntersectionTheory 158
C.1 Preliminaries 158
C.2 TheIntersectionPairing 160
C.3 TheAdjunctionFormula 163
C.4 CoveringRelations 166
C.5 TheIntersectionProductofBuildings 168
C.6 ComparisonwiththeECHLiterature 174
References 177
Index 182
Preface
The main portion of this book is a lightly revised set of expository lecture
notes written originally for a five-hour minicourse on the intersection theory
of punctured holomorphic curves and its applications in 3-dimensional con-
tact topology, which I gave as part of the LMS Short Course “Topology in
LowDimensions”atDurhamUniversity,August26–30,2013.Theselectures
were aimed primarily at students, and they required only a minimal back-
ground in holomorphic curve theory since the emphasis was on topological
rather than analytical issues. The original appendices were relatively brief,
theirpurposebeingtoprovideaquicksurveyofanalyticalbackgroundmate-
rial on holomorphic curves that I needed to refer to in the lectures without
assuming that students already knew it. In revising the manuscript for publi-
cation,Ihaveaddedamotivationalintroduction,andtakentheopportunityto
inserttwofurtheradditionsthatIfeltwerelackingfromtheexistingliterature,
as a result of which the appendices have become considerably more substan-
tial.One(AppendixB)isacompleteproofoflocalpositivityofintersections,
includingjustenoughbackgroundmaterialonellipticregularityforastudent
familiarwithdistributionsandSobolevspacestoconsiderit“self-contained”;
this notably includes a weak version of the Micallef–White theorem, which
somereadersmay,Ihope,findeasiertocomprehendthanthedeeperresultin
[MW95]thatinspiredit.Theother(AppendixC)isaquicksurveyofSiefring’s
intersectiontheoryofpuncturedholomorphiccurves,puttingtheessentialfacts
and formulas in as compact a form as possible for the benefit of researchers
who need a ready reference. Most of what is in Appendix C also appears in
Lectures 3 and 4, but the latter are written in a more pedagogical style that
developsthestructureofthetheorybasedonafewcoreideas–whichispre-
sumablyhelpfulifyourgoalistounderstandwhythemainresultsaretrue,but
lesssoifyoujustneedtolookupaspecificformula,andAppendixCisthere
tohelpinthatcase.
viii Preface
Intersectiontheoryhasplayedaprominentroleinthestudyofclosedsym-
plectic 4-manifolds since Gromov’s paper [Gro85] on pseudoholomorphic
curves, leading to a myriad of beautiful rigidity results that are either not
accessibleornottrueinhigherdimensions.Inthelast15years,thehighlynon-
trivial extension of this theory to the case of punctured holomorphic curves,
duetoSiefring[Sie08,Sie11],hasledtosimilarlybeautifulresultsaboutcon-
tact 3-manifolds and their symplectic fillings. These lectures begin with an
overview of the closed case and an easy application (McDuff’s characteriza-
tionofsymplecticruledsurfaces),andthenexplaintheessentialsofSiefring’s
intersection theory and how to use it in the real world. As a sample applica-
tion,Lecture5concludesbydiscussingtheclassificationofsymplecticfillings
ofplanarcontactmanifoldsviaLefschetzfibrations[Wen10b].
Howtousetheselecturenotes:Iexpectavarietyofaudiencestofindthese
lecture notes useful for a variety of reasons. Since they were written with an
audience of students in mind, I did not want to assume too much previous
knowledge of symplectic/contact geometry or holomorphic curves, and most
of the text reflects that. On the other hand, I also expect a certain number
of readers to be experienced researchers who already know the essentials of
holomorphiccurvetheory–includingtheadjunctionformulaintheclosedcase
–butwouldspecificallyliketolearnabouttheintersectiontheoryforpunctured
curves.Forreadersinthiscategory,IrecommendstartingwithAppendixCfor
an overview of the basic facts, and then turning back to Lectures 3 and 4 for
details whenever necessary. If, on the other hand, you are a student and still
getting to know the field of symplectic and contact topology, you’ll probably
wishtostartfromthebeginning.
Orifyoureallywanttochallengeyourself,feelfreetoreadthewholething
backward.