Table Of ContentLecture Notes in Mathematics 2216
Chris Wendl
Holomorphic
Curves in Low
Dimensions
From Symplectic Ruled Surfaces to
Planar Contact Manifolds
Lecture Notes in Mathematics 2216
Editors-in-Chief:
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BernardTeissier,Paris
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Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Chris Wendl
Holomorphic Curves
in Low Dimensions
From Symplectic Ruled Surfaces
to Planar Contact Manifolds
123
ChrisWendl
Institutf:urMathematik
Humboldt-Universit:atzuBerlin
Berlin,Germany
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-91369-8 ISBN978-3-319-91371-1 (eBook)
https://doi.org/10.1007/978-3-319-91371-1
LibraryofCongressControlNumber:2018943453
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book.
Preface
This book is an expanded version of a set of lecture notes for a minicourse
I gave at IRMA Strasbourg in October 2012 as part of the MasterClasseson
HolomorphicCurvesandApplicationstoEnumerativeGeometry,Symplecticand
ContactTopology. The focus of the minicourse was on certain specifically low-
dimensional aspects of the theory of pseudoholomorphic curves, which lend a
distinctive flavor to the study of symplectic and contact manifolds in dimensions
fourandthree,respectively.Whilemostofthesetopicsarecoveredtosomeextent
elsewhereintheliterature(notablyin[MS12],thestandardreferencein thefield),
they usually do not take center stage, and I have sometimes encountered experts
in symplectic topology who seem only vaguely aware of why holomorphiccurve
methodsaresomuchmorepowerfulindimensionfourthaninhigherdimensions.
Asaconvincingdemonstrationofthispower,Ichosetoexplainthemainresults
of McDuff’s classic paper [McD90] characterizing rational and ruled symplectic
4-manifolds. First proved around the end of the 1980s, these results are now
considered fundamental in the study of symplectic 4-manifolds, and their proofs
are quite beautiful and natural and, from a modern perspective, not conceptually
difficult.A beginner,however,mightfind themunfairlyintimidatingif attempting
toreadtheoriginalpapersonthesubject,whichwerewrittenbeforemanyofwhat
we would now call the “standard” techniques had been fully developed. My goal
therefore was to present these proofs in the most elegant way that I could, using
modern techniques that I regard as essential for researchers in the field to learn.
Since I had limited time and did not want to get bogged down with analysis,
most of the necessary analytical background on holomorphic curves was stated
without proofs, though I have endeavored in this book at least to give precise
statements of all required results and brief informal explanationsof why they are
true,withreferencestoothersourceswherethedetailsmaybefound.Incontrastto
other available treatments of this subject (e.g., in [MS12, LM96b]), I have placed
considerableemphasisonthenaturalroleplayedbyLefschetzpencilsandfibrations,
a distinctlytopological(ratherthananalytical)topicwhichhasexerteda similarly
largeinfluenceonsymplectictopologysincethe1990s.
vii
viii Preface
Asecondobjectiveoftheoriginalminicourseandofthisbookisaddressedinthe
lasttwochapters,whichdiscusscontacttopology.Myownmotivationtounderstand
McDuff’s rational/ruled paper came largely from this direction, as it had become
clear through the work of Hofer-Wysocki-Zehnder and others that the Gromov-
McDufftechniqueoffoliatingsymplectic4-manifoldsbyholomorphiccurvesalso
had many deep implications for contact 3-manifolds. In my own research, these
implicationshavebeenmostapparentthroughtherelationshipbetweenholomorphic
curves and Lefschetz fibrations on symplectic fillings, which connects naturally
with the study of open book decompositions on contact manifolds. The so-called
planarcontactmanifolds—thosewhicharesupportedbyplanaropenbooks—have
turnedoutto playan analogousrolein the three-dimensionalcontactworldto the
one played by rational and ruled surfaces in dimension four. The last chapter is
an attempt to illustrate this analogy, focusing in particular on two problems of
fundamentalimportance in the field: the existence of closed Reeb orbits (i.e., the
Weinstein conjecture)andthe classification of symplecticfillings. I have included
also in Chap.8 some general discussion of the context for these problems and
the historical development of the techniques used to study them, reaching from
Conley and Zehnder’s 1982 solution of the Arnol’d conjecture on the torus to
the introduction in 2000 and (as of this writing not yet complete) subsequent
development of symplectic field theory. These last two chapters are meant as a
survey,sotheyallowthemselvestheluxuryofnotpresentingcompleteproofs,but
covercorrespondinglymoreground.
Chapter 7 is a later addition that was not part of the original minicourse, but
fits in thematically with the material of the first six chapters. The subject here is
therelationshipbetweenMcDuff’scharacterizationofrational/ruledsymplectic4-
manifolds and the Gromov-Witten invariants, in particular the beautiful theorem
thatasymplectic4-manifoldissymplecticallyuniruledifandonlyifitisablowup
of a rational or ruled surface. Several results of fundamental importance can be
understood as consequences of this theorem, e.g., that the class of (blowups of)
symplectic rational or ruled surfaces is invariant under birational equivalence and
thattheminimalblowdownofasymplectic4-manifoldisuniqueunlessitisrational
orruled.Onegoodreasontowritethischapterwasthatwhilethetheorem“uniruled
ñ rational/ruled”has evidently been known to experts for at least 20 years, I am
notaware of any previoussource in the literature thatboth containsthe statement
andexplainswhyitistrue.Moreseriously,McDuff’spaper[McD92]onimmersed
spheres,whichcarriesoutthehardpartoftheproof,seemsnottohavepenetrated
the public consciousness nearly as much as its predecessor [McD90]. One reason
forthisissurelythatthemainproofin[McD92]issignificantlymoreintricatethan
anything in [McD90], and another is that the result was superseded a short time
later by developments from Seiberg-Witten theory. Since I had never planned to
discuss Seiberg-Wittentheory in my minicourse, it therefore seemed natural after
writing the notes that I should try to supplementthem with a readable accountof
thecontentsof[McD92],andwhileIcannotsaywithanycertaintywhetherIhave
succeeded,theoutcomeofthateffortisChap.7ofthisbook.Sinceitwasrelevant,I
tooktheopportunitytoaddagentleintroductiontoGromov-Wittentheory,andthe
Preface ix
restrictiontodimensionfourallowedmetodothisinawaythatsomereadersmay
findlessintimidatingthanthestandardpresentationin[MS12].Itshouldbeadded
that my explanationof [McD92] would have been completely impossible without
someextremelyvaluableinputfromMcDuffherself,whobecameenthusiasticabout
thiseffortbeforeIhadquiteunderstoodwhatIwasgettingmyselfinto.Theproof
I’vewrittenupinSect.7.3isessentiallyonethatsheexplainedto meaftershesat
downtoreconsidertheoriginalargumentof[McD92].
There are several topics that might have seemed natural to include but have
beenglaringlyomitted:foremostamongtheseisthesubstantialcontributionmade
by Seiberg-Witten theory to the classification of symplectic structures on rational
and ruled surfaces, including work of Taubes [Tau95, Tau00], Lalonde-McDuff
[LM96a], and Li and Liu [LL95, Li99, Liu96]. I will mention a few such results
in Sects.1.2 and 7.3.1, mainly for the sake of cultural knowledge, but without
any serious attempt to explain why they are true. It would have been even more
unrealistic to attempt a nontrivial discussion of Seiberg-Witten theory in contact
geometry; thus, my exposition says almost nothing about Taubes’s solution to
the Weinstein conjecture in dimension three [Tau07], nor its connections to the
SFT-like invariant known as embedded contact homology (see [Hut10]) and its
manyimpressiveapplications.Theinterestingtopicoffiniteenergyfoliations(see,
e.g., [HWZ03, Wen08, FS]) will be alluded to briefly but then forgotten, and my
discussionofLefschetzfibrationsonsymplecticfillingswillnecessarilyomitmany
additionalapplicationsfor which they can profitablybe used, e.g., in the study of
Steinmanifolds[ÖS04a]andLagrangianintersectiontheory[Sei08b].Ihavetried
atleasttosupplysuitablereferenceswhereverpossible,sothereadershouldnever
assumethatwhatIhavetosayonanygiventopicisallthatcanbesaid.
The target reader for this book is assumed to have at least a solid background
in basic differential geometry and algebraic topology (including homological
intersection numbers and the first Chern class), as well as some basic literacy
concerning symplectic manifolds (Darboux’s theorem, Moser’s stability theorem,
the Lagrangianneighborhoodtheorem, etc.) as found e.g. in the early chaptersof
[MS17]. Ihavetriedto avoidexplicitlyrequiringpriorknowledgeofholomorphic
curves—hence the technical overview in Chap.2—but readers who already have
suchknowledgewillprobablyfindithelpful(andsomeofthosewillbecontentto
skipmostofChap.2).
Acknowledgments
I would like to thank Emmanuel Opshtein, IRMA Strasbourg, and the CNRS for
bringingabouttheworkshopthatgaverisetothelecturenotesonwhichthisbook
is based.Many thanksalso to Patrick Massotand Paolo Ghigginifortheir careful
readingandhelpfulcommentsonvariouspreliminaryversions,JankoLatschevfor
providing the proof of Proposition 8.10, and especially to Dusa McDuff for her
x Preface
invaluableexplanationsofthepaper[McD92]andformanyconstructivecomments
onthefirstdraftofChap.7.
MuchofthewritingofthisbookwascarriedoutatUniversityCollegeLondon,
whereIwassupportedinpartbyaRoyalSocietyUniversityResearchFellowship.
Berlin,Germany ChrisWendl
