Table Of ContentLecture Notes in Mathematics 2140
Tim Kirschner
Period Mappings
with Applications
to Symplectic
Complex Spaces
Lecture Notes in Mathematics 2140
Editors-in-Chief:
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B.Teissier,Paris
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Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Tim Kirschner
Period Mappings with
Applications to Symplectic
Complex Spaces
123
TimKirschner
MathematischesInstitut
UniversitaRtBayreuth
Bayreuth,Germany
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-17520-1 ISBN978-3-319-17521-8 (eBook)
DOI10.1007/978-3-319-17521-8
LibraryofCongressControlNumber:2015948870
Mathematics SubjectClassification(2010):14F05(also18F20,32C35),32C20(also14D05,14D07),
32S35,14J32(or32Q25),18G40
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To myparents
Preface
Through this book, I intend to deliver to readers three chapters of state-of-the-art
mathematics positioned at the crossroad of algebraic geometry—more precisely,
thehomologicalalgebraofsheavesofmodulesonringedspaces—andthetheoryof
complexanalyticspaces.Frommyperspectiveastheauthor,thefocalpointofthe
textisChap.1.
The first chapter explores the territory surroundingNicholas Katz’s and Tadao
Oda’s conception of a Gauß-Manin connection defined on the relative algebraic
de Rham cohomology sheaf. Here, I attempt to employ Katz’s and Oda’s idea in
the realm of complex analytic spaces—to my knowledge for the first time in the
literature. Note that Katz and Oda work with schemes rather than with analytic
spaces.FromtheGauß-Maninconnection,Ifabricateperiodmappings,andthenice
thing,Ihopethereaderwillagree,isthatpropertiesoftheGauß-Maninconnection
thustranslateaspropertiesoftheperiodmappings.Alloftheseareexplainedhere
ingreatdetail.
Forthefirstchapter,Ihaveabroadaudienceinmind.So,if youareacomplex
geometer,thenthiswillmostlikelymeetyourneeds.Fromthepointwhereperiod
mappingsmaketheirappearance(Sect.1.7),thegeometricobjects(i.e.,theringed
spaces) under investigation are complex spaces, or even complex manifolds. In
case you have seen period mappings for families of compact complex manifolds
ofKählertype,orifyouhaveencounteredavariationofHodgestructure,mostof
my resultswill have a familiar ring.Keep in mind,though,that the assumptionsI
make in Chap.1 are certainly very much differentfrom what you are likely to be
usedto.Specifically,Istudyfamiliesofcomplexmanifoldsthatareneithercompact
norofKählertype.
Ifyouarenotinterestedincomplexmanifolds,butyouareinterestedinalgebraic
schemes, or in analytic spaces over a valued field other than that of complex
numbers,thenmorethanhalfofChap.1stillmayinterestyou.Asamatteroffact,
IhaveconceivedthecentralSect.1.5ofChap.1sothatitappliestoarbitraryringed
spaces. You might even pursue this suggested setup further—namely, to ringed
topoi—if you so wish. Also, the exposition is fully intended to be accessible to
you,evenifyouarejustgettingacquaintedwithalgebraicgeometry.Accordingly,
vii
viii Preface
Ihaveputalotofeffortintomakingmyexpositionaselementaryaspossible.The
prerequisitesarekepttoabareminimum.Readersareassumedtobefamiliarwith
sheavesofringsandsheavesofmodulesontopologicalspaces,aswellaswiththeir
morphisms and basic operations—that is, the tensor product, the sheaf hom, the
pushforward,andthepullback.Iusehomologicalalgebratotheextentthatweuse
suchtermsascomplexesofmodules,cohomology,andrightderivedfunctors.Last
but not least, readers are also assumed to understandthe basic jargon of category
theory—forinstance,inthecaseofcommutativediagrams.
Moving from Chap.1 to Chap.2 to Chap.3, the geometric objects and ideas
become increasingly concrete and specific. Already in Chap.2, general ringed
spaces have been replaced completely by complex analytic spaces. More to the
point,geometricnotionssuchastheKählerproperty,analyticsubsets,dimension,or
theflatnessofamorphismcomeintoplay.InChap.3,whichtreatstheapplications
materialhintedatinthebooktitle,thegeometrybecomesevenmorespecialaswe
lookatsymplecticcomplexspaces.
A nice feature of this book is that its three chapters are almost entirely
independent of one another. In fact, the chapters are really designed to be read
as stand-alone. When you open up Chap.2, you will find a problem set, which is
entirelydisjointfromChap.1.Specifically,notasingleresultestablishedinChap.1
is neededfor Chap.2, althoughI recycle a couple of notations. While I think that
Chap.2isofgreatinterestinitsownright,youshouldreadtheintroductorySect.2.1
andsee foryourself.With thegeometricobjectsnarrowing,the targetaudienceof
thetextisboundtoshiftsomewhatasitprogresses.IhopethatChap.2,asintended,
hits the core of what “algebraic methodsin the global theory of complex spaces”
areallabout.Asfarastheprerequisitesgo,Iassumeacertainreaderfamiliarityin
working with complex spaces. In particular, I assume that the reader knows what
is the Frölicher spectral sequence of a complex space, or better of a morphism
of complex spaces. Other than that, I have tried consistently to explain all the
techniquesthat I employ—especiallythat of local cohomologyand that of formal
completions.Readersshouldbeabletofollowtheproofseventhoughtheyhaven’t
previouslyseenthementionedtechniquesinactionbefore.
If you feel that Chap.1 contains a lot of general nonsense, no offense is
taken, and if you find Sect.2.1 only mildly intriguing, maybe Chap.3 can yet
convince you of the contrary. Symplectic complex spaces have been, perhaps for
the past two to three decades, and are, highly fashionable objects of research in
complex geometry. The interest in them stems, naturally, from their connection
with the all overshadowing hyperkähler manifolds. For context, let me refer you
to the individual sections of Chap.3. Clearly, symplectic complex manifolds—
moreprecisely,irreduciblesymplecticcomplexmanifolds—dominatethebusiness.
Singularsymplecticcomplexspaceshave,nevertheless,beenpresentinthetheory
from very early on, albeit only implicitly at first. In 1983, for instance, Beauville
already describes his Hilbert scheme of points on a K3 surface as a resolution of
singularitiesofwhatiscalledasingularsymplecticspacetoday.
Preface ix
Chapter 3 of this book serves a twofold purpose. First, it provides you with
a systematic introduction to the global theory of symplectic complex spaces. An
entiresectionaloneisdevotedtothedefinitionandtheelementarypropertiesofthe
Beauville-Bogomolov-Fujikiquadraticformin the possibly singular context.This
is,tomyknowledge,novelintheliterature.Second,itseekstoprovetwoverycon-
temporary theorems about what might be called “irreducible symplectic complex
spaces”—namely,alocalTorellitheoremandtheso-calledFujikirelation.Inorder
todeducethelocalTorellitheorem,IinvoketheconclusionsofChaps.1and2.Apart
fromthat,Chap.3is,again,fullyindependentofitspredecessors.Besides,inorder
to understandthe proofofmy localTorellitheorem,an understandingofChaps.1
and2isn’tstrictlynecessary.OnecansimplytakethefinalresultsofChaps.1and2
forgrantedandgofromthere.
The way it is presented here, the Fujiki relation is a consequence of the local
Torellitheorem.So,whenyouthinkthatsymplecticcomplexspacesaregreat,you
haveyourmotivationtotacklewhatIdoinChaps.1and2.Ontheonehand,Imay
suggestthat,forsingularspaces,myline ofargument,whichreliesheavilyonthe
resultsofChaps.1and2,istheonlyargumentfortheFujikirelationyetproposed.
On the otherhand,the Fujikirelationis an importanttool,which hasprovenvital
foramultitudeofapplicationsinthecontextofmanifolds.
Personally, I think that the proof of the Fujiki relation for singular symplectic
spaces affords a prime motivation for the more abstract theory in Chaps.1 and 2.
I believe, however, that Chaps.1 and 2—combined as in Chap.3, or each one on
its own—mayhold a far greaterpotential, a potentialthat goesbeyondthe notion
of symplectic spaces. So, my hope would be that you can find new, interesting
applicationsoftheseideas.Onthatnote,enjoythebookandhappyresearching!
Acknowledgements Firstandforemost,IwouldliketothankSpringer-Verlagforacceptingmy
manuscriptforpublication.IcanstillrememberwhenIwasayoung(er)studentofmathematics
standing, fully amazed, in front of the yellow wall of Lecture Notes in Mathematics in the
university library. Having myself become the proud owner of several titles of the series in
subsequentyears,Iamnowmorethanhappytobeabletoauthoravolume.Iwouldliketothank
UteMcCroryfromtheSpringereditorialofficeforheraccompanyingthisprojectfrombeginning
toendwithinfinitepatience.IwouldliketothankeveryoneattheSt.OberholzCaféinBerlin—
Ansgar Oberholz and Sasa Reuter in particular—for accommodating a math geek and decisive
non-entrepreneur likeme as a member of their coworking space. For six weeks inthesummer
of2014,yoursecondfloorhasbeenatruesecondhome.Theseweeksweredevotedexclusively
tomyworkingonthismanuscript.Manyfundamental ideasaboutthepresentationhaveformed
there.Last butnotleast,IwouldliketothankThomasPeternell forsupervising myPhDthesis
from2008to2012andforsupportingmydoingmathematicsvigorouslyeversince.Theresults
youfindinthisbookstem,asamatteroffact,entirelyfrommyPhDyears.Ifyouwanttotrack
howmyeditingofthematerialhasevolvedsince2012,youcanstillfindmyoriginalthesis,aswell
asasecond,reformattedversiononline.
Bayreuth,Germany TimKirschner
March2015
