Tannakian Categories of Perverse Sheaves on Abelian Varieties

181 Pages·2013·0.88 MB·English

INAUGURAL-DISSERTATION
zur
ErlangungderDoktorwu¨rde
ander
Naturwissenschaftlich-MathematischenGesamtfakulta¨t
derRuprecht-Karls-Universita¨tHeidelberg
TANNAKIAN CATEGORIES OF PERVERSE SHEAVES
ON ABELIAN VARIETIES
vorgelegtvon
Dipl.-Math. ThomasKra¨mer
ausFrankenberg(Eder)
Tagdermu¨ndlichenPru¨fung:
Betreuer: Prof. Dr. RainerWeissauer

Zusammenfassung. Die vorliegende Dissertation bescha¨ftigt sich mit
Tannaka-Kategorien,welchedurchFaltungperverserGarbenaufabelschen
Varieta¨tenentstehen. DieKonstruktiondieserKategorienistengverbunden
mit einem kohomologischen Verschwindungssatz, der ein Analogon zum
Satz von Artin-Grothendieck darstellt und als Spezialfall die generischen
Verschwindungssa¨tze von Green und Lazarsfeld entha¨lt. Die erhaltenen
Tannaka-Gruppen bilden interessante geometrische Invarianten, welche in
vielen Situationen eine Rolle spielen — nachdem die Grundlagen fu¨r das
StudiumdieserGruppenbereitgestelltsind,wirdalseinwichtigesBeispiel
derThetadivisoreinerprinzipalpolarisiertenkomplexenabelschenVarieta¨t
behandelt. Durch Entartungsargumente wird die Tannaka-Gruppe fu¨r eine
generische abelsche Varieta¨t bestimmt, welche wenigstens in Dimension 4
eine neue Antwort auf das Schottky-Problem liefert. Das Faltungsquadrat
des Thetadivisors in Dimension 4 ha¨ngt eng zusammen mit einer Familie
glatterFla¨chenvomallgemeinenTyp,undeinStudiumdieserFamiliefu¨hrt
auf eine Variation von Hodge-Strukturen mit Monodromiegruppe W(E ),
6
diemitderPrym-Abbildungverbundenist. DieDissertationschließtabmit
einerRekursionsformelfu¨rdengenerischenRangderdurchFaltungenvon
KurvenentstehendenBrill-Noether-GarbenaufJacobischenVarieta¨ten.
Abstract. WestudyTannakiancategoriesthatarisefromconvolutionsof
perverse sheaves on abelian varieties. The construction of these categories
iscloselyintertwinedwithacohomologicalvanishingtheoremwhichisan
analogoftheArtin-Grothendiecktheoremandcontainsasaspecialcasethe
generic vanishing theorems of Green and Lazarsfeld. The arising Tannaka
groups form a powerful new tool applicable in many different geometric
contexts—afterprovidingtheframeworkforthestudyofthesegroups,we
considerasanimportantexamplethethetadivisorofacomplexprincipally
polarizedabelianvariety. Usingdegenerationtechniqueswedeterminethe
associatedTannakagroupforagenericabelianvariety,andweshowthatin
dimension4thisyieldsaTannakiansolutiontotheSchottkyproblem. The
convolutionsquareofthethetadivisorindimension4iscloselyrelatedtoa
familyofsurfacesofgeneraltype,andadetailedstudyofthisfamilyleads
to a variation of Hodge structures with monodromy groupW(E ) which is
6
connected with the Prym map. To conclude the dissertation, we provide a
recursiveformulaforthegenericrankofBrill-Noethersheaveswhicharise
fromtheconvolutionofcurvesonJacobianvarieties.

Acknowledgements
Itisapleasureformetothankmyadvisor,ProfessorR.Weissauer,for
sharing his beautiful ideas in many productive mathematical discussions,
for his general open-mindedness in all occuring questions, for his careful
reading of the manuscript and for various helpful suggestions concerning
the exposition. I am also grateful to my roommates Johannes Schmidt,
ThorstenHeidersdorfandUweWeselmannandtotheothermembersofthe
Mathematical Institute at Heidelberg for the pleasant environment which
theycreatedduringthepastyears. FurthermoreIwouldliketothankseveral
people outside Heidelberg who enlivened the atmosphere on conferences
and expressed their interest in my work, among them Anders Lundman,
BasharDudin,ChristianSchnell,DanPetersen,DmitryZakharov,Giovanni
Mongardi,KatharinaHeinrich,NicolaTarascaandPaulHacking. Iamalso
indebted to the Studienstiftung des deutschen Volkes and to all members
of its staff for their flexibility and for the financial and ideal support from
whichIprofitedalot. Finally,mymostspecialthanksgotomyfamilyand
friends for their encouragement and patience during my work, but perhaps
evenmoreforthemultifarious,pleasantandinterestingdistractionsfromit
withoutwhichIwouldnothavebeenabletothinkfreely.

Contents
Introduction 1
Somecommonlyusednotations 7
Chapter1. Vanishingtheoremsonabelianvarieties 9
1.1. Themainresult 9
1.2. Arelativegenericvanishingtheorem 11
1.3. Kodaira-Nakano-typevanishingtheorems 12
1.4. Negligibleperversesheaves 15
1.5. Thespectrumofaperversesheaf 17
Chapter2. Proofofthevanishingtheorem(s) 23
2.1. Thesetting 23
2.2. Charactertwistsandconvolution 26
2.3. Anaxiomaticframework 30
2.4. TheAndre´-Kahnquotient 32
2.5. SuperTannakiancategories 35
2.6. Perversemultiplier 36
2.7. Themainargument 38
Chapter3. Perversesheavesonsemiabelianvarieties 41
3.1. Vanishingtheoremsrevisited 42
3.2. Somepropertiesofconvolution 46
3.3. Thethicksubcategoryofnegligibleobjects 47
3.4. Tannakiancategories 51
3.5. Nearbycycles 54
3.6. Specializationofcharacters 58
3.7. VariationofTannakagroupsinfamilies 61
Chapter4. TheTannakagroupofthethetadivisor 69
4.1. TheSchottkyproblemingenus4 71
4.2. Nondegeneratebilinearforms 73
4.3. Localmonodromy 74
4.4. Degenerationsofabelianvarieties 79
4.5. Proofofthemaintheorem 85
4.6. SingularitiesoftypeA 88
k
v
vi Contents
Chapter5. AfamilyofsurfaceswithmonodromyW(E ) 93
6
5.1. Intersectionsofthetadivisors 95
5.2. VariationsofHodgestructures 102
5.3. Integralcohomology 106
5.4. Ne´ron-Severilattices 109
5.5. TheupperboundW(E ) 115
6
5.6. Negligibleconstituents 119
5.7. Jacobianvarieties 123
5.8. Thedifferencemorphism 127
5.9. Asmoothglobalfamily 131
Chapter6. ThegenericrankofBrill-Noethersheaves 135
6.1. Brill-Noethersheaves 135
6.2. Aformulaforthegenericrank 137
6.3. Computationsinthesymmetricproduct 139
6.4. ThefibresofmultipleAbel-Jacobimaps 142
6.5. Bettipolynomials 144
6.6. Somenumericalvalues 146
AppendixA. Reductivesupergroups 151
AppendixB. Irreduciblesubgroupswithboundeddeterminant 157
AppendixC. AnirreduciblesubgroupofW(E ) 163
6
AppendixD. Doublytransitivepermutationgroups 165
Bibliography 169
Introduction
The geometry of a smooth complex projective curve C with Jacobian
varietyJac(C)isgoverned,afterthechoiceofabasepointonthecurve,by
the Abel-Jacobi morphism. In particular, the latter defines for each n∈N
amorphismCn=C×···×C−→Jac(C)whosefibrescorrespondtolinear
seriesofdivisorsonthecurve,andthelociwherethefibredimensionjumps
arethesubvarietiesWr ⊆Jac(C)ofspecialdivisorsthathavebeenstudied
n
extensivelyinclassicalBrill-Noethertheory[6].
ForsmoothcomplexprojectivevarietiesY ofhigherdimensiononestill
hastheAlbanesemorphism
f : Yn = Y×···×Y −→ X = Alb(Y),
n
butherethesituationismuchhardertodescribeingeneral. Forexample,the
image f (Y)(cid:44)→X can have any dimension between zero and dim(X), and
1
there is no obvious substitute for Brill-Noether theory in general. In some
sense,thefirstthreechaptersofthisthesisprovideapossibleframeworkfor
such a substitute. More specifically, the properties of f we are interested
n
in (such as the loci on X where the fibre dimension jumps) can be studied
viathehigherdirectimages
Rifn∗(CYn) for i≥0,
whichareconstructiblesheavesinthesensethatthereexistsastratification
of X into finitely many locally closed subvarieties over which they restrict
to locally constant sheaves. In general the geometry of these higher direct
image sheaves is rather involved, as we will see in a particular example in
chapter5. However,weproposeasimpledescriptionofsuchdirectimages
intermsoftherepresentationtheoryofareductivealgebraicgroupthatcan
beattachedtoY viatheTannakianformalism[33].
ThebasisforthisTannakiandescriptionwillbeavanishingtheoremfor
constructible sheaves on abelian varieties which can be seen as an analog
of the Artin-Grothendieck affine vanishing theorem [7, exp. XIV, cor. 3.2]
andcontainsasaspecialcasethegenericvanishingtheoremsofGreenand
Lazarsfeld [47]. For any character χ :π (X,0)−→C∗ of the fundamental
1
group, let us denote by L the corresponding local system on X. Then in
χ
chapters 1 and 2 which are based on joint work with R. Weissauer [68],
1
2 Introduction
we show that for any constructible sheaf F on X and a sufficiently general
character χ onehas
Hi(X,F⊗CLχ) = 0 forall i > dim(Supp(F)).
AnindependentproofofthisresulthasalsobeengivenbyC.Schnell[94],
using the Fourier-Mukai transform for holonomic D-modules. Our proof
isofadifferentflavourandiscloselyintertwinedwiththedefinitionofthe
Tannakiancategoriesweareinterestedin. Itisbasedonanabstractquotient
construction for semisimple tensor categories — we only use D-modules
at a single place in section 1.4 where we classify all perverse sheaves of
Eulercharacteristiczero(whichwillbepreciselytheobjectswhichbecome
isomorphic to zero under the above quotient construction). At present we
donotknowwhetherthisclassificationextendstoalgebraicallyclosedbase
fields ofpositive characteristic p>0, but otherwise allour arguments also
workforl-adicconstructiblesheavesonabelianvarietiesoverthealgebraic
closureofafinitefieldF forprimenumbersl (cid:54)= p.
p
OurTannakianresultsarebestformulatedintheframeworkofperverse
sheaves which has its historic roots in the theory of D-modules [57] and
in the sheaf-theoretic construction of intersection cohomology for singular
varieties [10]. Let us for convenience briefly recall some basic definitions
and notations from loc. cit. For any complex algebraic variety Z, we will
denote by Db(Z,C) the derived category of bounded C-sheaf complexes
c
whose cohomology sheaves are constructible for some stratification of Z
intoZariski-locallyclosedsubsets. Thisisatriangulatedcategory,andone
candefinethefullabeliansubcategory
Perv(Z,C) ⊂ Db(Z,C)
c
ofperversesheavestobethecoreofthemiddleperverset-structure. More
explicitly,asheafcomplexK issaidtobesemi-perverseifitscohomology
sheaves H −i(K) satisfy the support estimate dim(SuppH −i(K))≤i for
alli∈Z,anditisaperversesheafiffbothK anditsVerdierdualD(K)are
semi-perverse. Forexample,theperverseintersectioncohomologysheafis
bydefinitiontheintermediateextension
δ = j (C [dim(Z)]) ∈ Perv(Z,C)
Z !∗ U
where j:U (cid:44)→Z denotestheinclusionofasmoothopendensesubset. This
perverse sheaf is self-dual with respect to Verdier duality, and it does not
depend on the choice of the smooth open dense subset. Thus for smooth
varieties Z the perverse intersection cohomology sheaf δ coincides with
Z
theconstantsheafuptoadegreeshift. IfZ isaclosedsubvarietyofX with
embedding i:Z (cid:44)→X, then the direct image i :Perv(Z,C)(cid:44)→Perv(X,C)
∗
isafullyfaithfulfunctor,andinthiscasewewillbyabuseofnotationalso
writeδ fortheperversesheafi (δ )∈Perv(X,C).
Z ∗ Z