Groups of Autoequivalences of Derived Categories of Smooth Projective Varieties

71 Pages·2005·0.52 MB·English

Groups of Autoequivalences of Derived
Categories of Smooth Projective Varieties
David Ploog
Dissertation
eingereicht beim Institut für
Mathematik und Informatik
der Freien Universität Berlin
Januar 2005

Contents
Introduction 4
Notation 6
1 GeneralitiesonFourier-Mukaitransforms 7
1.1 DerivedcategoriesandFourier-Mukaikernels . . . . . . . . . . . . . . . 7
1.2 Comparisonwithcohomology . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Reflectionsalongsphericalobjects . . . . . . . . . . . . . . . . . . . . . . 16
1.4 OntheconditionK = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Y
2 K3surfaces 24
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 AutoequivalencesforK3surfaces . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 TheOrientationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Cohomologicallytrivialautoequivalences . . . . . . . . . . . . . . . . . . 32
3 Finitegroupactionsandderivedcategories 38
3.1 Linearisationsforfinitegroups . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 EquivariantFourier-MukaifunctorsandderivedMcKaycorrespondence 43
3.3 Groupsofequivariantautoequivalences . . . . . . . . . . . . . . . . . . . 46
3.4 Anapplication: Hilbertschemesofpoints . . . . . . . . . . . . . . . . . . 52
4 AutoequivalencesforabelianvarietiesandKummersurfaces 54
4.1 Derivedcategoriesofabelianvarieties: resultsofMukaiandOrlov . . . 54
4.2 Equivariantequivalencesfortheinvolutionaction . . . . . . . . . . . . . 58
4.3 TheKummercase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 GeneralisedKummervarieties . . . . . . . . . . . . . . . . . . . . . . . . 63
Aboutthegroundfield 65
Acknowledgements 66
References 67
3
Introduction
Thisstudyfocusesonautomorphismgroupsofderivedcategoriesofsmoothprojective
varieties. Part of the motivation comes from considering the derived category of such
varietiesassomethingliketheultimatecohomologicalinvariant. Inparticular,onecan
comparethederivedlevelwithmoretraditionalinvariantslikeK-theoryortopological
cohomology.
ItwasMukaiwhofirstgaveanexplicitexampleofasignificantequivalencebetween
derivedcategoriescomingfromgeometry[61]. Heshowedthatthederivedcategories
of an abelian variety and its dual are equivalent. The technique introduced by him to
provethishasbeengeneralisedsincethenandnowadaysgoesbythenameofFourier-
Mukai transforms. In a further step, Bondal and Orlov gave another link connecting
geometryandderivedcategories[11],[12]: theyshowedhowvarietiesofFanotypeor
with ample canonical sheaf can be reconstructed from their derived categories. Thus
attentionwasdrawntothecaseofvarietieswithtrivialcanonicalbundle. Orlovproved
aderivedversionoftheTorellitheoremforK3surfaces[71]andalsoastructuretheorem
for derivedcategoriesofabelian varieties [72]. Meanwhile, Kawamata provided some
evidence to the conjecture that two birational smooth projective varieties with trivial
canonical sheaves have equivalent derived categories [49]. Bridgeland proved this in
dimensionthree[16].
Oneshouldpointoutwhygroupsofequivalencesareimportantandinteresting. Let
us just note for the moment that all the fundamental results mentioned above bring
forthexpressionsintermsofthesegroups. Forexample,thereconstructiontheoremof
BondalandOrlovalsoimpliesthatthegroupsofautoequivalencesonvarietiesofFano
type or with ample canonical sheaf as small as possible [12]. In concrete terms, this
meansthateveryautoequivalenceisacompositionofanautomorphismofthevariety,
thetwistbysomeline bundle,andashiftofcomplexes. Ontheotherhand,considera
principally polarisedabelian variety. ThenMukai’sinitialobservationcanbereformu-
lated saying that the Poincaré bundle yields a genuinely different autoequivalence on
thisabelian variety.
Thereisacertainconnectionbetweenthetopicsaddressedhereandsomeaspectsof
theoreticalphysics[26]. Goingalittlebit moreintodetail,itisnowgenerallyaccepted
bystringtheoriststhatD-branesformatriangulatedcategory. InthecaseoftheIIBthe-
ory,whichlivesonaCalabi-Yaumanifold,theD-branesaresheavesor,moregenerally,
objectsofthederivedcategory. Kontsevich’shomologicalmirror symmetryconjecture
states that a symplectic manifold is the mirror of some Calabi-Yau variety precisely if
a certain triangulated category constructed from the symplectic data is equivalent to
thederivedcategoryonthealgebraicside[52]. Whilenoteventhefinaldefinitionsare
completelyclear onthesymplecticside,thisconjecturehas nonethelessgiven impetus
toresearchinbothsymplecticandalgebraicgeometry.
4
Thisstudyconsistsoffourchapters,andwenowsummarisetheirmaintopics. Inthe
first chapter, we review facts about derived categories and Fourier-Mukai transforms
from thegeneraltheory. This expositionmainly collects resultswhich are standard by
now. A point worthy of notice might be the section on spherical objects as it contains
asimpleproofthatthefunctorsderivedfromthemare equivalences. Anauxiliary fact
we could not find in the literature is that Fourier-Mukai kernels of equivalences are
alwayssimple.
Having covered the basics, we turn to K3 surfaces. The main result is that every
otherHodgeisometryofthefullcohomologylatticecanbeliftedtoanautoequivalence
ofthederivedcategory. Explicitly,wecancomparethegroupofautoequivalenceswith
thegroupofintegralHodgeisometriesusingahomomorphismfromtheformertothe
latter. Thetheoremstatesthat theimage is a subgroupofindex at mosttwo. Whether
thehomomorphismisactually surjectiveornotisnotknownatpresent. Furtheronin
thissection,werephraseandreconsiderthisprobleminsomeways.
Inthethirdpart,weconsiderfinitegroupactionsonsmoothprojectivevarietiesand
develop a theory of groups of equivariant autoequivalences. The aim is to compare
invariant Fourier-Mukai kernels and equivariant ones. How close they are depends
on the group action of course but, nonetheless, there are quite general results. As an
immediate application, we can showthat birational Hilbert schemesofpointsona K3
surface have equivalent derived categories. From this, we deduce that there are only
finitely many Hilbert schemes of points on K3 surfaces up to birational isomorphism.
ThistiesinwithKawamata’sphilosophyalludedtoabove.
The final part deals with Kummer surfaces. Since these are formed from abelian
surfaces by a quotient construction with the cyclic group of order two, our results on
equivariant autoequivalence groupsapply particularly well. We explain what portion
of the group of Kummer autoequivalences actually comes from the abelian surface.
TherearealsocertainremarksongeneralisedKummervarieties.
5
Notation and conventions
Varietiesandmorphisms
Ifnotmentionedexplicitlyotherwise,byavarietywemeanasmoothprojectivescheme
over the field C. Nonetheless, we write k(y) for the skyscraper sheaf of some point
y Y; points are always closed. Canonical projections are denoted throughout by
∈
p : Y Z Y or p ,p : Y Y Y. Becausethis notationis soeasyto understand,
Y 1 2
× → × →
themapswillsometimesappearwithoutfurtherintroduction.
Complexes
Complexesare writtenin the form [ Ai Ai+1 ...] and denotedby A or A ,
•
··· → → →
dependingonemphasis. Thehomologyofacomplex Aindegreei iswrittenashi(A).
Thereasonfor thisis toavoid anyconfusionwith sheafcohomology. Accordingly,the
dimension of sheaf cohomology spaces will always be denoted by dim(Hi( )). The
·
complex A[1]denotes Ashiftedoneplacetotheleft,sothat A[1]i = Ai+1.
Derivedcategories
We write D(Y) := Db(Coh(Y)) for the bounded category of coherent sheaves on a
variety Y. The category of coherent sheaves is throughout considered as the full sub-
categoryofcomplexeshavingonlyhomologyindegree0. Derivedfunctorswillappear
without special symbols to denote derivation. For example, considering a morphism
f : Y Z, the functor f : D(Y) D(Z) is exact. Taking a sheaf A Coh(Y), its
→ ∗ → ∈
directimage f (A) D(Z) isgenerallyagenuinecomplex,andthehomologysheaves
∗ ∈
hi(f (A)) = Rif (A) Coh(Z)aretheusualhigherdirectimages.
∗ ∗ ∈
As usual, we set Homi (A,B) = Hom (A,B[i]) for complexes A,B D(Y).
D(Y) D(Y) ∈
Forsheaves A,B Coh(Y) wehave Hom (A,B[i]) = Exti (A,B) by[30, §III.5]. We
∈ D(Y) Y
distinguish between the total homomorphism space Hom∗(A,B) := i ZHomi(A,B)
andthecomplexHom•(A,B) := i ZHomi(A,B)[ i]withzerodiffer⊕en∈tials.
⊕∈ −
Ifafunctor F isleftadjointtoafunctorG,wewillsometimeswritethisas F G.
⊣
Remark 0. If C is a smooth curve, then A = hi(A)[ i] for all A D(C) by [31,
∼ i Z − ∈
∈
Proposition6.3]or[42,Corollary3.14]. Thuswearemainlyinterestedindimension>1.
L
Cohomologyandintersectionrings
LetY beavarietyand Abeacoherentsheaf(orcomplex)onY. Asiscommonbynow,
we write v(A) := ch(A)√td H (Y,Q) for the Mukai vector of A. The cycle class of
Y ∗
∈
asubvariety Z Y ofcodimensionciswritten[Z] H2c(Y,Z). Inparticular, wehave
⊂ ∈
1 = [Y] H0(Y,Z) and [pt] H2dim(Y)(Y,Z), thelatternotdependingonaparticular
∈ ∈
point of Y. The same notation will apply to the intersection (or Chow) rings CH∗(Y),
sothat1 = [Y] CH0(Y) and [p] CHdim(Y)(Y). Notethatthelattercycle class inthe
∈ ∈
Chowringdependsonthe(rationalequivalenceclassofthe)point p Y. Theproduct
∈
oftwoclassesorcyclesz ,z issometimesdenotedz .z .
1 2 1 2
6
1 Generalities on Fourier-Mukai transforms
1.1 Derived categoriesand Fourier-Mukaikernels
Inthissection,wewillnottrytoreproducethegeneraltheorybut ratherassemblethe
important facts from a concrete point of view. Let Y be a smooth projective variety
over C. Then, an interesting homological invariant of Y is D(Y) := Db(Coh(Y)), the
bounded derived category of coherent sheaves. While relatives of D(Y), using quasi-
coherentsheavesorunboundedcomplexes,areimportant,theywillnotappearinthis
study. A virtue of insisting on ’smooth, projective’varieties and ’bounded’complexes
of’coherent’sheavesisthatD(Y)turnsouttobeafiniteC-lineartriangulatedcategory,
i.e. allHom’sarefinitedimensionalC-vectorspaces. ThisfollowsbasicallyfromSerre’s
theorem that sheaf cohomology is finite dimensional under these circumstances [36,
TheoremIII.5.2]. WeareinterestedintheautomorphismsofD(Y),thus
Aut(D(Y)) := ϕ : D(Y) D(Y)exactequivalence .
∼
{ → }
Itisobviousthatthecompositionofexactfunctorsisagainexact;furthermorethequasi-
inverseofanexactequivalenceisexact. HenceAut(D(Y))isindeedagroup. Ingeneral,
thereare threetypesofautoequivalencesthatwealways have at ourdisposal,namely
shiftsofcomplexes,automorphismsofthevarietyandlinebundletwists:
Z Aut(D(Y)), n [n],
→ 7→
Aut(Y) Aut(D(Y)), f f = (f 1) ,
− ∗
Pic(Y) → Aut(D(Y)), L7→ M∗ , M (E) := L E.
L L
→ 7→ ⊗
Allthreerepresentationsareinjective. Theycombinetoasemidirectproductsubgroup
Aut(Y)⋉Pic(Y) Z Aut(D(Y)), as (M f ) (M f ) = M (f f ) .
× ⊂ L1 ◦ 1∗ ◦ L2 ◦ 2∗ L1⊗f1∗L2 ◦ 1 2 ∗
Actually,thisisthedistinguishedsubgroupcontainingallautoequivalenceswhichmap
skyscrapersheavesofpointstothemselves,uptoshift;seeRemark1.9.
WeturntoSerrefunctorsandadjoints. ThenotionofSerrefunctorhasbeenaxiomati-
callydefinedfortriangulatedcategoriesin[10]butwesticktotheconcretesetting.
(1) The Serre functor for D(Y) is given by S : D(Y) D(Y),A A ω [dim(Y)]
Y Y
→ 7→ ⊗
(andsimilaronmorphisms). Itisobviouslyanexactequivalence.
(2) BySerredualityintheformdescribedin[35],S isanequivalencewithfunctorial
Y
isomorphismsHom(A,B) Hom(B,S(A)) forall A,B D(Y). Theseproperties
∼ ∗
→ ∈
characteriseS uniquelyuptoisomorphism;see[42,§1.1].
Y
(3) The Serre functor S commutes with exact equivalences. This follows from the
Y
uniquenessin(2)andtheusualYonedaargument.
(4) If F : D(Y) D(Z) isa functor(with Z anothersmoothprojectivevariety)which
→
hasarightadjoint G : D(Z) D(Y), thenthefunctor S 1 G S isaleftadjoint
→ Y− ◦ ◦ Z
for F. Thisisagainanimmediateconsequenceof(2).
7
Before we presenta couple of facts about adjoint functors, we introduce some general
notions. Acontravariant cohomologicalfunctor H : D(Y)opp Mod(C) totheabelian
→
category of C-vector spaces is of finite type if ∑ dimH(A[i]) < ∞ for all A D(Y). A
i ∈
triangulated category is called saturated if every contravariant cohomological functor
of finite type is representable. The next theorem is crucial for the existence of adjoint
functors:
Theorem 1.1 (Bondal, Van den Bergh). For a smooth projective variety Y, the bounded
derived categoryofcoherentsheaves D(Y)issaturated.
Proof. The theoremis proven in [9, AppendixA] togetherwith vast generalisations to
thesingularaswellastothenoncommutativecase.
Thefollowingremarkssettlethequestionofadjointsinoursettingcompletely.
(1) Anadjointfunctorofanexactfunctorisagainexact[71,Lemma1.2].
(2) Any exact functor F : D(Y) D(Z) has a right adjoint. To see this, consider for
→
any B D(Z) the cohomological functor H := Hom(F( ),B) : D(Y) Mod(C);
∈ · →
it is of finite type because D(Y) has finite dimensional total Hom’s. As D(Y) is
saturated, there is a unique object A D(Y) representing H. Setting G(B) := A
∈
defines a functor G : D(Z) D(Y) by the Yoneda lemma. We conclude F G
→ ⊣
fromHom(F( ),B) = H( ) = Hom( ,A) = Hom( ,G(B)).
· · · ·
(3) Finally, the last remark concerning Serre functors applies and we find that every
exactfunctor F : D(Y) D(Z) possessesbothleftandrightadjoints.
→
OncewerecognisethederivedcategoryD(Y) asaninterestinginvariant ofthevariety
Y, the relations among those categories will be of importance. As has been hinted at
previously,weregardD(Y)asafinerrelativeoftheK-groupK(Y),ChowringCH∗(Y)
or topological cohomology H (Y,Q). Most notably for Chow rings and cohomology,
∗
these structures have been classically studied by the means of correspondences. The
basic idea is that an object on the product gives rise to a map, e.g.a cohomologyclass
η H (Y Z,Q) yields the map H (Y,Q) H (Z,Q), α p (η.p α). Here, pull-
∈ ∗ × ∗ → ∗ 7→ Z∗ Y∗
back, push-forward and product are those from topology. Maps of this sort provide
all kinds of interesting examples. Since we also have the three basic operations in the
derivedcategoriesD(Y)atourdisposal,wecanestablishasimilarnotion:
Definition1.2. LetYandZbesmoothprojectivevarietiesandP D(Y Z)anyobject.
∈ ×
ThenthecorrespondingFourier-Mukaitransformwithkernel Pistheexactfunctor
FMP : D(Y) → D(Z), A 7→ pZ∗(P⊗pY∗A).
The functor FM is called a Fourier-Mukai equivalence if it is an exact equivalence be-
P
tweentriangulatedcategories.
8
Remark1.3. MukaiwasthefirsttoemployFourier-Mukaitransformsin[61]. Therehe
used the Poincaré bundle as kernel to show that the derived categories of an abelian
variety and of the dual abelian variety are equivalent. His use of the name Fourier
stems from the comparison with Volterra operators in functional analysis which are
formallycorrespondencesbetweenappropriate L2-spaces;see[62].
Remark 1.4. The calculus of correspondences [29, §14] carries over from the classical
versions. To quotejust one instance, thecomposite transformof FM : D(X) D(Y)
Q
→
and FM : D(Y) D(Z) haskernel P⋆Q := p (p P p Q) D(X Z), where
P → XZ∗ Y∗Z ⊗ ∗XY ∈ ×
p ,p ,p are thecanonical projectionsfrom X Y Z totheindicated factors. So
XY XZ YZ
× ×
wehaveFMP FMQ = FMP⋆Q : D(X) D(Z)forthecorrespondingfunctors.
◦ →
AnicefeatureofthesetransformsisthattheadjointsofFM : D(Y) D(Z)areagain
P
→
Fourier-Mukai transforms. Their kernels are PL := P p ω [dim(Z)] and PR :=
∨ ⊗ ∗Z Z
P p ω [dim(Y)], respectively. Thisfollowsfrom f f f! := S f S 1 fora
m∨o⊗rphY∗ismYf : Y Y whereS andS aretheSerre∗fu⊣nc∗to⊣rs. InparYt2i◦cul∗ar◦, ifY−F1M is
1 → 2 Y1 Y2 P
fullyfaithfulandbothadjointscoincide,thenFM isaFourier-Mukaiequivalencewith
P
quasi-inverseFM−P1 ∼= FMPL : D(Z) →∼ D(Y).
We will use the familiar facts about Fourier-Mukai transforms without much ado
whenevertheyarejustconsequencesofthetransformsbeingderivedcorrespondences.
Remark1.5. NotethatwecouldalsohaveusedPtoproduceafunctorinthedirection
D(Z) D(Y). In order to keepthe notation light, we will refrain from specifying the
→
directioninthesymbolFM . Thefunctorinthereversedirectionwill,however,appear.
P
In those cases, it will be denoted by FMt. The assignment FM FMt is a direction
P P 7→ P
reversinginvolutionalthoughFMt isnotthetransposeonthelevelofcohomology. We
P
will be careful to introduce a kernel P always togetherwith a prescribed direction for
FM ,sothatthenotationmakessense.
P
Examples1.6. WegivesomeexamplesforFourier-Mukaiequivalences. Theproofsfor
thefirstthreeinstancesarestraightforward.
(1) The structure sheaf of the diagonal ∆ Y Y is the kernel for the identity, i.e.
⊂ ×
FM = id : D(Y) D(Y). Analogously, weget theshift functors: FM = [n]
O∆ →∼ O∆[n]
for all n Z. In the same vein, let ∆ : Y Y Y be the diagonal embed-
∈ → ×
ding and L a line bundle on Y. Then, the kernel L∆ := ∆ L gives FML∆ = ML :
∗
D(Y) D(Y),A A L.
∼
→ 7→ ⊗
(2) Foramorphism f : Y Z,thestructuresheafofthegraphΓ(f) := (f,id ) 1(∆ )
Z − Z
→
yieldsFM = f . NotethatFMt = f .
OΓ(f) ∗ OΓ(f) ∗
(3) Consideran ellipticcurve E withafixedorigin p andidentify E and Eˆ = Pic0(E)
0
via ϕ : E Eˆ,x (x p ) (subtraction ofdivisors). ThenormalisedPoincaré
∼ E 0
→ 7→ O −
line bundle := (∆ p E E p ) has a universal propertywhich
E E 0 0
can be exprePssed bOy×FM (−k({x))}=×ϕ(x−). I×t is{ea}sy to see, using Cohomology and
BaseChange,thatFM PFM = ( 1) [ 1]. Inparticular,FM Aut(D(E)).
∗
P ◦ P − − P ∈
9
(4) Let us consider two Fourier-Mukai equivalences F = FM : D(Y) D(Y ) and
∼ ′
G = FM : D(Z) D(Z ). Then their product gives a fuFrther equ→ivalence F
∼ ′
G = FMG⊠ : D(Y→ Z) ∼ D(Y′ Z′) whose quasi-inverse has kernel L ⊠ ×L.
TakinganFotGherFour×ier-M→ukaitran×sformFM : D(Y) D(Z), wegettheFrelatiGon
P
→
FM = G FM Ft with Ft := FMt ,i.e.acommutativediagram
(F G)(P) P
× ◦ ◦ F
FM
D(Y) P //D(Z)
OO
Ft=FMt G
F (cid:15)(cid:15)
D(Y ) //D(Z )
′ FM ′
(F G)(P)
×
whichisjustarestatementofthefollowingisomorphismonthekernellevel:
⊠
(F×G)(P) = pY′Z′,∗(F G ⊗pY∗ZP) = pY′Z′,∗(p∗ZZ′G⊗ pY∗ZP⊗ pY∗′YF),
the last term being the kernel of FM FM FMt . As a special case, consider
P
G ◦ ◦ F
two automorphisms f : Y Y, g : Z Z. Then, the new kernel (f,g) P gives
∼ ∼ ∗
→ →
FM = g FM f .
(f,g) P ∗ P
∗ ◦ ◦ ∗
ThenotionofFourier-Mukaitransformcanalsoberegarded,inamorefancymanner,
asafunctor
FM : D(Y Z) Fun(D(Y),D(Z)), P FM
P
× → 7→
whereFun(D(Y),D(Z)) isthecategoryconsistingofexactfunctorsfromD(Y)toD(Z)
with natural transformations as morphisms. This category can be made graded by
defining the shift of F Fun(D(Y),D(Z)) as F [1] = [1] F. Yet declaring
∈ ◦ D(Y) ∼ D(Z) ◦
a triangle F F F F [1] to be distinguished in Fun(D(Y),D(Z)) if the trian-
′ ′′ ′
→ → →
gles F (A) F(A) F (A) F (A)[1] in D(Z) are distinguished for all A D(Y)
′ ′′ ′
→ → → ∈
doesnotproduceatriangulatedcategory—sinceconesarenotfunctorial. See[21]fora
similar,butmoresophisticatedviewonthistopic.
We will mainly be interested in equivalences between derived categories. A basic,
but highly non-trivial, theorem of Orlov states that all such equivalences actually are
Fourier-Mukaitransforms,and thuscan be describedusingkernels. Theprecisestate-
mentfollows:
Theorem1.7(Orlov). LetY andZ betwosmoothprojective varieties and F : D(Y) D(Z)
→
an exact fully faithful functor. Then there exists an object P D(Y Z) such that F = FM
∈ × ∼ P
areisomorphic functors. Moreover, Pisuniqueuptoisomorphism.
Proof. Fortheproofseetheoriginal[71,Theorem2.2]orKawamata’s generalisationto
orbifold-typestacks,[50,Theorem1.1]. Theoriginalstatementincludedtheadditional
assumptionthat F should possessadjoint functors. This is howeverautomatically ful-
filledinviewofTheorem1.1anditscorollary.
10