The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

65 Pages·2016·0.99 MB·English

by  Arend Bayer

Invent.math.(2016)206:869–933
DOI10.1007/s00222-016-0665-5
The space of stability conditions on abelian
threefolds, and on some Calabi-Yau threefolds
ArendBayer1 · EmanueleMacrì2,3 ·
PaoloStellari4
Received:15October2014/Accepted:12April2016/Publishedonline:9May2016
©TheAuthor(s)2016.ThisarticleispublishedwithopenaccessatSpringerlink.com
Abstract Wedescribeaconnectedcomponentofthespaceofstabilitycondi-
tions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the
crepant resolution of) a finite quotient of an abelian threefold. Our proof
includesthefollowingessentialsteps:
1. We simultaneously strengthen a conjecture by the first two authors and
Toda,andprovethatitfollowsfromamorenaturalandseeminglyweaker
statement.ThisconjectureisaBogomolov-Giesekertypeinequalityinvolv-
B
ArendBayer
[email protected]
http://www.maths.ed.ac.uk/∼abayer/
EmanueleMacrì
[email protected]
http://nuweb15.neu.edu/emacri/
PaoloStellari
[email protected]
http://users.unimi.it/stellari/
1 SchoolofMathematicsandMaxwellInstitute,UniversityofEdinburgh,JamesClerk
MaxwellBuilding,PeterGuthrieTaitRoad,EdinburghEH93FD,UK
2 DepartmentofMathematics,TheOhioStateUniversity,231W18thAvenue,
Columbus,OH43210,USA
3 PresentAddress:DepartmentofMathematics,NortheasternUniversity,360
HuntingtonAvenue,Boston,MA02115,USA
4 DipartimentodiMatematica“F.Enriques”,UniversitàdegliStudidiMilano,ViaCesare
Saldini50,20133Milan,Italy
123
870 A.Bayeretal.
ingthethirdCherncharacterof“tilt-stable”two-termcomplexesonsmooth
projectivethreefolds;weextenditfromcomplexesoftilt-slopezerotoarbi-
trarytilt-slope.
2. Weshowthatthisstrongerconjectureimpliestheso-calledsupportproperty
of Bridgeland stability conditions, and the existence of an explicit open
subsetofthespaceofstabilityconditions.
3. Weproveourconjectureforabelianthreefolds,therebyreprovingandgen-
eralizingaresultbyMaciociaandPiyaratne.
Importantinourapproachisamoresystematicunderstandingonthebehaviour
of quadratic inequalities for semistable objects under wall-crossing, closely
relatedtothesupportproperty.
Mathematics Subject Classification Primary 14F05; Secondary 14J30 ·
18E30
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870
2 Review:tilt-stabilityandtheconjecturalBGinequality . . . . . . . . . . . . . . . . . 879
3 ClassicalBogomolov-Giesekertypeinequalities . . . . . . . . . . . . . . . . . . . . 882
4 Generalizingthemainconjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
5 Reductiontosmallα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890
6 TiltstabilityandétaleGaloiscovers . . . . . . . . . . . . . . . . . . . . . . . . . . . 894
7 Abelianthreefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
8 ConstructionofBridgelandstabilityconditions . . . . . . . . . . . . . . . . . . . . . 903
9 Thespaceofstabilityconditionsonabelianthreefolds . . . . . . . . . . . . . . . . . 914
10ThespaceofstabilityconditionsonsomeCalabi-Yauthreefolds . . . . . . . . . . . . 916
Appendix1:Supportpropertyviaquadraticforms . . . . . . . . . . . . . . . . . . . . . 921
Appendix2:Deformingtilt-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
1 Introduction
In this paper, we determine the space of Bridgeland stability conditions on
abelian threefolds and on Calabi-Yau threefolds obtained either as a finite
quotient of an abelian threefold, or as the crepant resolution of such a quo-
tient. More precisely, we describe a connected component of the space of
stabilityconditionsforwhichthecentralchargeonlydependsonthedegrees
H3−i ch ( ), i = 0,1,2,3, of the Chern character1 with respect to a given
i
polarization H,andthatsatisfythesupportproperty.
1 Inthecaseofcrepantresolutions,wetaketheCherncharacterafterapplyingBKR-equivalence
[8]betweenthecrepantresolutionandtheorbifoldquotient.
123
Thespaceofstabilityconditionsonabelianthreefolds… 871
Stability conditions on threefolds via a conjectural Bogomolov-Gieseker
typeinequality
The existence of stability conditions on three-dimensional varieties in gen-
eral, and more specifically on Calabi-Yau threefolds, is often considered the
biggest open problem in the theory of Bridgeland stability conditions. Until
recentworkbyMaciociaandPiyaratne[29,30],theywereonlyknowntoexist
onthreefoldswhosederivedcategoryadmitsafullexceptionalcollection.Pos-
sibleapplicationsofstabilityconditionsrangefrommodularitypropertiesof
generatingfunctionsofDonaldson-Thomasinvariants[43,45]toReider-type
theoremsforadjointlinearseries[6].
In [11], the first two authors and Yukinobu Toda, also based on dis-
cussions with Aaron Bertram, proposed a general approach towards the
construction of stability conditions on a smooth projective threefold X.
The construction is based on the auxiliary notion of tilt-stability for two-
term complexes, and a conjectural Bogomolov-Gieseker type inequality for
the third Chern character of tilt-stable objects; we review these notions
in Sect. 2 and the precise inequality in Conjecture 2.4. It depends on
the choice of two divisor classes ω,B ∈ NS(X)R with ω ample. It was
shown that this conjecture would imply the existence of Bridgeland sta-
bility conditions,2 and, in the companion paper [6], a version of an open
case of Fujita’s conjecture, on the very ampleness of adjoint line bundles on
threefolds.
Ourfirstmainresultisthefollowing,generalizingtheresultof[29,30]for
thecasewhen X hasPicardrankone:
Theorem1.1 TheBogomolov-Giesekertypeinequalityfortilt-stableobjects,
Conjecture2.4,holdswhen X isanabelianthreefold,andωisarealmultiple
ofanintegralampledivisorclass.
ThereareCalabi-Yauthreefoldsthatadmitanabelianvarietyasafiniteétale
cover;wecallthemCalabi-Yauthreefoldsofabeliantype.Ourresultapplies
similarlyinthesecases:
Theorem1.2 Conjecture 2.4 holds when X is a Calabi-Yau threefold of
abeliantype,andωisarealmultipleofanintegralampledivisorclass.
Combined with the results of [11], these theorems imply the existence of
Bridgelandstabilityconditionsineithercase.ThereisonemoretypeofCalabi-
Yau threefolds whose derived category is closely related to those of abelian
threefolds: namely Kummer threefolds, that are obtained as the crepant reso-
lutionofthequotientofanabelianthreefold X bytheactionofafinitegroup
2 Notincludingtheso-called“supportproperty”reviewedfurtherbelow.
123
872 A.Bayeretal.
G.Usingthemethodof“inducing”stabilityconditionsonthe G-equivariant
derivedcategoryofXandtheBKR-equivalence[8],wecanalsotreatthiscase.
Overallthisleadstothefollowingresult(whichwewillmakemoreprecisein
Theorem1.4).
Theorem1.3 BridgelandstabilityconditionsonX existwhenX isanabelian
threefold,oraCalabi-Yauthreefoldofabeliantype,oraKummerthreefold.
Supportproperty
ThenotionofsupportpropertyofaBridgelandstabilityconditioniscrucialin
ordertoapplythemainresultof[13],namelythatthestabilityconditioncan
be deformed; moreover, it ensures that the space of such stability conditions
satisfieswell-behavedwall-crossing.
Inordertoprovethesupportproperty,wefirstneedaquadraticinequality
foralltilt-stablecomplexes,whereasConjecture2.4onlytreatscomplexes E
withtilt-slopezero.WestatesuchaninequalityinConjecture4.1forthecase
whereω,B areproportionaltoagivenampleclass H:
√
Conjecture4.1Let(X,H)beasmoothpolarizedthreefold,andω = 3αH,
B = βH, for α > 0, β ∈ R. If E ∈ Db(X) is tilt-semistablewith respect to
ω,B,then
(cid:2)(cid:2) (cid:3) (cid:3) (cid:2) (cid:3)
2 2
α2 H2chB(E) −2H3chB(E)H chB(E) +4 H chB(E)
1 0 2 2
−6H2chB(E)chB(E) ≥ 0,
1 3
wherechB := e−Bch.
InTheorem4.2,weprovethatthisgeneralizedconjectureisinfactequiv-
alenttotheoriginalConjecture2.4.Moreover,inTheorem8.7weprovethat
itimpliesasimilarquadraticinequalityforobjectsthatarestablewithrespect
to the Bridgeland stability conditions constructed in Theorem 1.3, thereby
obtainingaversionofthesupportproperty.
To be precise, we consider stability conditions whose central charge
Z: K(X) → Cfactorsvia
(cid:4) (cid:5)
v : K(X) → Q4, E (cid:7)→ H3ch (E),H2ch (E),H ch (E),ch (E) .
H 0 1 2 3
(1)
(In the case of Kummer threefolds, we apply the BKR-equivalence before
takingtheCherncharacter.)Weprovethesupportpropertywithrespecttov ;
H
thisshowsthatastabilityconditiondeformsalongasmalldeformationofits
centralcharge,ifthatdeformationstillfactorsviav .
H
123
Thespaceofstabilityconditionsonabelianthreefolds… 873
We discuss the relation between support property, quadratic inequalities
forsemistableobjectsanddeformationsofstabilityconditionssystematically
in Appendix 1. In particular, we obtain an explicit open subset of stability
conditionswheneverConjecture4.1issatisfied,seeTheorem8.2.
Thespaceofstabilityconditions
IneachofthecasesofTheorem1.3,weshowmoreoverthatthisopensubset
isaconnectedcomponentofthespaceofstabilityconditions.Wenowgivea
descriptionofthiscomponent.
InsidethespaceHom(Q4,C),considertheopensubsetVoflinearmaps Z
whose kernel does not intersect the (real) twisted cubic C ⊂ P3(R) parame-
trized by (x3,x2y, 1xy2, 1y3); it is the complement of a real hypersurface.
2 6
Such a linear map Z induces a morphism P1(R) ∼= C → C∗/R∗ = P1(R);
wedefinePbethecomponentofVforwhichthismapisanunramifiedcover
oftopologicaldegree+3withrespecttothenaturalorientations.LetP(cid:6) beits
universalcover.
WeletStab (X)bethespaceofstabilityconditionsforwhichthecentral
H
charge factors via the map v as in equation (1) (and satisfying the support
H
property).
Theorem1.4 Let X be an abelian threefold, or a Calabi-Yau threefold of
abelian type, or a Kummer threefold. Then Stab (X) has a connected com-
H
ponentisomorphictoP(cid:6).
Approach
Wewillnowexplainsomeofthekeystepsofourapproach.
Reductiontoalimitcase
The first step applies to any smooth projective threefold. Assume that ω,B
areproportionaltoagivenamplepolarization H of X.WereduceConjecture
4.1 to a statement for objects E that are stable in the limit as ω(t) → 0 and
νω(t),B(t)(E) → 0;if B := limB(t),theclaimisthat
(cid:7)
e−Bch(E) ≤ 0. (2)
X
Thereductionisbasedonthemethodsof[26]:asweapproachthislimit,either
Eremainsstable,inwhichcasetheaboveinequalityisenoughtoensurethatE
satisfiesourconjectureeverywhere.Otherwise,E willbestrictlysemistableat
123
874 A.Bayeretal.
somepoint;wethenshowthatallitsJordan-Hölderfactorshavestrictlysmaller
“H-discriminant”(whichisavariantofthediscriminantappearingintheclas-
sicalBogomolov-Giesekerinequality).Thisallowsustoproceedbyinduction.
Abelianthreefolds
Inthecaseofanabelianthreefold,wemakeextensiveuseofthemultiplication
bym mapm: X → X inordertoestablishinequality(2).Thekeyfactisthat
∗
if E istilt-stable,thensoism E.
To illustrate these arguments, assume that B is rational. Via pull-back we
can then assume that B is integral; by tensoring with O (B) we reduce to
X
the case of B = 0. We then have to prove that ch (E) ≤ 0; in other words,
3
we have to prove an inequality of the Euler characteristic of E. To obtain a
contradiction,assumethatch (E) > 0,andconsiderfurtherpull-backs:
3
χ(O ,m∗E) = ch (m∗E) = m6ch (E) ≥ m6. (3)
X 3 3
However,bystabilitywehaveHom(O (H),m∗E) = 0;moreover,ifD ∈ |H|
X
isageneralelementofthelinearsystemofH,classicalarguments,basedonthe
Grauert-Mülich theorem and bounds for global sections of slope-semistable
sheaves,giveaboundoftheform
h0(m∗E) ≤ h0((m∗E)| ) = O(m4)
D
Similarboundsforh2 leadtoacontradictionto(3).
Supportproperty
As pointed out by Kontsevich and Soibelman in [21, Sect. 2.1], the support
property is equivalent to the existence of a real quadratic form Q: Q4 → R
suchthat
(a) Thekernelofthecentralcharge(asasubspaceofR4)isnegativedefinite
withrespectto Q,and
(b) Everysemistableobject E satisfies Q(v (E)) ≥ 0.
H
The inequality in Conjecture 4.1 precisely gives such a quadratic form. We
therefore need to show that this inequality is preserved when we move from
tilt-stabilitytoactualBridgelandstabilityconditions.
We establish a more basic phenomenon of this principle in Appendix1,
whichmaybeofindependentinterest:ifastabilityconditionsatisfiesthesup-
portpropertywithrespectto Q,andifwedeformalongapathforwhichthe
centralchargesallsatisfycondition(a),thencondition(b)remainspreserved
123
Thespaceofstabilityconditionsonabelianthreefolds… 875
underthisdeformation,i.e.,itispreservedunderwall-crossing.Theessential
argumentsinvolveelementarylinearalgebraofquadraticforms.
Tilt-stabilitycanbethoughtofasalimitingcaseofapathinthesetofsta-
bilityconditionsweconstruct.InSect.8weshowthattheprincipledescribed
in the previous paragraph similarly holds in this case: we show that a small
perturbation of the quadratic form in Conjecture 4.1 is preserved under the
wall-crossingsbetweentilt-stabilityandanyofourstabilityconditions,thereby
establishingthedesiredsupportproperty.
Connectedcomponent
InAppendix1,wealsoprovideamoreeffectiveversionofBridgeland’sdefor-
mationresult.Inparticular,theproofofthesupportpropertyyieldslargeopen
setsofstabilityconditions,whichcombinetocoverthemanifoldP(cid:6)described
above.
In Sect. 9, we show that this set is in fact an entire component. The proof
is based on the observation that semi-homogeneous vector bundles E with
c (E)proportionalto H arestableeverywhereonP;theirChernclasses(up
1
torescaling)aredenseinC.
Thisfactisveryuniquetovarietiesadmittingétalecoversbyabelianthree-
folds.Inparticular,whileConjecture4.1impliesthatP(cid:6)isasubsetofthespace
ofstabilityconditions,oneshouldingeneralexpectthespacetobemuchlarger
thanthisopensubset.
Applications
Ourworkhasafewimmediateconsequencesunrelatedtoderivedcategories.
Although these are fairly specific, they still serve to illustrate the power of
Conjecture4.1.
Corollary1.5 Let X be a Calabi-Yau threefold of abelian type. Given α ∈
Z>0,let L beanamplelinebundleon X satisfying
• L3 > 49α,
• L2D ≥ 7αforeveryintegraldivisorclassDwithL2D > 0andLD2 < α,
and
• L.C ≥ 3α foreverycurveC ⊂ X.
Then H1(L ⊗ I ) = 0 for every 0-dimensional subscheme Z ⊂ X of length
Z
α.
Inaddition,if L = A⊗5 foranamplelinebundle A,then L isveryample.
Proof Since Conjecture 2.4 holds for X by our Theorem 1.2, we can apply
Theorem4.1andRemark4.3of[6]. (cid:12)(cid:13)
123
876 A.Bayeretal.
Settingα = 2weobtainaReider-typecriterionforL tobeveryample.The
statement for A⊗5 confirms (the very ampleness case of) Fujita’s conjecture
forsuch X.ThebestknownboundsforCalabi-Yauthreefoldssaythat A⊗8 is
veryampleifL3 > 1[18,Corollary1], A⊗10isveryampleingeneral,andthat
A⊗5 induces a birational map [33, Theorem I]. For abelian varieties, much
strongerstatementsareknown,see[37,38].
Corollary1.6 Let X beoneofthefollowingthreefolds:projectivespace,the
quadricinP4,anabelianthreefold,oraCalabi-Yauthreefoldofabeliantype.
Let H be a polarization, and let c ∈ Z>0 be the minimum positive value of
H2D for integral divisor classes D. If E is a sheaf that is slope-stable with
respectto H,andwith H2c (E) = c,then
1
3cch (E) ≤ 2(H ch (E))2.
3 2
The assumptions hold when NS(X) is generated by H, and c (E) = H. We
1
refer to Example 4.4 and Remark 4.5 for a proof and more discussion. Even
for vector bundles on P3, this statement was not previously known for rank
biggerthanthree.
ItisaspecialcaseofConjecture4.1.EvenwhenXisacompleteintersection
threefold and E = I ⊗ L is the twist of an ideal sheaf of a curve C, this
C
inequalityisnotknown,see[49].
Openquestions
GeneralproofofConjecture4.1
While Conjecture 4.1 for arbitrary threefolds remains elusive, our approach
seemstogetabitcloser:inourproofofTheorem1.1(inSects.2,3,4,5,6,7),
onlySect.7isspecifictoabelianthreefolds.Onecouldhopetogeneralizeour
construction by replacing the multiplication map m with ramified coverings.
This would immediately yield the set P(cid:6) as an open subset of the space of
stabilityconditions.
StrengtheningofConjecture4.1
In order to construct a set of stability conditions of dimension equal to the
rankofthealgebraiccohomologyofX,wewouldneedastrongerBogomolov-
Giesekertypeinequality,dependingonch andch directly,notjustonH2ch
1 2 (cid:4) (cid:5)1
and H ch .Wepointoutthattheobviousguess,namelytoreplace H2ch 2
2 1
byH ch2·H3,and(H ch )2byanappropriatequadraticformonH4(X),does
1 2
notworkingeneral:forα → +∞,suchaninequalityfailsfortorsionsheaves
supportedonadivisor D with HD2 < 0.
123
Thespaceofstabilityconditionsonabelianthreefolds… 877
Higherdimension
Ourworkalsoclarifiestheexpectationsforhigherdimensions.Thedefinition
ofPdirectlygeneralizestodimensionn in(cid:4)anobviousway,byreplacingth(cid:5)e
twistedcubicwiththerationalnormalcurve xn,xn−1y, 1xn−2y2,..., 1 yn .
2 n!
LetP(cid:6) → P denotethecorrespondinguniversalcovering.
n n
Conjecture1.7 Let(X,H)beasmoothpolarizedn-dimensionalvariety.Its
spaceStab (X)ofstabilityconditionscontainsanopensubsetP(cid:6) ,forwhich
H n
skyscrapersheavesofpointsarestable.Inthecaseofabelianvarieties,P(cid:6) ⊂
n
Stab (X)isaconnectedcomponent.
H
Suchstabilityconditionscouldbeconstructedbyaninductiveprocedure;thei-
thinductionstepwouldbeanauxiliarynotionofstabilitywithrespecttoaweak
notion of central charge Z depending on Hnch ,Hn−1ch ,...,Hn−i ch .
i 0 1 i
Semistableobjectswouldhavetosatisfyaquadraticinequality Q involving
i
chi+1.Thepreciseformof Qi woulddependontheparametersofthestability
condition; it would always be contained in the defining ideal of the rational
normalcurve,andthekernelofZ wouldbesemi-negativedefinitewithrespect
i
to Q .
i
Onecouldhopetoprovesuchinequalitiesfori < nusingasecondinduction
bydimension:forexample,aninequalityforch forstableobjectsonafourfold
3
wouldfollowfromaMehta-Ramanathantyperestrictiontheorem,showingthat
such objects restrict to semistable objects on threefolds. As a first test case,
oneshouldtrytoprovethatagiventilt-stableobjectonathreefoldrestrictsto
aBridgeland-stableobjectonadivisorofsufficientlyhighdegree.
Relatedwork
Asindicatedabove,thefirstbreakthroughtowardsconstructingstabilitycon-
ditionsonthreefolds(withoutusingexceptionalcollections)isduetoMaciocia
and Piyaratne, who proved Theorem 1.1 in the case of principally polarized
abelian varieties of Picard rank one in [29,30]. Their method is based on
an extensive analysis of the behavior of tilt-stability with respect to Fourier-
Mukaitransforms;inadditiontoconstructingstabilityconditions,theyshow
theirinvarianceunderFourier-Mukaitransforms.
Ourapproachisverydifferent,asitonlyusestheexistenceoftheétaleself-
mapsgivenbymultiplicationwithm.Nevertheless,therearesomesimilarities.
Forexample,acrucialstepintheirargumentsusesrestrictiontodivisorsand
curvestocontrolacertaincohomologysheafoftheFourier-Mukaitransform
of E, see the proof of [29, Proposition 4.15]; in Sect. 7 we use restriction
of divisors explicitly and to curves implicitly (when we use Theorem 7.2) to
controlglobalsectionsofpull-backsof E.
123
878 A.Bayeretal.
Asmentionedearlier,itiseasytoconstructstabilityconditionsonanyvari-
etyadmittingacompleteexceptionalcollection;however,itisstilladelicate
problemtorelatethemtotheconstructionproposedin[11].Thiswasdonein
[11,26]forthecaseofP3,andin[39]forthecaseofthequadricinP4;these
aretheonlyothercasesinwhichConjecture2.4isknown.
Thereisanalternativeconjecturalapproachtowardsstabilityconditionson
the quintic hypersurface in P4 via graded matrix factorizations, proposed by
Toda [46,47]. It is more specific, but would yield a stability condition that
is invariant under certain auto-equivalences; it would also lie outside of our
setP(cid:6).HisapproachwouldrequireastrongerBogomolov-Giesekerinequality
already for slope-stable vector bundles, and likely lead to very interesting
consequencesforgeneratingfunctionsofDonaldson-Thomasinvariants.
Conjecture2.4canbespecializedtocertainslope-stablesheaves,similarto
Corollary1.6;see[11,Conjecture7.2.3].ThisstatementwasprovedbyToda
forcertainCalabi-Yauthreefolds,includingthequintichypersurface,in[48].
AnothercaseofthatconjectureimpliesacertainCastelnuovo-typeinequality
between the genus and degree of curves lying on a given threefold; see [49]
foritsrelationtoboundsobtainedviaclassicalmethods.
Our results are at least partially consistent with the expectations formu-
lated in [36]; in particular, semi-homogeneous bundles are examples of the
Lagrangian-invariantobjectsconsideredbyPolishchuk,aresemistableforour
stabilityconditions,andtheirphasesbehaveaspredicted.
Planofthepaper
Appendix1maybeofindependentinterest.Wereviewsystematicallytherela-
tionbetweensupportproperty,quadraticinequalitiesforsemistableobjectsand
deformationsofstabilityconditions,andtheirbehaviourunderwall-crossing.
Sections 2 and 3 and Appendix 2 review basic properties of tilt-stabilty,
itsdeformationproperties(fixingasmallinaccuracyin[11]),theconjectural
inequalityproposedin[11]andvariantsoftheclassicalBogomolov-Gieseker
inequalitysatisfiesbytilt-stableobjects.
InSect.4weshowthatamoregeneralformofConjecture2.4isequivalent
totheoriginalconjecture,whereasSect.5showsthatbothconjecturesfollows
fromaspeciallimitingcase.
ThislimitingcaseisprovedforabelianthreefoldsinSect.7;inthefollowing
Sect.8weshowthatthisimpliestheexistenceoftheopensubsetP(cid:6)ofstabilty
conditionsdescribedabove.Section9showsthatinthecaseofabelianthree-
folds,P(cid:6) isinfactaconnectedcomponent,andSect.10extendstheseresults
to(crepantresolutions)ofquotientsofabelianthreefolds.
123