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.(2021)“StablePairInvariantsofLocalCalabi–Yau4-folds,”
InternationalMathematicsResearchNotices,,,–46
doi:
StablePairInvariantsofLocalCalabi–Yau4-folds
Downloadedfrom-article/doi/
YalongCao1,MartijnKool2andSergejMonavari2,∗
1KavliInstituteforthePhysicsandMathematicsoftheUniverse
(WPI),TheUniversityofTokyoInstitutesforAdvancedStudy,The
UniversityofTokyo,Kashiwa,Chiba277-8583,Japanand2Mathematical
Institute,UtrechtUniversity,,3508TAUtrecht,The
Netherlands
∗Correspondencetobesentto:e-mail:s.******@
In2008,Klemm–PandharipandedefinedGopakumar–VafatypeinvariantsofaCalabi–
Yau4-foldsXusingGromov–,Cao–Maulik–Todaproposeda
isthetotalspaceofthesumoftwolinebundlesoverasurfaceS,andallstable
pairsareschemetheoreticallysupportedonthezerosection,weexpressstablepair
application,weobtainnewverificationsoftheCao–Maulik–Todaconjecturesforlow-
degreecurveclassesandfindconnectionstoCarlsson–
verificationsinvolvegenuszeroGopakumar–Vafatypeinvariantsrecentlydetermined
inthecontextofthelog-localprinciplebyBousseau–Brini–,using
thevertexformalism,weprovideafewmoreverificationsoftheCao–Maulik–Toda
conjectureswhenthickenedcurvescontributeandalsoforthecaseoflocalP3.
1Introduction
–Yau4-folds
Gromov–Witteninvariantsarerationalnumbers,whicharevirtualcountsofstable
,they
ReceivedJune16,2020;RevisedFebruary4,2021;AcceptedFebruary22,2021
©TheAuthor(s),
pleasee-mail:journals.******@:.
.
–Yau4-folds,Klemm–Pandharipande[26]defined
Gopakumar–VafatypeinvariantsusingGromov–Wittentheoryandconjecturedtheir
,letXbeasmoothprojectiveCalabi–Yau4-–
Witteninvariantsvanishforgenusg2fordimensionalreasonsandoneonlyneedsto
://-article/doi/
ThegenuszeroGromov–WitteninvariantsofXforclassβ∈H2(X,Z)aredefined
:M(X,β)→∈H4(X,Z),
0,1
onedefines
GW(γ)=ev∗(γ).
0,βvir
[M0,1(X,β)]
ThegenuszeroGopakumar–Vafatypeinvariants
n0,β(γ)∈Q()
aredefinedin[26]bytheidentity
∞
GW(γ)qβ=n(γ)d−2qdβ,
0,β0,β
β>0β>0d=1
wherethesumisoverallnon-zeroeffectiveclassesinH2(X,Z).Forthegenusonecase,
thevirtualdimensionofM1,0(X,β)iszeroandonedefines
GW1,β=1∈Q.
[M1,0(X,β)]vir
ThegenusoneGopakumar–Vafatypeinvariants
n1,β∈Q()
aredefinedin[26]bytheidentity
∞σ(d)1
GWqβ=nqdβ+n(c(X))log(1−qβ)
1,β1,βd240,β2
β>0β>0d=1β>0
1
−mlog(1−qβ1+β2),
24β1,β2
β1,β2:.
StablePairInvariantsofLocalCY4-folds3
whereσ(d)=i|diandmβ1,β2∈Zarecalledmeetinginvariants,whichcanbe
inductivelydeterminedbythegenuszeroGromov–[26],bothof
theinvariants()and()
andmirrorsymmetry,theycalculatetheGromov–WitteninvariantsofXinnumerous
://-article/doi/
hasbeenprovedbyIonel–Parkerusingsymplecticgeometry[25,].
–Yau4-folds
StablepairswereintroducedingeneralbyLePotier[39]andusedbyPandharipande–
Thomastodefinevirtualinvariantsofsmoothprojectivethree-folds[35–37].Stablepair
invariantsofthree-foldsarerelatedtoGromov–Witteninvariantsbythecelebrated
GW/PTcorrespondence[31,35],whichhasbeenprovedinmanycasesbyPandhari-
pande–Pixton[33,34].
In[14],Cao–Maulik–Todastudiedstablepairtheoryofasmoothprojective
Calabi–Yau4-
interpretationoftheGopakumar–Vafatypeinvariants()and().In[13,16],
theauthorsalsoproposedasheaftheoreticalinterpretationof()and()using
Donaldson–Thomastypecountinginvariantsofone-dimensionalstablesheavesonX.
LetPn(X,β)bethemodulispaceofstablepairs{s:OX→F}withch(F)=
(0,0,0,β,n).Thereexistsavirtualclass
vir
[Pn(X,β)]∈H2nPn(X,β),Z,()
inthesenseofBorisov–Joyce[3],whichdependsonthechoiceofanorientationofa
certain(real)linebundleoverP(X,β)[7].Forγ∈H4(X,Z),wedefineprimaryinsertions
n
τ:H4(X,Z)→H2(P(X,β),Z),τ(γ)=π(π∗γ∪ch(F)),
nP∗X3
whereπXandπPareprojectionsfromX×Pn(X,β)tothecorrespondingfactorsand
I•={O→F}
istheuniversalstablepaironX×Pn(X,β).Notethatch3(F)isPoincarédualtothe
:.
.
definedby
P(γ):=τ(γ)n.()
n,βvir
[Pn(X,β)]
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Whenn=0,wesimplydenotethisinvariantbyP0,,0:=1andn0,0(γ):=0.
.([14])LetXbeasmoothprojectiveCalabi–Yau4-folds,β∈H2(X,Z),
γ∈H4(X,Z),andn
n
Pn,β(γ)=P0,β0·n0,βi(γ),
β0+β1+···+βn=βi=1
β0,β1,...,βn0
wherethesumisoveralleffectivedecompositionsofβ.
.([14])LetXbeasmoothprojectiveCalabi–Yau4-
choicesoforientationssuchthat
n1,β
Pqβ=Mqβ,
0,β
β0β>0
k−k
whereM(q)=k1(1−q)denotestheMacMahonfunction.
-crossingformulainthecategoryof
D0–D2–D8boundstatesinCalabi–Yau4-folds[15],
[14],theseconjectureswereverifiedinthefollowingcases(modulo
someminorassumptionsinsomeofthecases).Ineachcase,
verifiedforn=1.
•Xisageneralsexticandβ=[],2[],where⊆Xisaline.
•XisaWeierstrassellipticfibrationandβ=r[F],where[F]isthefibreclass
andr>0(=1).
•X=Y×E,whereYisasmoothprojectiveCalabi–Yauthree-fold,Eisan
ellipticcurve,andβisthepush-forwardofanirreducibleclassonY×{pt}.
•X=Y×E,whereYisasmoothprojectiveCalabi–Yauthree-fold,Eis
anellipticcurve,andβ=r[E],where[E]isthefibreclassandr>0
().:.
StablePairInvariantsofLocalCY4-folds5
WhenXiseitherthetotalspaceofasmoothprojectiveFanothree-fold,orthetotal
spaceofO(−1)⊕O(−2)onP2,orO(−1,−1)⊕O(−1,−1)onP1×P1,themoduli
spacesPn(X,β).
Inthissetting,theconjectureswereverifiedinsomecasesforirreduciblecurveclasses
in[14].Downloadedfrom-article/doi/
Oneofthemaingoalsofthispaperistoprovidemoreverificationsforthese
localgeometriesformoregenerallowdegreecurveclasses.
LetSbeasmoothprojectivesurfaceandletL1andL2betwolinebundlesonSsatisfying
L⊗L∼=∼=⊕LoverSisanon-properCalabi–Yau4-
12S12
folds,(X,β)ofstable
pairs(F,s)withχ(F)=nandsuchthatFhasproperschemetheoreticsupportinclass
β∈H2(X,Z).AlthoughPn(X,β)isingeneralnon-proper,itcanbeproperinseveral
interestingcases().Thenwecandefinevirtualclasses()and
correspondingstablepairinvariants().
(S,L,L)=(P2,O(−1),O(−2))and(P1×P1,O(−1,−1),O(−1,−1)),
12
themodulispacePn(X,β)isprojectiveforalln,β().
(S,L,L)=(P1×P1,O(−1,0),O(−1,−2)),P(X,β)isingeneralnon-
12n
,letH:={pt}×P1,takeβ=[H],andn=χ(O)=
11H1
N∼=∼=O⊕O⊕O(−2)hassectionsinthe1stfibredirection,soH⊆P1×P1⊆Xcan
H1/X1
moveoffthezerosectionP1×P1⊆XandP(X,[H])isnon-,
11
forH:=P1×{pt}andβ=[H],wehaveβ·L<0andβ·L<0,soP(X,[H])isprojective
221212
.
WhenSistoric,thelocalsurfaceXistoricandthevertexformalismfor
calculatingstablepairinvariantsofXhasbeendevelopedin[9,10]inanalogywith
[36].LetT⊆(C∗)4denotethe3-dimensionalsubtoruspreservingtheCalabi–Yauvolume
form,thenthefixedlocusP(X,β)Tconsistsoffinitelymanyisolatedreducedpoints
n
[9,],thoughthenumberoffixedpointsistypicallyverylargemaking
calculationsusingthevertexformalismcumbersome.
Althoughweperformafewnewcalculationsusingthevertexformalismas
well,wemainlyfocusonanotherapproach,:.
.
WeconsiderthecasewhenallstablepairsonXareschemetheoreticallysupportedon
thezerosectionι:S
→X,thatis,wehaveanisomorphism
ι∗:Pn(S,β)∼=∼=Pn(X,β).
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Underthisisomorphism,wehave()
virβ·L2+nvir
[Pn(X,β)]=(−1)·e−RHomπP(F,FL1)·[Pn(S,β)],()
S
where[P(S,β)]viristhevirtualclassofthepairsobstructiontheoryonS,e(·)denotes
n
Eulerclass,πPS:S×Pn(S,β)→Pn(S,β)istheprojection,RHomπP=RπPS∗◦RHom,
S
andFistheuniversalone-dimensionalsheafonS×P(S,β).Thesign(−1)β·L2+n=
n
(−1)β·c1(Y)+n,whereY=Tot(L),comesfromapreferredchoiceoforientationon
S1
Pn(X,β)whichwasdiscussedinasimilarsituationin[6].
Inordertouse()forcalculations,weneedthefactthatPn(S,β)isisomorphic
,assumeb1(S)=0anddenoteby|β|
→|β|theuniversalcurve,then[37,
]gives
P(S,β)∼=∼=Hilbm(C/|β|),
n
whereHilbm(C/|β|)denotestherelativeHilbertschemeofmpointsonthefibresof
C→|β|and
m=n+g(β)−1=n+1β(β+K).
2S
Thisisomorphismwasexploitedinordertodeterminethesurfacecontributionto
stablepairinvariantsoflocalsurfacesTotS(KS)in[30].TherelativeHilbertscheme
Hilbm(C/|β|)isanincidencelocusinasmoothambientspace
Hilbm(C/|β|)⊆S[m]×|β|,
whereS[m],Hilbm(C/|β|)
iscutouttautologicallybyasectionofavectorbundleonS[m]×|β|aswerecall
:.
StablePairInvariantsofLocalCY4-folds7
intersectionnumbersonS[m]×|β|,ormoreprecisely,onthe“virtual”ambientspace
S[m]×Pχ(β)−1,where
χ(β):=χ(OS(β)).
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Inwhatfollows,Z⊆S×S[m]denotestheuniversalsubschemeandIisthecorrespond-
,thecorrespondingtautologicalbundleis
definedby
L[m]:=pq∗L,
∗
wherep:Z→S[m]andq:Z→,weconsiderthe“twisted
tangentbundle”[17]
TS[m](L):=R (S,L)⊗O−RHomπ(I,IL),()
whereπ:S×S[m]→S[m],wedenotethetotalChernclass
bycandthetautologicallinebundleonPχ(β)−1byO(1).Weprovethefollowingresult
().
(S)=pg(S)=0andL1,L2∈
Pic(S)suchthatL⊗L∼=∼=∈H(S,Z)andn0arechosensuchthat
12S2
P(X,β)∼=∼=P(S,β)forX=Tot(L⊕L).Denoteby[pt]∈H4(X,Z)thepull-backofthe
nnS12
Poincaré(X,β)beendowedwiththeorientationasin
().Then
nχ(L1(β))χ(L2(β))
β·L2+n[m]h(1+h)(1−h)c(TS[m](L1))
Pn,β([pt])=(−1)cm(OS(β)(1))[m][m]∨,
S[m]×Pχ(β)−1c(L1(β)(1))·c((L2(β)(1)))
whenβ2:=n+g(β)−1andh:=c(O(1)).Moreover,P([pt])=0when
1n,β
β2<0.
ThemainassumptioninthistheoremisPn(X,β)∼=∼=Pn(S,β).For(S,L1,L2)with
Sminimalandtoric,L⊗L∼=∼=KwithL−1,L−1non-trivialandnef,weclassifyall
12S12
casesforwhichn0,Pn(X,β)∼=∼=Pn(S,β),andPn(S,β)isnon-empty(,
).NotethatPn(X,β)∼=∼=Pn(S,β)moreorlessfo