Table Of ContentA FLOP FORMULA FOR DONALDSON-THOMAS INVARIANTS
HUA-ZHONGKE
DepartmentofMathematics,SunYat-SenUniversity,Guangzhou,510275,China
6
[email protected]
1
0 Abstract:LetX andX′benonsingularprojective3-foldsrelatedbyaflopofadisjoint
2 unionof( 2)-curves. WeproveaflopformularelatingtheDonaldson-Thomasinvariants
−
of X tothoseofX ,whichimpliessomesimplerelationsamongBPSstatecounts. Asan
n ′
a application,weshowthatifX satisfiestheGW/DTcorrespondenceforprimaryinsertions
J and descendants of the point class, then so does X . We also propose a conjectural flop
′
3 formulaforgeneralflops.
1 Keywords: Donaldson-Thomas, flop, ( 2)-curve, BPS state counts, GW/DT corrre-
−
spondence
]
G MR(2010)SubjectClassification: 14N35
A
. CONTENTS
h
t 1. Introduction 1
a
m 2. PreliminariesonDonaldson-Thomasinvariants 3
3. Proofofmainresult 5
[
4. BPSstatecounts 12
2 5. GW/DTcorrespondence 14
v
References 16
8
5
7
2
0 1. INTRODUCTION
.
1
TheDonaldson-Thomastheoryofanonsingularprojective3-foldX countsthenumber
0
ofstablesheavesonX [DT,Th]. Inparticular,whenconsideringidealsheavesofcurves,
6
1 thetheorygivesvirtualnumbersofembeddedcurvesinX. Anothercurvecountingtheory
: onXisthemuchstudiedGromov-Wittentheory,whichessentiallycountsstablemapsfrom
v
curveswithmarkedpointstoX.In[MNOP1,MNOP2],Maulik,Nekrasov,Okounkov,and
i
X Pandharipandeproposeda remarkableconjecturethat the Gromov-Wittentheory of X is
r equivalenttotheDonaldson-ThomastheoryofXinasubtleway. Thissuggeststhatmany
a
phenomenoninonetheoryhavecounterpartsintheothertheory.
Theabovementionedcurvecountingtheoriesaredeformationinvariant.Afundamental
probleminGromov-Wittentheoryisto investigatethetransformationofGromov-Witten
invariants under birational surgeries [Ru]. The first breakthroughis the work of Li and
Ruan[LR],whoshowedthat,for3-folds,theprimaryGromov-Wittentheoriesareinvari-
ant under general flops. It is also importantto study the effect of biraional surgeries on
Donaldson-Thomastheory. HuandLi[HL]usedthedegenerationformulatounderstand
the change of Donaldson-Thomas invariants under flops of a disjoint union of ( 1, 1)
− −
curveswhichareallnumerallyequivalent. ForgeneralflopsbetweenCalabi-Yau3-folds,
Toda[T2]usedthecategoricalmethodtoestablishedaflopformula(seealso[Ca]).
Inthispaper,weproveaflopformulainDonaldson-Thomastheoryforflopsofadis-
jointunion( 2)-curves,andderivesomeinterestingrelationsonBPSstatecounts. Asan
−
application,we givepositiveevidenceforthe conjecturalGW/DT correspondence. Here
1
2 HUA-ZHONGKE
anembeddedcurveina3-foldisa( 2)-curve[Re]ifitisanonsingularrationalcurvewith
−
normalbundleoftype ( 1, 1)or(0, 2). Our flopformulageneralizesthe resultofHu
− − −
andLi[HL],sincea( 1, 1)-curveisa( 2)-curve,andwedonotassumethatthecurves
− − −
arenumericallyequivalent.
Throughoutthispaper,letXandX benonsingularprojective3-foldsoverC,whichare
′
relatedbyaflop f : X d X ofsomecontraction[Ko]. Then f isabirationalmap,andit
′
is biregularoutsideof a subvarietyof codimensiontwo in X, calledthe centerof f. The
centerof f is a disjointunionof treesofrationalcurves, andithasa neighborhoodwith
trivialcanonialbundle.Wehaveanaturalisomorphismofgroups
(cid:27)
F :H (X,Z) H (X ,Z),
2 2 ′
−→
defined as follows. For any β H (X,Z), we can choose a real 2-dimensional pseudo-
2
submanifoldΣrepresentingβin∈X,whichliesinthecomplementofthecenterof f. Now
Fβisrepresentedby f(Σ)inX ,whichliesinthecomplementofthecenterof f 1. Simi-
′ −
larly,byconsideringPoincare´dualsofclassesofdegree>3,wealsohaveanisomorphism
H>3(X,Q) H>3(X ,Q),
′
→
whichcanbeextendedtoanisomorphismofcohomologygroups
(cid:27)
H (X,Q) H (X ,Q),
∗ ∗ ′
−→
byrequiringthisisomorphismtopreservethePoincare´pairing.Theisomorphismwillalso
bedenotedbyF byabuseofnotation. LetCen(f)bethesubgroupofH (X,Z)generated
2
bythecyclesofirreduciblecurvesinthecenterof f. Themainresultofthispaperisthe
following.
Proposition1.1. Let f beaflopofadisjointunionof( 2)-curves.Supposethatγ , ,γ
1 m
− ··· ∈
H∗(X,Q)(m>0)havesupportsawayfromthecenterof f,andd1, ,dm Z>0. Thenwe
··· ∈
have
m m
(1) β∈HP2(X,Z)vβZD′T(X;q|Qi=1τ˜di(γi))β = β∈HP2(X,Z)vβZD′T(X′;q|iQ=1τ˜di(Fγi))Fβ,
vβZD′T(X;q|)β vβZD′T(X′;q|)Fβ
β Cen(f) β Cen(f)
∈P ∈P
(2) ZD′T(X;q|)β = ZD′T(X′;q|)−Fβ, ∀β∈Cen(f).
Remark1.2. Weremarkthatwecanchoosethesupportofγ awayfromthecenterof f if
i
degγ >2.
i
WesketchtheproofofProposition1.1,thedetailofwhichwillbegiveninSection3.By
abeautifulresultofReid[Re],wecandecomposetheflop f of( 2)-curvesintoasequence
−
ofblow-upsof( 2)-curvesfollowedbyasequenceofblow-downs.Sinceblow-upscanbe
−
describedintermsofsemi-stabledegenerations,itfollowsthatwecanusethedegeneration
formula[LW]andtheabsolute/relativecorrespondence[HLR,MP]torelateinvariantsof
X tothoseoftheblow-upof X (see (9)). Therefore,inprinciple,theDonaldson-Thomas
invariantsof X canberelatedtothoseof X . Duetothedenominatorsin (1), weneedto
′
understandthetransformationoftheinvariantsattachedtoclassesinCen(f)underblow-
ups. To this end, we give a detailed analysisof the change of effectivenessof classes in
Cen(f)underblow-up(seeLemma3.1and3.2).
Proposition1.1relatestheDonaldson-Thomasinvariantsof X tothoseof X ina non-
′
trivialway.In[HHKQ]and[Ke],weobtainedsomeblow-upformulaeforGromov-Witten
andstablepairtheorieswhichcontainsomeextrafactors,andwediscoveredthatthesefor-
mulaeimplysomeinterestingrelationamongBPSstatecounts. Inthispaper,weconsider
thechangeofBPSstatecountsofDonaldson-Thomastheoryunderflops. Proposition1.1
impliesthefollowingsimpleflopformulaeforBPSstatecounts.
AFLOPFORMULAFORDONALDSON-THOMASINVARIANTS 3
Corollary1.3. Let f beaflopofadisjointunionof( 2)-curves.Supposethatγ , ,γ
1 m
− ··· ∈
H (X,Q)(m>0),andg Z. Thenwehave
∗
∈
(3) ngX,β(γ1,··· ,γm)=ngX,′Fβ(Fγ1,··· ,Fγm), ∀β∈H2(X,Z)\Cen(f);
(4) nX =nX′ , β Cen(f) 0 .
g,β g, Fβ ∀ ∈ \{ }
−
The Donaldson-Thomastheory of X counts embedded curves on X only in a virtual
sense. AfundamentalproblemintheDonaldson-Thomastheoryistounderstandthehid-
den enumerative meanings of the invariants. It is conjectured that BPS state counts are
enumerative. It is interestingto understandCorollary1.3 fromthe pointof view of enu-
merativegeometry.
Asanotherapplication,weinvestigatetheconjecturalGW/DTcorrespondence. Inthe
primary case, the correspondence is established for several classes of 3-folds, including
toric3-folds[MOOP],andCalabi-Yau3-foldswhicharecompleteintersectionsinproducts
ofprojectivespaces[PP,T1]. However,inthedescendentcase,notmuchisknown. The
followingresultgivesfurtherpositiveevidencetotheMNOPconjecture.
Corollary1.4. Let f beaflopofadisjointunionof( 2)-curves. AssumethatX satisfies
−
the GW/DT correspondence for primary insertions and descendants of the point class.
ThensodoesX .
′
WeobservethatToda’sflopformulae(Theorem1.2in[T2])areanalogoustoProposi-
tion1.1,andwecancheckthatCorollary1.3and1.4alsoholdforgeneralflopsbetween
Calabi-Yau 3-folds. Based on the established flop formulae of [HL], [T2] and ours, we
proposethefollowingconjecture.
Conjecture1.5. Theformulae(1)and(2)holdforgeneralflops.
Weexpectthatthedegenerationformulawillplayaroleintheproofoftheconjecture.
Note thatan embeddednonsingularrationalcurvein a 3-foldis locally floppableonlyif
ithasnormalbundleoftype( 1, 1),(0, 2)or(1, 3)[La]. However,unlikethecaseof
− − − −
( 2)-curves,itisdifficulttodescribeageneralflopof(1, 3)-curvesintermsofblow-ups
− −
andblow-downs(see[Pi]forsomeexplicitexamples).
Mostoftheresultsobtainedinthispaperalsoholdinthestablepairtheory[PT],since
the behaviorof stable pairinvariantsunderdegenerationis similar to thatof Donaldson-
Thomas theory. We also have correspondingcorollarieson BPS state counts and GW/P
correspondence,andconjecturalflopformulaeforgeneralflopsinthestablepairtheory.
Anoutlineofthispaperisasfollows. InSection2,wereviewsomebasicmaterialson
Donaldson-Thomasinvariants. InSection3,werecallReid’sresulttodecomposetheflop
underconsiderationintoasequenceofblow-upsfollowedbyasequenceofblow-downs,
andusethedegenerationformulatoproveProposition1.1.InSection4,wegiveaworking
definitionoftheBPSstatecountsforDonaldson-ThomastheoryandproveCorollary1.3.
InSection5,wereviewtheconjecturalGW/DTcorespondenceandproveCorollary1.4.
2. PRELIMINARIES ON DONALDSON-THOMAS INVARIANTS
Inthissection,webrieflyreviewsomebasicmaterialsonDonaldson-Thomasinvariants
andfixnotations.Wereferreadersto[DT,LW,MNOP1,MNOP2,Th]fordetails.
Donaldson-Thomas theory is defined via integration over the moduli space of ideal
sheavesofX. Hereanidealsheafisatorsion-freesheafofrank1withtrivialdeterminant.
EachidealsheafIdeterminesasubschemeY X viatheexactsequence
⊂
0 I O O 0.
X Y
→ → → →
Inthisnote,wewillconsideronlythecasedimY 6 1. Theonedimensionalcomponents
ofY (weightedbytheirintrinsicmultiplicities)determineanelement,
[Y] H (X,Z).
2
∈
4 HUA-ZHONGKE
Forn Zandβ H (X,Z),letI (X,β)bethemodulispaceofidealsheavesIsatisfying
2 n
∈ ∈
χ(O )=n, [Y]=β,
Y
where χ is the holomorphic Euler characteristic. From the deformation theory, I (X,β)
n
carriesavirtualfundamentalclassofdegree c (X).
β 1
Ford Z>0andγ H∗(X,Q),thedescendRantinsertionτ˜d(γ)isdefinedasfollows.Let
∈ ∈
π : X I (X,β) X,
X n
× →
π : X I (X,β) I (X,β)
I n n
× →
betautologicalprojections.LetI betheuniversalsheafoverX I (X,β). Theoperation
n
×
(−1)d+1πI∗(cid:18)π∗X(γ)·ch2+d(I)∩π∗I(·)(cid:19):H∗(In(X,β),Q)→H∗(In(X,β),Q)
is the action of τ˜ (γ). The Donaldson-Thomasinvariantswith descendantinsertions are
d
definedasthevirtualintegration
m m
τ˜ (γ) = τ˜ (γ),
hYi=1 di i in,β Z[In(X,β)]virYi=1 di i
whered1 ,dm Z>0,andγ1, ,γm H∗(X,Q). Heretheintegralisthepush-forward
··· ∈ ··· ∈
toapointoftheclass
τ˜ (γ) τ˜ (γ )([I (X,β)]vir).
di i ◦··· dm m n
ThepartitionfunctionoftheDonaldson-Thomasinvariantsisdefinedby
m m
Z X;q τ˜ (γ) = τ˜ (γ) qn,
DT(cid:18) |Yi=1 di i (cid:19)β Xn ZhYi=1 di i in,β
∈
andthereducedpartitionfunctionisobtainedbyformallyremovingthedegreezerocon-
tributions,
m
Z X;q τ˜ (γ)
ZD′T(cid:18)X;q|Yi=m1 τ˜di(γi)(cid:19)β = DT(cid:18)ZDT|XQi=;1qdi i (cid:19)β.
(cid:18) |(cid:19)0
LetS Xbeanonsingulardivisor.Forn Zandnonzeroβ H (X,Z)with [S]>0,
⊂ ∈ ∈ 2 β
letI (X/S,β)bethemodulispaceofrelativeidealsheaves,whichcarriesavirtuRalfunda-
n
mentalclassofdegree c (X). Wehavethefollowingnaturalmorphism
β 1
R
ǫ :I (X/S,β) Hilb(S, [S])
n
→ Z
β
Thepull-backofcohomologyclassesofHilb(S, [S])givesrelativeinsertions.
β
Let us briefly recall Nakajima basis for the cRohomologyof Hilbertschemes of points
ofS. Let δ bea basisof H (S,Q)with dualbasis δi . Foranycohomologyweighted
i ∗
{ } { }
partitionη with respectto the basis δ , Nakajima constructeda cohomologyclassC
i η
{ } ∈
H∗(Hilb(S, η),Q). TheNakajimabasisof H∗(Hilb(S,d),Q)istheset Cη η=d. We refer
| | { }| |
readersto[Na]formoredetails.
ThepartitionfunctionoftherelativeDonaldson-Thomasinvariantsisdefinedby
m m
Z X/S;q τ˜ (γ)η = qn τ˜ (γ) ǫ C ,
DT(cid:18) |Yi=1 di i | (cid:19)β Xn Z Z[In(X/S,β)]virYi=1 di i · ∗ η
∈
AFLOPFORMULAFORDONALDSON-THOMASINVARIANTS 5
andthereducedpartitionfunctionisobtainedbyformallyremovingthedegreezerocon-
tributions
m
Z X/S;q τ˜ (γ)η
ZD′T(cid:18)X/S;q|Yi=m1 τ˜di(γi)|η(cid:19)β = DT(cid:18)ZDT X|/iQS=1;qdi i | (cid:19)β
(cid:18) ||(cid:19)0
Let ∆ C be the unit disc, and let π : χ ∆ be a nonsingular4-fold over D, such
thatχ = π⊂1(t) (cid:27) X fort , 0,andχ isaunio→noftwoirreduciblenonsingularprojective
t − 0
3-foldsX andX intersectingtransversallyalonganonsingularprojectivesurfaceS. (We
1 2
can also considerthe generalcase wherethe centralfiberhas severalirreduciblecompo-
nents,butwerestrictourselvestothissimplecaseforsimplicityofpresentation.)Consider
thenaturalinclusionmaps
i :X =χ χ, i :χ χ,
t t 0 0
−→ −→
andthegluingmap
g=(j , j ):X X χ .
1 2 1 2 0
−→
a
Wehave
H2(X,Z) it∗ H2(χ,Z) i0∗ H2(χ0,Z) g∗ H2(X1,Z) H2(X2,Z),
−→ ←− ←− ⊕
wherei isanisomorphismsincethereexistsadeformationretractfromχtoχ (see[Cl]).
0 0
Also, sin∗ce the family χ A1 comes from a trivial family, it follows that each γ
→ ∈
H (X,Q)hasgloballiftingssuchthattherestrictionγ(t)onχ isdefinedforallt.
∗ t
ThedegenerationformulafortheDonaldson-Thomastheoryexpressestheabsolutein-
variantsofX viatherelativeinvariantsof(X ,S)and(X ,S):
1 2
m
Z X;q τ˜ (γ)
D′T(cid:18) |Yi=1 di i (cid:19)β
( 1)η ℓ(η)z(η)
= XZD′T(cid:18)X1/S;q|Yi∈P1τ˜di(j∗1γi(0))|η(cid:19)β1 · − |q|−|η| ·ZD′T(cid:18)X2/S;q|Yi∈P2τ˜di(j∗2γi(0))|η∨(cid:19)β2,
ℓ(η)
wherez(η) = Aut(η) η, η is definedbytakingthe Poincare´ dualsof thecohomol-
i ∨
| |· i=1
ogyweightsofη,andthQesumisovercohomologyweightedpartitionsη,degreesplittings
i β = i (j β + j β ), and marking partitions P P = 1, ,m . In particular, if
t 0 1 1 2 2 1 2
(η∗,β1,β2)∗ha∗snontriv∗ialcontributioninthedegenerati`onformu{la,·th··enw}ehavethefollow-
ingdimensionconstraint:
vdim P (X /S,β )+vdim P (X /S,β )=vdim P (X,β)+2η.
C n 1 1 C n 2 2 C n
| |
3. PROOF OF MAIN RESULT
Inthissection,wegiveadetailedproofoProposition1.1.WefirstrecallReid’sresultto
decomposeaflopofadisjointunionof( 2)-curvesintoasequenceofblow-upsfollowed
−
by a sequence of blow-downs, and then use the degeneration formula to prove our flop
formula. We refer readers to [Re] for explicit local description of the flop of a single
( 2)-curve,andto[Ko,KM]forgeneralmaterialsonbirationalgeometryof3-folds.
−
LetC , ,C betheirreduciblecomponentsofthecenterof f. Wecancontractthese
1 l
curvestoo·b·t·ainacontractionψ:X X¯,andthenthesecurvesgenerateanextremalface
→
inNE(X). ThewidthofC inXisdefinedbyReidasfollows[Re]:
i
w := width(C X)
i i
⊂
:= sup k thereexistsaschemeS (cid:27)C Spec(C[ǫ]/ǫk)suchthatC S X .
i i
{ | × ⊂ ⊂ }
6 HUA-ZHONGKE
SinceC is isolated, it follows that 1 6 w < . Note that ψ(C) X¯ is a hypersurface
i i i
∞ ∈
singularitygivenby
x2+y2+z2+t2wi =0.
Inparticular,C isa( 1, 1)-curveifandonlyifw =1.
i i
− −
Withoutlossofgenerality,assumethat
w > >w >1.
1 l
···
Letw=w ,andford =1, ,w,set
1
···
k :=sup iw >d .
d i
{| }
Then
16k 6 6k =l.
w 1
···
Write X = X and C = C . Then proceeding inductively, we obtain a sequence of
0 i 0,i
blow-ups:
Xw φw−1 Xw 1 φw−2 φ1 X1 φ0 X0.
−−−→ − −−−→···−→ −→
Hereford =0,1, ,w 2,φ istheblow-upofX alongthe( 2)-curvesC , ,C .
··· − d d − d,1 ··· d,kd+1
Let
Ed+1,i :=φ−d1(Cd,i)(cid:27)( FF02,, ii==k1d,+·1··+,1kd,+2, ,kd+1.
···
Fori = 1, ,kd+2,Cd+1,i Ed+1,i istheuniquenonsingularrationalcurvewithnegative
··· ⊂
selfintersectionnumber,whichisalsoa( 2)-curveinXd+1with
−
width(Cd+1,i Xd+1)=wi d 1, d =1, ,w 2.
⊂ − − ··· −
Moreover,φ istheblow-upofX alongthe( 1, 1)-curvesC , ,C ,and
w−1 w−1 − − w−1,1 ··· w−1,kw
E :=φ 1 (C )(cid:27)F , i=1, ,k .
w,i −w−1 w−1,i 0 ··· w
Ford = 1, ,w 1andi = 1, ,kd+1,thestricttransformofEd,iunderφd,denoted
byE˜d,i,isisom··o·rphic−toEd,i. Moreo·v··er,E˜d,i Ed+1,iisanonsingularrationalcurve,which
hasnegativeselfintersectionnumberonE˜d,i∩,andselfintersectionnumber2onEd+1,i. In
particular,E˜ E isa(1,1)-curveonE (cid:27)F . NotethatE˜ isnotaffectedbyblow-
w 1,i w,i w,i 0 d,i
upsφd+1, ,−φw∩1,andcanbeviewedasanembeddedsurfaceinXw. Ford =1, ,w 1
andi= kd·+·1·+1,− ,kd,Ed,i isnotaffectedbyblow-upsφd, ,φw 1,andcanb·e··view−ed
··· ··· −
asanembeddedsurfaceinX .
w
Write E = E . Sinceeach E (cid:27) F hasarulingnotcontractedbyφ ,itfollows
w,i w′,i w′,i 0 w 1
−
that we can blow down X along these rulings for all i simultaneously to obtain φ :
w ′w 1
X X . Proceedinginductively,wealsohaveasequenceofblow-downs: −
w → w′ 1
−
Xw −φ−′w−→−1 Xw′−1 −φ−′w−→−2 ···−φ→′1 X1′ −φ→′0 X0′.
Ford =0,1, ,w 2,let
··· −
C :=φ (E ), i=1, ,k
w′−1−d,i ′w−1−d w′−d,i ··· w−d
and
φ φ (E˜ )(cid:27)F , i=1, ,k ,
Ew′−1−d,i =( φ′w′w−−11−−dd◦◦······◦◦φ′w′w−−11(Eww−−11−−dd,,ii)(cid:27)F20 i=kw−··d·+1w,−·d·· ,kw−1−d.
ThenC isa( 2)-curveinX with
w′ 1 d,i − w′ 1 d
− − − −
width(C X )=w w+1+d,
w′−1−d,i ⊂ w′−1−d i−
andC E is the uniquenonsingularrationalcurvewith negativeself inter-
sectionw′−n1−udm,ib⊂er. w′S−in1−cde,ifor i = 1, ,k , each E (cid:27) P (O O( 2)) has a
fiber ruling, and for i = k + 1,··· ,kw−d , eachw′E−1−d,i haCsw′−a1−rd,uiling⊕not−contracted
w d w 1 d w 1 d,i
− ··· − − − −
AFLOPFORMULAFORDONALDSON-THOMASINVARIANTS 7
byφ ,itfollowsthatwecanblowdownX alongtheserulingssimultaneouslyto
w 1 d w′ 1 d
obtain−φ− : X X . − −
Now′wfo−2r−dd=0w′,−11,−d →,w w′−12,−dthebirationalmap
··· −
f :=φ φ φ 1 φ 1 :X dX
d ′d◦···◦ ′w 1◦ −w 1◦···◦ −d d d′
− −
is a flop of ( 2)-curvesC , ,C , where C is flopped to C . In particular, we
haveX = X −and f = f . d,1 ··· d,kd+1 d,i d′,i
′ 0′ 0
DegenerateXalongC , ,C simultaneously,andwehave
1 l
···
m
Z X;q τ˜ (γ)
D′T(cid:18) |Yi=1 di i (cid:19)β
= XZD′T(cid:18)X1/E1;q|Yi=m1 τ˜di(φ∗0γi)|η∨1,··· ,η∨l (cid:19)β˜Yi=l1 (−1)|ηiq|−|ηℓ(i|ηi)z(ηi)ZD′T(Pi/Di;q||ηi)βi,
l
wherewehaveassumedthatthesupportofγ isawayfrom C,and
i i
i=1
S
l
E := E ,
1 1,i
[i=1
P := P (N O ), (N isthenormalbundleofC inX)
i Ci Ci ⊕ Ci Ci i
D := P (N 0 ).
i Ci Ci ⊕{ }
Bydimensionconstraint,wefindthatη = =η = . So
1 l
··· ∅
β˜ E =β D =0.
1 i i
· ·
Forβ˜,notethatφ inducesanaturalinjectionvia’pull-back’of2-cycles
0
φ! = PD φ PD :H (X,Z) H (X ,Z),
1 X1 ◦ ∗0◦ X 2 → 2 1
wheretheimageofφ! isthesubsetofH (X ,Z)consistingof2-cycleshavingintersection
0 2 1
numberzerowithE ,andsowehaveβ˜ Imφ!. Forβ,notethat
1 ∈ 0 i
H (P,Z)=Z[C] Zf,
2 i i i
⊕
wherewehaveusedtheidentificationC (cid:27) P ( 0 O ), and f istheclassofa linein
thefiberofPi. Soβi Di =0impliesthatiβi CZi>{0[C}⊕i],siCniceβi iseiffective.Therefore
· ∈
m
Z X;q τ˜ (γ)
D′T(cid:18) |Yi=1 di i (cid:19)β
m l
= Z X /E ;q τ˜ (φ γ) Z (P/D;q ) .
β′β+′n∈1H[C2(1XX]+,Z·)··,nnli[∈CZl>]=0β D′T(cid:18) 1 1 |Yi=1 di ∗0 i |(cid:19)φ!0β′Yi=1 D′T i i || ni[Ci]
In particular, since the irreduciblecurvesin the center of f generatean extremalface in
NE(X),itfollowsthatforβ Cen(f),wehave
∈
l
Z X;q = Z X /E ;q Z (P/D;q ) .
D′T(cid:18) |(cid:19)β lX D′T(cid:18) 1 1 ||(cid:19)φ!0β′Yi=1 D′T i i || ni[Ci]
β′+ ni[Ci]=β
i=1
β′P∈Cen(f)
8 HUA-ZHONGKE
Thereforewehaveobtainedthefollowing:
m m
vβZ X;q τ˜ (γ) = vβZ X /E ;q τ˜ (φ γ)
β∈HX2(X,Z) D′T(cid:18) |Yi=1 di i (cid:19)β β∈HX2(X,Z) D′T(cid:18) 1 1 |Yi=1 di ∗0 i |(cid:19)φ!0β
l
· vd[Ci]ZD′T(Pi/Di;q||)d[Ci],
Yi=1 Xd>0
l
(5β)XCen(f)vβZD′T(X;q|)β = β XCen(f)vβZD′T(cid:18)X1/E1;q||(cid:19)φ!0β·Yi=1 Xd>0vd[Ci]ZD′T(Pi/Di;q||)d[Ci],
∈ ∈
whichimpliesthat
m m
vβZ X;q τ˜ (γ) vβZ X /E ;q τ˜ (φ γ)
(6) β∈HP2(X,Z) D′T(cid:18) |Qi=1 di i (cid:19)β = β∈HP2(X,Z) D′T(cid:18) 1 1 |Qi=1 di ∗0 i |(cid:19)φ!0β.
vβZ (X;q)
β∈CPen(f) D′T | β β Cen(f)vβZD′T(cid:18)X1/E1;q||(cid:19)φ!β
∈P 0
NowdegenerateX alongE , ,E simultaneously,andweobtain
1 1,1 1,l
···
m
Z X ;q τ˜ (φ γ)
D′T(cid:18) 1 |Yi=1 di ∗0 i (cid:19)φ!1β
= XZD′T(cid:18)X1/E1;q|Yi=m1 τ˜di(φ∗0γi)|η∨1,··· ,η∨l (cid:19)β˜Yi=l1 (−1)|ηiq|−|ηℓ(i|ηi)z(ηi)ZD′T(P1,i/D1,i;q||ηi)βi,
where
P := P (N O ), (N isthenormalbundleofE inX )
1,i E1,i E1,i ⊕ E1,i E1,i 1,i 1
D := P (N 0 ).
1,i E1,i E1,i ⊕{ }
Bydimensionconstraint,wefindthatη = =η = . Sowehave
1 l
··· ∅
m
Z X ;q τ˜ (φ γ)
D′T(cid:18) 1 |Yi=1 di ∗0 i (cid:19)φ!1β
m l
= Z X /E ;q τ˜ (φ γ) Z (P /D ;q ) ,
φ!0ββ′+′∈(πH1β2,1i(·)XE∗,β1Z,1iX)+=,β·β·i·i∈+·DH(πi2=1(,P0l)1∗,iβ,Zl=)φ!0β D′T(cid:18) 1 1 |Yi=1 di ∗0 i |(cid:19)φ!0β′Yi=1 D′T 1,i 1,i || βi
wherewehaveusedtheidentificationE (cid:27)P ( 0 O ),andπ isthecomposition
1,i E1,i { }⊕ E1,i 1,i
P E ֒ X .
1,i 1,i 1
→ →
Inparticular,since
β +(φ ) (π ) β + +(φ ) (π ) β =β,
′ 0 1,1 1 0 1,l l
∗ ∗ ··· ∗ ∗
itfollowsthatforβ Cen(f),wehave
∈
Z X ;q
D′T(cid:18) 1 |(cid:19)φ!β
0
l
= Z X /E ;q Z (P /D ;q ) ,
β′+(φ0)∗(πβ1,i1·)Eβ∗1β′,∈i1X=C+βe··in··+D((f1φ),i0=)0∗(π1,l)∗βl=β D′T(cid:18) 1 1 ||(cid:19)φ!0β′Yi=1 D′T 1,i 1,i || βi
AFLOPFORMULAFORDONALDSON-THOMASINVARIANTS 9
Sowehaveobtained
m m
vβZ X ;q τ˜ (φ γ) = vβZ X /E ;q τ˜ (φ γ)
β∈HX2(X,Z) D′T(cid:18) 1 |Yi=1 di ∗0 i (cid:19)φ!0β β∈HX2(X,Z) D′T(cid:18) 1 1 |Yi=1 di ∗0 i |(cid:19)φ!0β
l
· vd[Ci] ZD′T(P1,i/D1,i;q||)βi,
Yi=1 Xd>0 (φβ0iβ)·Ei(∈1π,Hi1X=,2iβ)(Piβ·1Di,i=,1Z,di=)[C0i]
∗ ∗
(7) vβZ (X ;q) = vβZ X /E ;q
β XCen(f) D′T 1 | φ!0β β XCen(f) D′T(cid:18) 1 1 ||(cid:19)φ!0β
∈ ∈
l
· vd[Ci] ZD′T(P1,i/D1,i;q||)β,
Yi=1 Xd>0 (φβ0·)βE∈(1πH,i1X=2,i(β)P·D1β,i=1,,Zdi=)[C0i]
∗ ∗
whichimpliesthat
m
vβZ X ;q τ˜ (φ γ)
β∈HP2(X,Z) D′T(cid:18) 1 |iQ=1 di ∗0 i (cid:19)φ!0β
vβZ (X ;q)
β Cen(f) D′T 1 | φ!0β
∈P
m
vβZ X /E ;q τ˜ (φ γ)
(8) = β∈HP2(X,Z) D′T(cid:18) 1 1 |iQ=1 di ∗0 i |(cid:19)φ!0β.
vβZ X /E ;q
β Cen(f) D′T(cid:18) 1 1 ||(cid:19)φ!β
∈P 0
Thenfrom(6)and(8),wehave
m m
vβZ X;q τ˜ (γ) vβZ X ;q τ˜ (φ γ)
(9) β∈HP2(X,Z) D′T(cid:18) |iQ=1 di i (cid:19)β = β∈HP2(X,Z) D′T(cid:18) 1 |Qi=1 di ∗0 i (cid:19)φ!0β.
vβZ (X;q) vβZ (X ;q)
β Cen(f) D′T | β β Cen(f) D′T 1 | φ!0β
∈P ∈P
(cid:27)
UsingtheidentificationF :H (X,Z) H (X ,Z),wealsohave
2 2 ′
−→
m
vβZ X ;q τ˜ (Fγ)
β∈HP2(X,Z) D′T(cid:18) ′ |iQ=1 di i (cid:19)Fβ
vβZD′T(X′;q|)Fβ
β Cen(f)
∈P
m
vβZ X ;q τ˜ (((φ ) Fγ)
(10) = β∈HP2(X,Z) D′T(cid:18) 1′ |Qi=1 di ′0 ∗ i (cid:19)((φ′0)!Fβ.
β Cen(f)vβZD′T(X1′;q|)(φ′0)!Fβ
∈P
Nowweuseinductiononw=1,2,3, toprove(1)inProposition1.1. Forw=1,we
···
havethefollowingobservation.
Lemma3.1. Foranynonzeroβ Cen(f),φ!βisnoteffective.
∈ 0
Proof. Argue by contradiction, and then β = (φ ) φ!β is also effective. We can write
0 0
∗
l
β=i=1ai[Ci]withai ∈Z>0. NotethatF[Ci]=−[Ci′],andthen
P
l
(φ ) φ!β=Fβ= a[C ].
′0 ∗ 0 −Xi=1 i i′
10 HUA-ZHONGKE
l
Since a[C ]iseffective,itfollowsthat(φ ) φ!βisnoteffective,whichimpliesthatφ!β
i i′ ′0 0 0
i=1 ∗
isnotePffective. (cid:3)
Therefore,
β XCen(f)vβZD′T(X1;q|)φ!0β =1and β CXen(f 1)vβ′ZD′T(X1;q|)φ!0β′ =1.
∈ ′∈ −
Notethatin(9)and(10),wehave
φ γ =(φ ) Fγ andφ!β=(φ )!Fβ.
∗0 i ′0 ∗ i 0 ′0
Sointhecasew=1,(1)followsfrom(9)and(10).
Assumethatthecaseforw=W >1isproved.Thenforw=W+1,wehave
m m
β1∈HP2(X1,Z)vβ1ZD′T(X1;q|iQ=1τ˜di(φ∗0γi))β1 = β1∈HP2(X1,Z)vβ1ZD′T(X1′;q|iQ=1τ˜di(F1φ∗0γi))F1β1,
vβ1ZD′T(X1;q|)β1 vβ1ZD′T(X1′;q|)F1β1
β1∈CPen(f1) β1∈CPen(f1)
where F is the correspondence on (co)homology groups induced by f . We have the
1 1
followingkeyobservation.
Lemma3.2. LetS =Span [C ], ,[C ] . Foranyβ Cen(f) S,φ!βisnoteffective.
Z{ 1 ··· k2 } ∈ \ 0
Proof. Withoutlossofgenerality,assumethat
S ∩{[Ck2+1],··· ,[Cl]}={[Cl′],··· ,[Cl]}.
n
Arguebycontradiction,andwecanwriteφ!0β= j=1mj[Vj],wheremj ∈Z>0,andV1,··· ,Vn
P
aremutuallydistinctirreduciblecurvesinX . Since
1
β=(φ ) φ!β Cen(f) S,
0 ∗ 0 ∈ \
and [C ], ,[C] generate an extremal face in NE(X), it follows that, for each j, V is
1 l j
···
mapped onto a point or some C. In the former case, V is a fiber of one irreducible
i j
componentof E andthenV E < 0. Inthelattercase,V iscontainedin E andthen
1 j 1 j 1,i
V E 6 0. Moreover,we c·an find some V which is contained in some E (cid:27) F for
j 1,i j 1,i 0
·
l 6i6l,andthenV E <0. Insum,wehaveφ!β E <0,whichisabsurd. (cid:3)
′ j· 1,i 0 · 1
SinceCen(f )= φ!β:β S ,itfollowsthat
1 { 0 ∈ }
β1∈XCen(f1)vβ1ZD′T(X1;q|)β1 =β∈XCen(f)vφ!0βZD′T(X1;q|)φ!0β.
Nowwehave
vF1β1ZD′T(X1′;q|)F1β1
β1∈XCen(f1)
= vβ′1ZD′T(X1′;q|)β′1
β′1∈CXen(f1−1)
= v(φ′0)!β′ZD′T(X1′;q|)(φ′0)!β′
β CXen(f 1)
′∈ −
= v(φ′0)!FβZD′T(X1′;q|)(φ′0)!Fβ
β XCen(f)
∈
= vF1φ!0βZD′T(X1′;q|)(φ′0)!Fβ,
β XCen(f)
∈
