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A 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∗ialcontributioninthedegeneratiònformu{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) ∈