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Niveau: Supérieur, Doctorat, Bac+8
LECTURES ON GROMOV–WITTEN THEORY AND CREPANT TRANSFORMATION CONJECTURE Y.P. LEE ABSTRACT. These are the pre-notes for the Grenoble Summer School lec- tures June-July 2011. They aim to provide the students some background in preparation for the conference. Nominally, only the basic knowledge on moduli of curves covered in the first week is assumed, although I tac- itly assume the students either have heard one thing or two about the subjects, or are formidable learners. It is well-nigh impossible for a mere human to learn GWT in a week. In fact, I only know of two persons who have done it. Please help find the errors, typographical or mathematical. Without a moment's hesitation, I bet there are plenty. CONTENTS 1. Defining GWI 1 2. Some GWT generating functions and their structures 7 3. Givental's axiomatic GWT at genus zero 13 4. Relative GWI and degeneration formula 19 5. Orbifolds and Orbifold GWT 22 6. Crepant transformation conjecture 25 Appendix A. Quantization and higher genus axiomatic theory 31 Appendix B. Degeneration analysis for simple flops 34 References 44 1. DEFINING GWI The ground field is C. All cohomological degrees are Chow or “com- plex” degrees, and dimensions are complex dimensions. 1.1. Moduli of stable maps. The main reference is [4].

  • ??y ??y

  • class when

  • virtual class

  • axiom says

  • called stable

  • say say nothing

  • can easily

  • splitting axiom



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ABSTRACT. Thesearethepre-notesfortheGrenobleSummerSchoollec-
turesJune-July2011. Theyaimtoprovidethestudentssomebackground
in preparation for the conference. Nominally, only the basic knowledge
itly assume the students either have heard one thing or two about the
subjects, orareformidablelearners. Itiswell-nighimpossibleforamerehumanto
learnGWTinaweek. Infact,Ionlyknowoftwopersonswhohavedoneit.
Pleasehelpfindthe errors,typographicalormathematical. Withouta
1. DefiningGWI 1
2. SomeGWTgeneratingfunctionsandtheirstructures 7
3. Givental’s axiomaticGWTatgenuszero 13
4. RelativeGWIanddegenerationformula 19
5. Orbifolds andOrbifold GWT 22
6. Crepanttransformationconjecture 25
AppendixA. Quantization andhighergenusaxiomatic theory 31
AppendixB. Degenerationanalysisforsimpleflops 34
References 44
The ground field isC. All cohomological degrees are Chow or “com-
11.1. Moduliofstablemaps. Themain referenceis[4].
An n-pointed, genus g, prestable curve (C,x ,x ,...,x ) is a projective,1 2 n
LetSbeanalgebraicscheme. (Afamilyof)n-pointed,genusg,prestable
curve over S is a flat projective morphism π : C → S with n sections
1The referencesof thesepre-notesaremostlysurveyarticles. Forthose interestedinthe
originalpapers,pleaseasktheexperts. Wehave manyinthe School!
12 Y.P.LEE
x ,x ,...,x ,suchthateverygeometricfiberisann-pointed,genusg,prestable1 2 n
Let X be an algebraic scheme. A prestable map over S from n-pointed,
genus gcurvesto X isthefollowingdiagram
C −−−→ X

suchthatπ isdescribedaboveand f isamorphism. Twomaps f :C → Xi i
over S are isomorphic if there is an isomorphism g :C →C over S such1 2
∼that f ◦g = f .1 2
A prestablemap overC iscalled stable ifit has noinfinitesimalautomor-
phism. Aprestablemapover Siscalled stable ifthemaponeachgeometric
fiberofπ isstable.
Exercise1.1. Provethatthestabilityconditionisequivalenttothefollowing:
ForeveryirreduciblecomponentC ⊂ C,i
1∼(1) if C P and C maps to a point in X, then C contains at least 3=i i i
(2) if the arithmetic genus of C is 1 and C maps to a point, then Ci i i
invariant β := f ([C]) ∈ NE(X) in addition to g and n, where NE(X)∗
stands for Mori cone of the numerical (or homological) classes of effective
1cycles. Let M (X,β) bethemodulistackofthefunctordefinedabove.g,n
Theorem1.2(Kontsevich(i),Pandharipande(ii)). Themoduli M (X,β)g,n
(i) isaproper separated Deligne–Mumford stackoffinitetype(overC),and
(ii) hasaprojective coarse modulischeme.
1.2. Natural morphisms. As in moduli of curves, there are the forgetful
ft : M (X,β)→ M (X,β),g,n+1 g,ni
forgetting the i-th marked point and stabilize. As you must have learned
madefunctorial. Infact,
Exercise 1.3. ft : M (X,β)→ M (X,β) is isomorphictotheuniver-g,n+1 g,nn+1
salcurve. (Thisissimilar tothecaseofcurves.)
Theevaluation morphisms
ev : M (X,β)→ X,i g,n
arethemorphismswhich send[f : (C,x ,...,x )→ X] to f(x )∈ X.1 n iGWT AND CTC 3
Thestabilization morphism
st : M (X,β)→ Mg,n g,n
exists when M does. It assigns an (equivalence class of) stable curveg,n
¯[(C,x¯ ,...,x¯ )] to (that of) a stable map [f : (C,x ,...,x )→ X]. Some1 n 1 n
¯[(C,x ,...,x )].1 n
Exercise1.4. Formulatethestabilization fortheforgetfulmorphismswhich
forgetsonemarkedpoint. (Probably doneinthefirstweekalready.)
Hint: IntermsoffamiliesoverS:
⊗k∞¯C ֒→Proj ⊕ π ω ( x ) ,∗k=0 C/S∑ iS
whereω isthedualizinglinebundle.C/S
′ ′′(1.1) M ′ (X,β )× M ′′ (X,β )→ M (X,β),∑ ∑ g ,n +1 X g−g ,n +1 g,n1 1
′ ′′ ′ ′′β +β =βn +n =n
M (X,β) ←−−− D −−−→ M (X,β).g−1,n+2 g,n
 
 
y y(1.2)
X×X ←−−− X

Remark 1.5. The images of the gluing morphisms are in the “boundary”of
the the moduli. However, unlike the curve theory, the moduli are not of
pure dimension in general, and it doesn’t really make sense to talk about
the “divisors”. On the other hand, the virtual classes, which we will talk
aboutsoonalbeit inasuperficialway,are compatible withthegluingmor-
phisms. Thusthegluingdefinesvirtualdivisors.
1.3. Gromov–Witteninvariantsandtheaxioms. Givenaprojectivesmooth
variety X, Gromov–Witten invariants (GWIs) for X are numerical invari-
ants constructed via the auxiliary moduli spaces/stacks M (X,β). Theyg,n
arecalledinvariantsbecausetheyare(real)symplectic-deformation invariants
of X. We will say say nothing about symplectic perspective but to point
out that it does mean that GWIs are deformation invariant. Even though
haved. Infact,theycanbeasbadlybehavedasanyprescribedsingularities.
auxiliary spaces, thanks to the existence of and the functorial properties
virenjoyed by the virtual fundamental classes [M (X,β)] . A good, conciseg,n
expositionoftheconstructionofthevirtualclassescanbefoundinthefirst4 Y.P.LEE
2fewpagesof[7]and will not berepeatedhere. Instead,we will only state
somefunctorial properties(or axioms) theseinvariants, or equivalently the
One of the most important properties of the virtual class is the virtual
(1.3) vdim(M (X,β)) :=−K .β+(1−g)(dimX−3)+ng,n X
vir[M (X,β)] ∈ H (M (X,β))).g,n vdim g,n
Thewell-definedvirtualdimensioniscalled thegradingaxiom.
Before we go further, let’s see what these invariants look like. As for
M ,thereisauniversalcurveover M (X,β):g,n g,n
π :C→ M (X,β),g,n
which definesthe ψ-classes ψ,i = 1,...n, asfor themoduliofcurves. Thei
mostgeneralGWIcan bewrittenas
k ∗ ∗i(1.4) (ψ ev (α ))st (Ω),∏ ii i
vir[M (X,β)]g,n i
∗whereΩ∈ H (M ).g,n
Convention 1.6. (i) The above “integral” or pairing between cohomology
∗(ii)Whenthestabilizationmorphismisnotdefined,onecansetst (Ω) =
However,sometimesweareonlyconcernedwiththecasewhenΩ = 1
ki ∗hτ (α ),...,τ (α )i := (ψ ev (α )).k 1 k n g,n,β ∏ in i1 ivir[M (X,β)]g,n i
These are generally called gravitational descendents. When k = 0 for all i,i
theyare called primary invariants. Asyoucaneasilyguess,thedescendentsarethe“descendents”
of the primary fields. “Gravitation” is involved because ψ classes are the gravitational fields of the “topological
Proposition1.7. Asamatteroffact, thesetinvariants in(1.4)canbereduced to
a subset when all k = 0, and 3g−3+n≥ 0 (when the stabilization morphismi
Assumingthisproposition,wecanviewGWIsasmulti-linear maps
X ∗ ⊗n ∗I (β) : H (X) → H (M ),g,ng,n
2Noted added: Due to a change of heart of one organizer, the construction of virtual
fundamental classes was covered in these lectures. However, due to the time constraint, I
willnotbeabletoputthatlectureintothesenotes.GWT AND CTC 5
which will be called GW maps. This is the approach taken by Kontsevich,
Manin, Beherend etc.. Due to the symmetry of the marked points, I isg,n
S -invariant up to a sign. When all the cohomology classes are algebraicn
classes, therewillbenosign. Wewillignore thesign forsimplicity.
Phrased this way, the existence of virtual classes implies that the GW
This is called the motivic axiom. In other words, motivic axiom says that
The S -covariance axiom says that permuting the marked points will notn
changetheinvariants (uptoasign,whichisignored!)
Theeffectivity axiom saysthat if β is notan effective curve class, thenthe
correspondingGWIvanishes. Thisshouldbeobviousasthecorresponding
The fundamental class axiom says that for the forgetful morphisms: ft :i
M (X,β)→ M (X,β), thevirtualclass pull-backstovirtualclassg,n+1 g,n
∗ vir virft ([M (X,β)] = [M (X,β)] .g,n g,n+1i
Anotheristhemappingtoapointaxiom. Supposeβ = 0,thenitshouldbe
M (X,0) = M ×X.g,n g,n
This is a smooth DM stack with dimension 3g−3+n+dimX. However
the virtual dimension is 3g−3+n+(1−g)dimX. Therefore, the virtual
talclassexists. Inthiscase,thereisanobstruction bundle
∨E ⊗T → M ×X,X g,n
∨whereE isthedualoftheHodgebundle,whichasyouallknowhasrank
g. Thevirtualclassis
vir ∨[M (X,0)] = c (E ⊗T )∩[M ×X].g,n top X g,n
Notethatthevirtualclassisthefundamentalclasswhen g = 0.
The next two axioms are the splitting axiom and genus reduction axiom,
collectivelycalled gluingtailsaxiom.
Let φ be one of the gluing morphisms for moduli of curves, parallel to
∗thosein(1.1)and(1.2). Let{T}beabasisof H (X)(asavectorspace)andµR
µνg := T T isthematrix ofPoincare´ pairing. Let g betheinversema-µν µ νX
trix of g . From Kunneth formula we know that the class of the diagonalµν
∗ µνin H (X×X) is∑ g T ⊗T .µ νµ,ν6 Y.P.LEE
Theaxiomscan bewrittenasthefollowingtwoequations:
∗ nφ (I ((⊗ )α )) =g,n,β i ∑ ∑ ∑i=1
′ ′′ ′ ′′ µ,νβ +β =βn +n =n
′n µν n
′ ′ ′ ′′ ′′ ′′I (⊗′ α )⊗T g I (⊗′′ ′ )α ⊗Tµ νg ,n +1,β i g ,n +1,β i1 i =1 2 i =n +1
∗ n n µνφ (I ((⊗ )α )) = I (⊗ )⊗T ⊗T g .µ νg,n,β i ∑ g−1,n+2,βi=1 i=1
Exercise 1.8. Rephrase these axioms in two different ways. First, in terms
of the numerical invariants. (Easy!) Second, use GW maps, but without
∗introducingthebasisof H (X). (SothatitcanbeappliedtoChowgroups.)
In the remaining subsection,we will introduce the last axiom and show
∗¯Exercise1.9. Letψ := st (ψ )betheψ-classespulled-backfrom M . Con-i i g,n
vir vir¯(ψ−ψ )∩[M (X,β)] = [D ] ,i i g,n i
where D is the virtual divisor on M (X,β) defined by the image of thei g,n
(i) ′ ′′M (X,β )× M (X,β )→ M (X,β),∑ 0,2 X 0,n g,n
′ ′′β +β =β
(i) ′where the i-th marked point goes to M (X,β ) and the remaining n−10,2
′′pointsto M (X,β ).0,n
¯With this exercise, we can see that one can exchange ψ-classes with ψ-
classes, which are pulled-back from M , and the boundary divisors. Theg,n
GWI associated with boundary divisors can be written as GWI of lower
orderclasses,bythesplittingandgenusreductionaxioms,andothers. (Ex-
ercise: Whatisagoodinductiveorder?)
ToshowProposition1.7,westillneedtodealwiththecaseswhen g = 0
and n≤ 2, or (g,n) = (1,0), for which the stabilization morphisms, and
therefore the invariants, are not defined. We need the divisor axiom. Let D
beadivisoron X andft betheforgetfulmorphism,thenn+1
Z n
∗∗ ∗ ∗¯(ev (α )).st ft (Ω). ev ([D])∏ i n+1i n+1
vir[M (X,β)]g,n+1 i=1
Z Z n
∗ ∗=( D) (ev (α )).st (Ω).∏ i i
virβ [M (X,β)]g,n i=1
3Somepropertiesofvirtualclasses,whichwillbementionedlater,mustbeassumed. For
the time being, assume that the moduli are all connected, smooth, projective variety with
thecorrectdimension.GWT AND CTC 7
Exercise 1.10. (i) Convince yourself that the divisor axiom holds. You will
needto usethe fundamental class axiom. Thenuse the projection formula
forft .n+1
k k1 nhα ψ ,...,α ψ ,Di1 n g,n+1,β,n1
Z n
k−1k k k k1 n 1 i n= Dhα ψ ,...,α ψ i + hα ψ ,...,D.α ψ ,...,α ψ i ,1 n n g,n,β, ∑ 1 i n n g,n,β,1 1 i
β i=1
−1wherebyconventionψ := 0. Youwillneedthevirtualversionofthecom-
∗(1.6) ψ = ft (ψ )+E,i n+1 i i
E = M (X,0)× M (X,β)⊂ M (X,β),i X g,n g,n+10,3
wherei≤ n andthei-thand n+1-stpointslie inthefirstmodulifactor.
(iii) Prove that the divisor axiom will uniquely determine those GWIs
with3g−3+n < 0.
Now, we have all the tools to prove Proposition 1.7. Remember what
you learned about the comparisons of ψ-classes with respect to forgetful
morphisms. Theywillbeneededhere.
Exercise1.11. ProveProposition1.7.
Remark1.12. TheenumerativeinterpretationofGWIsisnotalwaysclear. Morally,
theprimaryinvarianthα ,...,αi shouldcountthenumberofn-pointed1 n g,n,β
genus g curves in X with degree β, such that the i-th marked point lies in
thei-thcycle,(thePoincare´ dualof)α ,allingeneralposition. Whengenusi
iszero,and X ishomogeneous,e.g.projective,theabove interpretationac-
2.1. Generating functions. It is often useful to form the generating func-
tions of GWIs. Indeed, most of the structures in GWT only reveal them-
selves in terms of generating functions. As Fulton said, this is a gift from physics. The rest
Thefirstoneisthegenusgdescendentpotential. Recallthedescendents
ki ∗hτ (α ),...,τ (α )i := (ψ ev (α )).k 1 k n g,n,β ∏ in i1 ivir[M (X,β)]g,n i
kAt each marked point, the insertion can be α⊗ψ for any k. Abstractly, we
∞ ∗(2.1) H :=⊕ H (X),t k=08 Y.P.LEE
kwithbasis{T ψ},eventhoughthosevectorsmighthaverelations. (Thinkµ
ofthisas the“universal” space,independentof β,n etc.. Theactualspaces
forinsertionsarequotients.) Let{t }bethedualcoordinatesand
µ kt := t T ψµ∑ k
beageneralvectorinH . Thegenus gdescendent potentialist
β βq qnF (t) := h⊗ ti := h t,...,ti .g ∑ g,n,β ∑ g,n,β| {z }n! n!
n,β n,β
βThe variables{q} are called Novikov variables. Since NE(X) is aβ∈NE(X)
β β β +β1 2 1 2cone,Novikovvariableshasaringstructureq q = q . Itiscalledthe
∗Convention2.1. DenoteΛtheNovikovringofX. Wewilluse H (X)[[q]] to
∗standfor H (X,Λ).
¯If we want to replace the ψ classes by ψ classes, we will have to make
surethatthetargetofthestabilization morphismexists. Let
µµ k¯ ¯ ¯s := s T , t := t T ψ .∑ µ ∑ µk
µ,k µ,k
Thegenus gancestorpotential isdefinedas
Zβq ⊗l ⊗m¯ ¯ ¯F (t,s) := t ⊗s ,g ∑
virl!m! [M (X,β)]g,l+ml,m,β
M (X,β)→ M → M .g,l+m g,l+m g,l
Asremarked above, the indices of the summation, l,m,β, must ensurenot
only the existence of M (X,β), but also of M . Who are the children of theseg,mg,m+l
2.2. Quantumrings. Thesplittingaxiomatgenuszero,combinedwiththe
permutationinvariance oftheinvariants, givetheassociativity ofthequan-
tumrings,aswewillproceedtoshow. Considerthegeneratingfunctionof
βq ⊗nF (s) := hs i .0 0,n,β∑ n!
4To be more precise, I will have to say that Novikov ring is the formal completion of
the semigroup ringC[NE(X)] in the I-adic topology, where I ⊂ C[NE(X)] is the ideal
generatedbynonzeroelementsin NE(X).GWT AND CTC 9
∗Defineaproductstructure∗on H (X)[[q]] by

∂ ∂ ∂ δǫT ∗ T := F (s) g T .µ s ν 0 ǫ∑ µ ν δ∂s ∂s ∂s
Exercise2.2. Showthat∗ givestheusualintersection/cupproduct.q=0
Exercise2.3. Showthattheassociativityoftheproduct∗isequivalenttothe
followingequation,oftencalled WDVVequationafterB.Dubrovin.

∂ ∂ ∂ ∂ ∂ ∂abF (s) g F (s)0 0µ ν a ǫ δ b∂s ∂s ∂s ∂s ∂s ∂s

∂ ∂ ∂ ∂ ∂ ∂ab= F (s) g F (s) .0 0µ δ a ǫ ν b∂s ∂s ∂s ∂s ∂s ∂s
Inotherwords,thefunctionontheLHSisinvariant underS action.4
ofstabilization andforgetfulmorphisms:
1∼M (X,β)→ M P .=0,n≥4 0,4
Then notice that the LHS of the WDVV equation corresponds to a virtual
boundarydivisorin M (X,β),whichisthepullbackofoneofthebound-0,n
arypointin M (cappedwiththevirtualclass),whiletheRHScorrespond0,4
1toanother. Since thepointclass inP are rational/homological equivalent
by definition, the invariants have to be equal, (assuming the axioms of the
Exercise 2.4. Check the above statements. What are the axioms one has to
∗This associative ring structureon H (X)[[q]] is oftencalled thebig quan-
tum cohomology, to distinguish itself from the small quantum cohomology,
wheretheringstructureis definedby∗ , orbetter,byrestricting s tothes=0
divisorial coordinates. The “equivalence” can be seen from the following
rExercise2.5. Let X =P ,andlet
0 1 r rs = s 1+s h+...+s h ,
′ands := s| 1 . Verifythats =0
1dℓ dsq e ⊗n′F (s) = hs i .g 0,n,dℓ∑ n!
WhataboutageneralX?10 Y.P.LEE
22.3. Computing GWIs I: WDVV. Let X = P . Let us proceed to find
Xhα ,...,αi =hα ,...,αi .1 n 1 nn,d g=0,n,dℓ
2Theinsertionclassesα canbe1,h,h = pt.j
Exercise 2.6. (i) Show, by the fundamental class axiom, if any α = 1, thej
invariant vanishesunless (n,d) = (3,0).
(ii) Show, by mapping to a point axiom, that if d = 0, the invariant van-
ishes unless n = 3. (In this case, quantum cohomology is classical coho-
If α = h, this can be taken cared of by divisor axiom, as in Exercise 2.5.j
Soweonlyhavetocalculate GWIsofthefollowingform
n⊗hpt i .n,d
Now the virtual dimension, in this case equal to the actual dimension, is
3d+n−1. Inorderfortheaboveinvariantsnottovanishapriori,wehave
torequirethecohomological degree,which is 2n, tobeequaltothevirtual
dimension. Thatis, n = 3d−1. Define
3d−3⊗N :=hpt i .d n=3d−1,d
Exercise2.7. ProvethatWDVVequationsinExercise2.3givethefollowing
recursiveequationsfor N :d

3d−4 3d−42 2 3(2.2) N = N N d d −d dd ∑ d d 22 1 2 11 3d −2 3d −11 1
d +d =d,d >01 2 i
By Remark 1.12, we know that N = 1, as there is exactly one line in X1
ItiseasytoseenowN arecompletelydeterminedby(2.2)andtheinitiald
condition N = 1.1
This sounds great, until we realize that there are very few cases which
canbecomputedbyWDVValone. Soletuslookforsomethingelse.
2.4. Equivariant cohomologyand localization. Suppose we have a T :=
∗C acting on X, a smooth projective variety. Then the universal T-bundle
∞ ∞ET := (C \{0})→ BT :=P .
Onecan constructtheassociative X-bundle
(2.3) X := X× ET→ BT.T T
Theequivariant cohomologyof T isdefinedtobe
∗ ∗H (X) := H (X ).TT
5Actually, these arenot algebraic schemes,sosome kindof approximationisneeded. It