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# LECTURES ON GROMOV–WITTEN THEORY AND CREPANT TRANSFORMATION CONJECTURE Description

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 .

• ??y ??y

• class when

• virtual class

• axiom says

• called stable

• say say nothing

• can easily

• splitting axiom

Subjects

##### Virtual class

Informations

LECTURES ON GROMOV–WITTEN THEORY AND
CREPANT TRANSFORMATIONCONJECTURE
Y.P.LEE
ABSTRACT. Thesearethepre-notesfortheGrenobleSummerSchoollec-
turesJune-July2011. Theyaimtoprovidethestudentssomebackground
in preparation for the conference. Nominally, only the basic knowledge
onmoduliofcurvescoveredintheﬁrstweekisassumed,althoughItac-
itly assume the students either have heard one thing or two about the
subjects, orareformidablelearners. Itiswell-nighimpossibleforamerehumanto
learnGWTinaweek. Infact,Ionlyknowoftwopersonswhohavedoneit.
moment’shesitation,Ibetthereareplenty.
CONTENTS
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
1. DEFINING GWI
The ground ﬁeld isC. All cohomological degrees are Chow or “com-
plex”degrees,anddimensionsarecomplexdimensions.
11.1. Moduliofstablemaps. Themain referenceis.
An n-pointed, genus g, prestable curve (C,x ,x ,...,x ) is a projective,1 2 n
connected,reduced,nodalcurveofarithmeticgenusgwithndistinct,non-
singular,orderedmarkedpoints.
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
12 Y.P.LEE
x ,x ,...,x ,suchthateverygeometricﬁberisann-pointed,genusg,prestable1 2 n
curvedeﬁnedabove.
Let X be an algebraic scheme. A prestable map over S from n-pointed,
genus gcurvesto X isthefollowingdiagram
f
C −−−→ X

πy
S
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
special(nodalandmarked)points;
(2) if the arithmetic genus of C is 1 and C maps to a point, then Ci i i
containsatleast1specialpoint.
Toformamodulistackofﬁnitetype,oneneedstoﬁxanothertopological
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
morphisms
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
intheFirstWeekoftheSchool,theabove“set-theoretic”descriptioncanbe
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
stabilizationmightbenecessarytoensurethestabilityofthepointedcurve
¯[(C,x ,...,x )].1 n
Exercise1.4. Formulatethestabilization fortheforgetfulmorphismswhich
Hint: IntermsoffamiliesoverS:
!
⊗k∞¯C ֒→Proj ⊕ π ω ( x ) ,∗k=0 C/S∑ iS
i
whereω isthedualizinglinebundle.C/S
Asinthecaseofmoduliofcurves,therearealsogluingmorphisms:
′ ′′(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
and
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
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
(ThisisR.Vakil’s“Murphy’sLaw”.)
However,theseGWIswillbehavemostlyliketheyaredeﬁnedviasmooth
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
2fewpagesofand will not berepeatedhere. Instead,we will only state
somefunctorial properties(or axioms) theseinvariants, or equivalently the
virtualclasses,havetosatisfy.
One of the most important properties of the virtual class is the virtual
dimension(orexpecteddimension,orRiemann–Rochdimension...)
(1.3) vdim(M (X,β)) :=−K .β+(1−g)(dimX−3)+ng,n X
and
vir[M (X,β)] ∈ H (M (X,β))).g,n vdim g,n
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
Z
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
andhomologyisdeﬁnedtobezeroifthetotaldegreeofcohomologyisnot
equaltothevirtualdimension.
∗(ii)Whenthestabilizationmorphismisnotdeﬁned,onecansetst (Ω) =
1.
However,sometimesweareonlyconcernedwiththecasewhenΩ = 1
Z
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
gravity”.
Proposition1.7. Asamatteroffact, thesetinvariants in(1.4)canbereduced to
a subset when all k = 0, and 3g−3+n≥ 0 (when the stabilization morphismi
isdeﬁned).
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
GWIsareconstructedoutofavirtualclass.
Kontsevich–Maninlists9axioms,sowehave7moretogo.
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
modulistackisempty.
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
easytoseethat
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
classisnotthefundamentalclass,eventhoughitissmoothandfundamen-
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
µ,ν
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
howtoproveProposition1.7.
∗¯Exercise1.9. Letψ := st (ψ )betheψ-classespulled-backfrom M . Con-i i g,n
3vinceyourselfthat
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
gluingmorphism
(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
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
¯Hereandelsewhere,ftaretheforgetfulmorphismsformoduliofcurves.
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
(ii)Provethedivisoraxiomfordescendents
(1.5)
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-
parisontheoremforψclassesunderforgetfulmorphisms,asforcurves,
∗(1.6) ψ = ft (ψ )+E,i n+1 i i
where
(i,n+1)
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-
tuallyholds.
2. SOME GWT GENERATING FUNCTIONS AND THEIR STRUCTURES
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
mathematiciansmightbeabletoﬁgureout....
Theﬁrstoneisthegenusgdescendentpotential. Recallthedescendents
looklike
Z
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
canthinkoftheinsertioncomefromaninﬁnitedimensionalvectorspace
∞ ∗(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
µ
forinsertionsarequotients.) Let{t }bethedualcoordinatesand
k
µ kt := t T ψµ∑ k
µ,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
ninsertions
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
4Novikovring.
∗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
¯wheretheψclassesarepullbacksfromthecompositionofstabilizationand
forgetfulmorphisms:
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
ancestors,Ioftenwonder?
2.2. Quantumrings. Thesplittingaxiomatgenuszero,combinedwiththe
permutationinvariance oftheinvariants, givetheassociativity ofthequan-
tumrings,aswewillproceedtoshow. Considerthegeneratingfunctionof
genuszeroprimaryinvariants
βq ⊗nF (s) := hs i .0 0,n,β∑ n!
n,β
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
δ,ǫ
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
Toshowtheassociativityofthe∗,onecanusethefollowingcomposition
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
virtualclasses).
Exercise 2.4. Check the above statements. What are the axioms one has to
use?
∗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
exercise.
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!
d,n
22.3. Computing GWIs I: WDVV. Let X = P . Let us proceed to ﬁnd
genuszeroprimaryinvariants
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-
mology.)
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
through2pointsingeneralposition.
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
5is
∞ ∞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
wasallworkedoutcarefullybyB.TotaroandEdidin–Graham(andothers).