MIRROR SYMMETRY FOR CONCAVEX VECTOR BUNDLES ON PROJECTIVE SPACES

PII. S0161171203112136 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

MIRROR SYMMETRY FOR CONCAVEX VECTOR BUNDLES ON PROJECTIVE SPACES

ARTUR ELEZI Received 20 December 2001

LetX⊂Ybe smooth, projective manifolds. Assume thatι:X P^s is the zero lo- cus of a generic section ofV⁺= ⊕i∈Iᏻ(ki), where all theki’s are positive. Assume furthermore thatᏺX/Y=ι^∗(V⁻), whereV⁻= ⊕j∈Jᏻ(−lj)and all thelj’s are nega- tive. We show that under appropriate restrictions, the generalized Gromov-Witten invariants ofXinherited fromYcan be calculated via a modified Gromov-Witten theory onP^s. This leads to local mirror symmetry on theA-side.

2000 Mathematics Subject Classification: 14N35, 14L30.

1. Introduction. LetV⁺= ⊕i∈Iᏻ^(ki)andV⁻= ⊕j∈Jᏻ⁽−lj)be vector bundles onP^s withkiandljpositive integers. Suppose thatX P^{ι s} is the zero locus of a generic section ofV⁺andY is a projective manifold such thatX^j Y with normal bundleᏺX/Y=ι^∗(V⁻). The relations between Gromov-Witten theories ofXandYare studied here by means of a suitably defined equivariant Gromov- Witten theory inP^s. We apply mirror symmetry to the latter to evaluate the gravitational descendants ofY supported inX.

Section 2 is a collection of definitions and techniques that will be used throughout this paper. In Section 3, using an idea from Kontsevich, we in- troduce a modified equivariant Gromov-Witten theory inP^s corresponding to V=V⁺⊕V⁻. The correspondingᏰ-module structure [4,11,22] is computed inSection 4. It is generated by a single function ˜JV. In general, the equivari- ant quantum product does not have a nonequivariant limit. It is shown in Lemma 4.3that the generator ˜JV does have a limitJV which takes values in H^∗P^m[[q, t]]. It is this limit that plays a crucial role in this work.

LetY be a smooth, projective manifold. The generator JY of the pureᏰ^- module structure ofY encodes one-pointed gravitational descendants ofY. It takes values in the completion ofH^∗Y along the semigroup (Mori cone) of the rational curves ofY. The pullback mapj^∗:H^∗Y→H^∗Xextends to a map between the respective completions. InTheorem 4.7, we describe one aspect of the relation between pure Gromov-Witten theory ofX^j Y and the modified Gromov-Witten theory ofP^s. Under natural restrictions, the pullbackj^∗(JY) pushes forward toJV. It follows that although defined onP^s,JV encodes the

gravitational descendants ofY supported inX, hence the contribution ofXto the Gromov-Witten invariants ofY.

The only way thatX remembers the ambient variety Y in this context is by the normal bundle, Y can therefore be substituted by a local manifold.

This suggests that there should be a local version of mirror symmetry (see the remark at the end ofSection 4). This was first realized by Katz et al. [15].

The principle of local mirror symmetry in general has yet to be understood.

Some interesting calculations that contribute toward this goal can be found in [6].

InSection 5, we give a proof of the mirror theorem which allows us to com- puteJV. A hypergeometric seriesIV that corresponds to the total space ofVis defined. The mirrorTheorem 5.1states thatIV=JVup to a change of variables.

Hence, the gravitational descendants ofY supported onX can be computed inP^s.

Two examples of local Calabi-Yau threefolds are considered inSection 6. For X=P¹andV=ᏻ⁽−1)⊕ᏻ⁽−1), we obtain the Aspinwall-Morrison formula for multiple covers. IfX=P²andV=ᏻ(−3), the quantum product ofY pulls back to the modified quantum product inP². The mirror theorem in this case yields the virtual number of plane curves on a Calabi-Yau threefold.

The rich history of mirror symmetry started in 1990 with a surprising con- jecture by Candelas et al. [5] that predicts the numberndof degreedrational curves on a quintic threefold. In [11], Givental presented a clever argument which, as shown later by Bini et al. in [4] and Pandharipande in [22], yields a proof of the mirror conjecture for Fano and Calabi-Yau (convex) complete in- tersections in projective spaces. Meanwhile, in a very well-written paper [20], Lian et al. used a different approach to obtain a complete proof of mirror the- orem for concavex complete intersections on projective spaces. An alternative proof of the convex mirror theorem has been given by Bertram [3]. In this pa- per, we use Givental’s approach to study the local nature of mirror symmetry and to present a proof of the concavex mirror theorem.

2. Stable maps and localization

2.1. Genus zero stable maps. LetM0,n(X, β)be the Deligne-Mumford mod- uli stack of pointed stable maps toX. For an excellent reference on the con- struction and its properties, we refer the reader to [10]. We recall some of the features onM0,n(X, β)and establish some notation. For each marking pointxi, letei:M0,n(X, β)→X be the evaluation map atxi, andᏸithe cotangent line bundle atxi. The fiber of this line bundle over a moduli point(C, x1, . . . , xn, f ) is the cotangent space of the curveC atxi. Letπk:M0,n(X, β)→M0,n−1(X, β) be the morphism that forgets the kth marked point. The obstruction the- ory of the moduli stackM0,n(X, β)is described locally by the following exact sequence:

0 →Ext⁰



ΩC



ⁿ

i=1

xi



,ᏻC



 →H⁰

C, f^∗T X

→᐀M

→Ext¹



ΩC



ⁿ

i=1

xi



,ᏻC



 →H¹

C, f^∗T X

→Υ →0.

(2.1)

(Here and thereafter, we are naming sheaves after their fibres.) To understand the geometry behind this exact sequence, we note that ᐀M =Ext¹(f^∗ΩX → ΩC,ᏻC) and Υ = Ext²(f^∗ΩX → ΩC,ᏻC) are, respectively, the tangent space and the obstruction space at the moduli point (C, x1, . . . , xn, f ). The spaces Ext⁰(ΩC(n

i=1xi),ᏻC)and Ext¹(ΩC(n

i=1xi),ᏻC)describe, respectively, the in- finitesimal automorphisms and infinitesimal deformations of the marked source curve. It follows that the expected dimension of M0,n(X, β) is

−KX·β+dimX+n−3.

A smooth projective manifoldXis calledconvexifH¹(P¹, f^∗T X)=0 for any morphismf:P¹→X. For a convexX, the obstruction bundleΥ vanishes and the moduli stack is unobstructed and of the expected dimension. Examples of convex varieties are homogeneous spacesG/P.

In general, this moduli stack may behave badly and have components of larger dimensions. In this case, a Chow homology class of the expected dimen- sion has been constructed [2, 18]. It is called the virtual fundamental class and denoted by[M0,n(X, β)]^virt. Although its construction is quite involved, we mainly use two relatively easy properties. The virtual fundamental class is preserved when pulled back by the forgetful mapπn. A proof of this fact can be found in [7, Section 7.1.5]. If the obstruction sheafΥis free, the virtual fundamental class refines the top Chern class ofΥ. This fact is proven in [2, Proposition 5.6].

2.2. Equivariant cohomology and localization theorem. The notion of equivariant cohomology and the localization theorem is valid for any com- pact connected Lie group. For a detailed exposition on this subject, we suggest [7, Chapter 9]. Below, we state without proof the results that are used in this work.

The complex torusT=(C^∗)^s+1is classified by the principalT-bundle ET=

C^∞+1−{0}_s+1

→BT=

CP^∞_s+1

. (2.2)

Letλi=c1(π_i^∗(ᏻ^{(1)))and λ:}=(λ0, . . . , λs). We useᏻ^(λi)for the line bundle π_i^∗(ᏻ(1)). Clearly,H^∗(BT )=C[λ]. IfT acts on a varietyX, we letXT:=X×T

ET.

Definition2.1. The equivariant cohomology ofXis H_T^∗(X):=H^∗

XT

. (2.3)

IfX=xis a point, thenXT=BTandH^∗_T(x)=C[λ]. For an arbitraryX, the equivariant cohomologyH_T^∗(X)is aC[λ]-module via the equivariant morphism X→x.

Letᐁbe a vector bundle overX. If the action ofT onXcan be lifted to an action onᐁ, which is linear on the fibers,ᐁis an equivariant vector bundle and ᐁTis a vector bundle overXT. The equivariant Chern classes ofEarec_k^T(ᐁ^):= ck(ᐁT). We useE(ᐁ)(ET(ᐁ)) to denote the nonequivariant (equivariant) top Chern class ofᐁ.

LetX^T= ∪j∈JXjbe the decomposition of the fixed point locus into its con- nected components. The componentsXjare smooth for alljandXjis smooth for allj and the normal bundle Nj ofXjinX is equivariant. Letij:Xj→X be the inclusion. The following form of the localization theorem will be used extensively here.

Theorem2.2. Letα∈H_T^∗(X)⊗C(λ). Then,

XT

α=

j∈J (X_j)T

i^∗_j(α) ET

Nj. (2.4)

A basis for the characters of the torus is given byεi(t0, . . . , ts)=ti. There is an isomorphism between the character group of the torus andH²(BT )sending εitoλi. We say thatthe weightof the characterεiisλi.

For an equivariant vector bundleᐁoverX, it may happen that the restriction ofᐁon a fixed-point componentXjis trivial (e.g., ifXj is an isolated point).

In that case,ᐁdecomposes as a direct sum⊕^mi=1µiof characters of the torus.

If the weight ofµiisρi, then the restriction ofc^T_k(ᐁ)onXjis the symmetric polynomialσk(ρ1, . . . , ρm).

Our interest here is forX=P^s. For any action ofT onP^s, we denote ᏼ:=H_T^∗P^s, ᏾=:ᏼ⊗C(λ). (2.5)

Consider the diagonal action ofT=(C^∗)^s+1onP^swith weights(−λ0, . . . ,−λs), that is,

t0, t1, . . . , ts

·

z0, z1, . . . , zs

=

t₀⁻¹z0, . . . , t_s⁻¹zs

. (2.6)

Then, P^sT =P(⊕iᏻ(−λi)). There is an obvious lifting of the action ofT on the tautological line bundle ᏻ(−1). It follows thatᏻ(k)is equivariant for all k. Let p = c^T₁(ᏻP^s(1)) be the equivariant hyperplane class. We obtain ᏼ = C[λ, p]/

i(p−λi)and᏾=C(λ)[p]/

i(p−λi). The locus of the fixed points consists of pointspj for j=0,1, . . . , s, wherepj is the point whosejth co- ordinate is 1 and all the other ones are 0. On the level of the cohomology, the mapi^∗_j sendsp toλj. A basis for᏾ as aC(λ)-vector space is given by

φj=

k≠j(p−λk)forj=0,1, . . . , s. Also,i^∗_j(φj)=

k≠j(λj−λk)=EulerT(Nj).

The localizationTheorem 2.2says that for any polynomialF (p)∈C(λ)[p]/

s

i=1(p−λi)

P^s_TF (p)=

j

F λj

k≠j

λj−λk

. (2.7)

Translating the target of a stable map, we get an action ofTonM0,n(P^s, d). In [17], Kontsevich identified the fixed-point components of this action in terms of decorated graphs. Iff:(C, x1, . . . , xn)→P^sis a fixed stable map, thenf (C) is a fixed curve inP^s. The marked points, collapsed components, and nodes are mapped to the fixed-pointspiof theT-action onP^s. A noncontracted compo- nent must be mapped to a fixed linepipjonP^s. The only branch points are the two fixed pointspiandpjand the restriction of the mapfto this component is determined by its degree. The graphΓcorresponding to the fixed-point compo- nent containing such a map is constructed as follows. The vertices correspond to the connected components off⁻¹{p0, p1, . . . , ps}. The edges correspond to the noncontracted components of the map. The graph is decorated as follows.

Edges are marked by the degree of the map on the corresponding component, and vertices are marked by the fixed point ofP^swhere the corresponding com- ponent is mapped to. To each vertex, we associate a leg for each marked point that belongs to the corresponding component. For a vertexv, letn(v)be the number of legs or edges incident to that vertex. Also, for an edgee, letdebe the degree of the stable map on the corresponding component. Let

ᏹΓ:=

v

M0,n(v). (2.8)

There is a finite group of automorphismsG_Γacting onM_Γ [7,12]. The order of the automorphism groupG_Γ is

a_Γ=

e

de·Aut(Γ). (2.9)

The fixed-point component corresponding to the decorated graphΓ is

M_Γ=ᏹΓ/G. (2.10)

Leti_Γ :M_Γ M0,n(P^s, d)be the inclusion of the fixed-point component cor- responding toΓ andN_Γ its normal bundle. This bundle isT-equivariant. Let α be an equivariant cohomology class in H_T^∗(M0,n(P^s, d)) and α_Γ :=i^∗_Γ(α).

Theorem 2.2says that

M0,n(P^s,d)T

α=

Γ (M_Γ)T

α_Γ a_ΓEulerT

N_Γ. (2.11)

Explicit formulas for EulerT(N_Γ)in terms of Chern classes of cotangent line bundles inH_T^∗(M_Γ)have been found by Kontsevich in [17].

2.3. Linear and nonlinear sigma models for a projective space. Two com- pactifications of the space of degreedmapsP¹→P^sare very important in this paper.Md:=M0,0(P^s×P¹, (d,1))is called the degree-dnonlinear sigma model ofP^s, andNd:=P(H⁰(P¹,ᏻ_P¹(d))^s+1)is called the degree-dlinear sigma model of the projective spaceP^s. An element inH⁰(P¹,ᏻ_P^{1(d))s+1is an(s}+1)-tuple of degreedhomogeneous polynomials in two variablesw0andw1. As a vector space,H⁰(P¹,ᏻ_P¹(d))^s+1 is generated by the vectorsvir =(0, . . . ,0, w₀^rw₁^d−r, 0, . . . ,0)fori=0,1, . . . , s and r=0,1, . . . , d. The only nonzero component of viris theith one.

The action ofT :=T×C^∗inP^s×P¹with weights(−λ0, . . . ,−λs)in theP^sfac- tor and(−,0)in theP¹factor gives rise to an action ofT inMdby translation of maps.T also acts inNdas follows. For ¯t=(t0, . . . , ts)∈T andt∈C^∗,

(¯t, t)· P0

w0, w1 , . . . , Ps

w0, w1

= t0P0

tw0, w1

, . . . , tsPs

tw0, w1 . (2.12) There is aT -equivariant morphismψ:MdNd. Here is a set-theoretical de- scription of this map (for a proof that it is a morphism, see [11] or [19]). Letqi

fori=1,2 be the projection maps onP^s×P¹. For a stable map(C, f )∈Md, let C0be the unique component ofCsuch thatq2◦f:C0→P¹is an isomorphism.

LetC1, . . . , Cn be the irreducible components ofC−C0 and di the degree of the restriction ofq1◦f onCi. Choose coordinates onC0P¹such thatq2◦ f (y0, y1)=(y1, y0). LetC0∩Ci=(ai, bi)andq1◦f=[f0:f1:···:fs]:C0 P^s. Then,

ψ(C, f ):= n i=1

biw0−aiw1

d_i

f0:f1:···:fs

. (2.13)

Letpir be the points ofNdcorresponding to the vectorsvir. The fixed-point loci of theT -action onNdconsists of the pointspir. We writeκfor the equi- variant hyperplane class ofNd. The restriction ofκ at the fixed pointpir is λi+r. The restriction of the equivariant Euler class of the tangent spaceT Nd

atpir is [19]

Eir=

(j,t)≠(i,r )

λi−λj+r−t

. (2.14)

Fixed-point components ofMd are obtained as follows. Let Γdⁱ_j be the graph of aT-fixed point component in M0,1(P^s, dj), where the marking is mapped topiandd1+d2=d. Let(d1, d2)be a partition ofd. We identifyM_Γi

d1×M_Γi d2

with a T -fixed point component M_dⁱ

1d₂ in Md in the following manner. Let (C1, x1, f1)∈M_Γi

d1

and(C2, x2, f2)∈M_Γi d2

. LetCbe the nodal curve obtained by

gluingC1withP¹atx1and 0∈P¹andC2withP¹atx2and∞ ∈P¹. Letf:C→ P^s×P¹mapC1to the sliceP^s× ∞by means off1andC2toP^s×0 by means off2. Finally, f mapsP¹ topi×P¹ by permuting coordinates and ψ maps M_dⁱ

1d₂topid₂∈Nd, hence the equivariant restriction ofψ^∗(κ)inM_dⁱ

1d₂isλi+ d2. The normal bundleN_Γi

d1d2

of this component in the above identification can be found by splitting it in five pieces: smoothing the nodes x1 and x2

and deforming the restriction of the map to C1, C2, P¹. Using Kontsevich’s calculations, Givental obtained [11]

1 ET

N_Γi

d1d2

= 1

k≠i

λi−λk 1 ET

N_Γi

d1

1 ET

N_Γi

d2

e^∗₁ φi

−

−−c1 e^∗₁ φi

−c2, (2.15) wherecj,j=1,2 is the first Chern class of the cotangent line bundle onM_Γi

dj. 3. A Gromov-Witten theory induced by a vector bundle

3.1. The obstruction class of a concavex vector bundle. The notion of concavex vector bundle is due to Lian et al. [19] and is central to this work.

Definition3.1. (1) A line bundleᏸ^{onXis calledconvexifH1(C, f∗(}ᏸ⁾⁾= 0 for any genus zero stable map(C, x1, . . . , xn, f ).

(2) A line bundleᏸonXis calledconcaveifH⁰(C, f^∗(ᏸ))=0 for any non- constant genus zero stable map(C, x1, . . . , xn, f ).

(3) A direct sum of convex and concave line bundles onXis called aconcavex vector bundle.

A concavex vector bundleVin a projective spaceP^shas the form V=V⁺⊕V⁻=

⊕i∈Iᏻki

⊕

⊕j∈Jᏻ−lj

, (3.1)

wherekiandljare positive numbers. DenoteE⁺:=E(V⁺)andE⁻:=E(V⁻).

Letd >0. Consider the following diagram:

M_0,n+1 P^s, d

π_n+1

e_n+1

P^s

M0,n P^s, d

.

(3.2)

SinceV is concavex, the sheaf

Vd:=V_d⁺⊕V_d⁻=π_n+1∗e^∗_n+1 V⁺

⊕R¹π_n+1∗e^∗_n+1 V⁻

(3.3) is locally free.

Definition3.2. The obstruction class corresponding toVis defined to be Ed:=E

Vd

=E V_d⁺

E V_d⁻

:=E⁺dE⁻d. (3.4) For aT-action onP^sthat lifts to a linear action on the fibers ofV=V⁺⊕V⁻, letE⁺:=ET(V⁺)andE⁻:=ET(V⁻). Assume thatE⁻is invertible.

Definition3.3. The modified equivariant integralωV :᏾→C(λ)corre- sponding toV is defined as follows:

ωV(α):=

P^mT

α∪E⁺

E⁻. (3.5)

Consider the trivial action ofT =(⊕i∈IC^∗)⊕(⊕j∈JC^∗)onP^s. In this case, P^sT =P^s×(⊕i∈IP^∞)×(⊕j∈JP^∞)and M0,n(P^s, d)_T =M0,n(P^s, d)×(⊕i∈IP^∞)× (⊕j∈JP^∞). It follows that ᏼ=H^∗(P^s,C[λ])and ᏾=H^∗(P^s,C(λ)). Letp de- note the equivariant hyperplane class. TheT-action lifts to a linear action on the fibers ofVwith weights((−λi)_i∈I, (−λj)_j∈J). Letqiandqjdenote the pro- jection maps onM0,n(P^s, d)_T. BothV_d⁺andV_d⁻areT-equivariant bundles and

V_d⁺

T=V_d⁺⊗

⊕i∈Iq^∗_iᏻP^∞

−λi , V_d⁻

T=V_d⁻⊗

⊕j∈Jq_j^∗ᏻP^∞

−λj

. (3.6)

The equivariant obstruction class is Ed:=ET

Vd

=ET V_d⁺

ET V_d⁻

=E⁺_dE⁻_d. (3.7) The modified equivariant integral for the trivial action ofT onP^sgives rise to a modified perfect pairing in᏾

a, bV:=ωV(a∪b). (3.8)

Let T0=1, T1=p, . . . , T^s =p^s be a basis of ᏾ as a C(λ)-vector space. The intersection matrix(gr t):=(Tr, TtV)has an inverse(g^{r t}). LetTⁱ=s

j=0g^ijTj

be the dual basis with respect to this pairing. Clearly, Tⁱ=Tm−i·E⁻

E⁺

. (3.9)

This implies that, inH^∗(P^s×P^s)⊗C(λ), we have s

i=1

Ti⊗Tⁱ=∆·

1⊗E⁻ E⁺

, (3.10)

where∆=s

i=0Ti⊗T_s−iis the class of the diagonal inP^s×P^s.

Recall that the morphismπk:M0,n(P^s, d)→M_0,n−1(P^s, d) forgets thekth marked point.

Lemma3.4. The forgetful morphisms satisfyπk∗(Ed)=Ed andπk∗(Ed)= Ed.

Proof. For simplicity, we consider the caseV=ᏻ^(k)⊕ᏻ⁽−l)andk=n. The general case is similar. LetMk=M0,k(P^s, d)andMn,n=Mn×M_n−1Mn. Consider the following equivariant commutative diagram:

M_n+1πn+1eµ_n+1πn, Mn,nα^β P^s, Mn π_nM_nπn

en, M_n−1.

(3.11)

We compute

πn+1∗e^∗_n+1ᏻ^(k)=πn+1∗π_n^∗en∗ᏻ^(k)=β_∗µ_∗µ^∗α^∗en∗ᏻ^{(k). (3.12)} By the projection formula,

µ_∗µ^∗α^∗en∗ᏻ(k)=α^∗en∗ᏻ(k)⊗µ_∗ ᏻM_n+1

. (3.13)

Since the mapµis birational andM_n+1is normalµ_∗(ᏻM_n+1)=ᏻMn,n, hence µ_∗µ^∗α^∗en∗ᏻ(k)=α^∗en∗ᏻ(k). (3.14) Substituting into (3.12) and applying base extension properties (πn is flat) yields

π_n+1∗e^∗_n+1ᏻ(k)=β_∗α^∗en∗ᏻ(k)=πn∗

πn∗en∗ᏻ(k)

. (3.15) For the case of a negative line bundle, we have

R¹πn+1∗e^∗_n+1ᏻ⁽−l)=R¹πn+1∗πn∗en∗ᏻ⁽−l)=R¹πn+1∗µ^∗α^∗en∗ᏻ⁽−l).

(3.16) We now use the spectral sequence

R^pβ_∗

R^qµ_∗Ᏺ ⇒R^p+qπ_n+1∗Ᏺ, (3.17) whereᏲis a sheaf ofᏻᏹn+1-modules. The mapµis birational. If we think ofMn

as the universal map ofMn−1, then the mapµhas nontrivial fibers only over pairs of stable maps inMnthat represent the same special point (i.e., node or marked point) of a stable map inMn−1. These nontrivial fibers are isomorphic toP¹. SinceᏲ=e^∗_n+1ᏻ⁽−l), we obtainR^qµ_∗Ᏺ=0 forq >0. It follows that this spectral sequence degenerates, giving

R¹π_n+1∗e^∗_n+1ᏻ(−l)=R¹β_∗µ_∗µ^∗α^∗en∗ᏻ(−l). (3.18)

Now, we proceed as in (3.14) to conclude R¹π_n+1∗e^∗_n+1ᏻ(−l)=πn∗

R¹πn∗en∗ᏻ(−l)

. (3.19)

The lemma is proven.

Remark3.5. The previous lemma justifies the omission ofnfrom the no- tation of the obstruction class.

3.2. Modified equivariant correlators and quantum cohomology. Letγi∈

᏾ for i=1, . . . , n and d >0. Introduce the following modified equivariant Gromow-Witten invariants:

˜Id

γ1, . . . , γn :=

M_0,n(P^m,d)_T

e^∗₁ γ1

∪···∪e_n^∗ γn

∪Ed∈C(λ). (3.20)

Now,M0,n(P^s,0)=M0,n×P^s and all the evaluation maps equal the projection q2to the second factor. The integrals in this case are defined as follows:

I˜0

γ1, . . . , γn

:=

M_0,n(P^s,0)

e^∗₁ γ1

∪···∪e^∗_n γn

∪q^∗₂ E(V )

∈C(λ). (3.21)

The modified equivariant gravitational descendants are defined similarly to Gromov-Witten invariants

˜Id

τk₁γ1, . . . , τknγn

:=

M_0,n(P^s,d)_T

c₁^k1 ᏸ1

∪e^∗₁ γ1

∪···∪c₁^kn ᏸn

∪e^∗_n γn

∪Ed. (3.22)

Lemma 3.4is essential in proving that the modified correlators satisfy the same properties, such as fundamental class property, divisor property, point mapping axiom, and so on, that the usual Gromov-Witten invariants do. The proofs are similar to the ones in pure Gromov-Witten theory. As an illustration, we prove one of these properties.

Fundamental class property. Letγn=1 andd≠0. The forgetful mor- phism πn:M0,n(P^s, d)→M0,n−1(P^s, d) is equivariant. UsingLemma 3.4, we obtain

e^∗₁ γ1

∪···∪e^∗_n−1 γ_n−1

∪e^∗_n(1)∪Ed

=π^∗ e^∗₁

γ1

∪···∪e^∗_n−1 γn−1

∪Ed

. (3.23)

Therefore,

˜Id

γ1, . . . , γn−1,1

= M_0,n(P^s,d)

π^∗ e^∗₁

γ1

∪···∪e_n−1^∗ γn−1

∪Ed

= πn∗(M_0,n(P^s,d))

e₁^∗ γ1

∪···∪e^∗_n−1 γn−1

∪Ed=0.

(3.24)

The last equality is because the fibers ofπnare positive dimensional. Ifd=0, by the point mapping property we know that the integral is zero unlessn=3.

In that case, ˜I0(γ1, γ2,1)= γ1, γ2.

We will now prove a technical lemma that will be very useful later. LetA∪B be a partition of the set of markings andd=d1+d2. LetD=D(A, B, d1, d2) be the closure inM0,n(P^s, d)of stable maps of the following type. The source curve is a unionC=C1∪C2of two lines meeting at a node x. The marked points corresponding toAare onC1, and those corresponding toBare onC2. The restriction of the mapf onCihas degreedifori=1,2.Dis a boundary divisor inM0,n(P^s, d). LetM1:=M0,|A|+1(P^s, d1)andM2:=M0,|B|+1(P^s, d2). Let ex and ˜ex be the evaluation maps at the additional marking in M1 and M2

andµ:=(ex,e˜x). The boundary divisorDis obtained from the following fibre diagram:

D

ν

ι M1×M2

µ

P^{s δ} P^s×P^s

(3.25)

whereνis the “evaluation map at the nodex” andδis the diagonal map.

Lemma3.6. For any classesγ1, . . . , γnin᏾,

D

n i=1

e^∗_i γi

Ed= s a=0 M₁

i∈A

e_i^∗ γi

e^∗_x Ta

Ed₁

× M₂

j∈B

e_j^∗ γj

˜ e^∗_x

T^a Ed₂

. (3.26) Proof. This lemma is the analogue of [10, Lemma 16]. The proof needs a minor modification. Letα:D→M0,n(P^s, d). Consider the normalization se- quence atx

0 →ᏻC →ᏻC ⊕ᏻC →ᏻx →0. (3.27) Twisting it byf^∗(V⁺)andf^∗(V⁻)and taking the cohomology sequence yield the following identities onD:

α^∗ E⁺_d

ν^∗ E⁺

=ι^∗ E_d⁺

1×E_d⁺

2

, (3.28)

α^∗ E⁻_d

=ι^∗ E_d⁻

1×E⁻_d

2

ν^∗ E⁻

. (3.29)

Combining (3.28) and (3.29), we obtain the restriction ofEd in the divisorD α^∗

Ed

=ι^∗

Ed₁×Ed₂ ν^∗

E⁻ E⁺

. (3.30)

Using formula (3.10), we obtain

ι_∗ν^∗ E⁻

E⁺

=µ^∗

1⊗E⁻ E⁺

µ^∗(∆). (3.31)

Therefore,

D

n i=1

e^∗_i γi

∪Ed

= M₁×M₂

n i=1

e^∗_i γi

∪Ed₁∪Ed₂∪µ^∗

1⊗E⁻ E⁺

∪µ^∗(∆)

= M₁×M₂

n i=1

e^∗_i γi

∪Ed₁∪Ed₂∪µ^∗

a

Ta⊗T^a

= m a=0





M₁ n₁

i=1

e^∗_i γi

∪e^∗_x Ta

∪Ed₁



×





M₂ n₂

j=1

e^∗_j γj

∪˜e^∗_x T^a

∪Ed₂



.

(3.32) The lemma is proven.

The same proof can be used to show that the previous splitting lemma is true for gravitational descendants as well.

Corollary3.7. The following modified topological recursion relations hold:

˜Id



τ_k1+1γ1, τk₂γ2, τk₃γ3, n i=4

τs_iωi





=

˜Id₁

τk₁γ1,

i∈I₁

τs_iωi, Ta

˜Id₂

T^a, τk₂γ2, τk₃γ3,

i∈I₂

τs_iωi

,

(3.33)

where the sum is over all splittingsd1+d2=dand partitionsI1∪I2= {4, . . . , n}

and over all indicesa.

Proof. LetAand Bbe two disjoint subsets of{1,2, . . . , n}. We denote by D(A, B)the sum of boundary divisorsD(E, F , d1, d2)such thatE, Fis a partition of{1,2, . . . , n}andA⊂E,B⊂F, andd1+d2=d. The notationD(A, B)reflects neither the numbernof marked points nor the degreedof the maps, but they will be clear from the context. Consider the morphismπ:M0,n(P^s, d)→M0,3

that forgets the map and all but the first 3 markings. Since M0,3is a point, the cotangent line bundle at the first marking is trivial. Butπ^∗(ᏸ1)=ᏸ1− D({1},{2,3}); therefore,ᏸ1=D({1},{2,3})inM0,n(P^s, d). Multiply both sides of the previous equation by3

j=1c1(ᏸj)^kj∪e^∗_j(γj)∪n

i=4c1(ᏸi)^si∪e^∗_i(ωi)∪Ed

and integrate. The corollary follows from the splitting lemma for gravitational descendants.

In the process of finding solutions to the WDVV equations, Kontsevich sug- gested the following modified equivariant Gromov-Witten potential:

˜Φ

t0, t1, . . . , tm

:=

n≥3

d≥0

1 n!˜Id

γ^⊗n

, (3.34)

whereγ=t0+t1p+···+tsp^s andti∈C(λ). Let ˜Φijk=∂³Φ/∂t˜ i∂tj∂tk. Definition3.8. The modified, equivariant quantum product on᏾is de- fined to be the linear extension of

Ti∗VTj:=

m k=0

Φ˜ijkT^k. (3.35)

Theorem3.9. The algebraQH_V^∗P^sT:=(᏾,∗V)is a commutative, associative algebra with unitT0.

Proof. A simple calculation shows that Φ˜ijk=

n≥0

d≥0

1 n!˜Id

Ti, Tj, Tk, γ^⊗n

. (3.36)

The commutativity of the modified, equivariant quantum product follows from the symmetry of the new integrals.T0is the unit due to the fundamental class property for the modified Gromov-Witten invariants. To prove the associativ- ity, we proceed as in [9, Theorem 4]. Let ˜Φijk=∂³˜Φ/∂ti∂tj∂tk. We compute

Ti∗VTj

∗VTk=

˜Φijeg^ef˜Φf klg^ldTd, Ti∗V

Tj∗VTk

=

˜Φjkeg^ef˜Φf ilg^ldTd.

(3.37)

Since the matrix (g^ld)is nonsingular, (Ti∗VTj)∗VTk=Ti∗V(Tj∗VTk) is equivalent to

e,f

Φ˜ijeg^efΦ˜f kl=

e,f

Φ˜jkeg^efΦ˜f il. (3.38)

Equation (3.38) is the WDVV equation for the modified potential ˜Φ. To prove this equation, letq,r,s,tbe four different integers in{1,2, . . . , n}. There exists an equivariant morphism

π:M0,n

P^s, d

→M0,4=P¹ (3.39)