GENERATING FUNCTIONS FOR THE NUMBER OF CURVES ON ABELIAN SURFACES

GENERATING FUNCTIONS FOR THE NUMBER OF CURVES ON ABELIAN SURFACES

JIM BRYANandNAICHUNG CONAN LEUNG

1. Introduction. LetXbe an Abelian surface and letCbe a holomorphic curve in Xrepresenting a primitive homology class. FornandgsatisfyingC·C=2g−2+2n, there is ag-dimensional space of curves of genusgin the class ofC. To define an enumerative problem, one must imposeg constraints on the curves. There are two natural ways to do this. One way is to count the number of curves passing throughg generic points, which we denoteNg,n(X,C). The second way is to count the number of curves in the fixed linear system|C|passing throughg−2 generic points, which we denoteN_g,n^FLS(X,C). We define (modified) Gromov-Witten invariants that compute the numbersNg,n(X,C)andN_g,n^FLS(X,C),and we prove that they do not depend on XorCbut are universal numbers henceforth denoted by Ng,nandN_g,n^FLS. Our main theorem computes these numbers as the Fourier coefficients of quasi-modular forms.

Note thatXdoes not contain any genus-0 curves, so implicitlyg >0 throughout.

Theorem1.1 (Main theorem). The universal numbersN_g,n^FLS andNg,nare given by the following generating functions:

∞ n=0

Ng,nq^n+g−1=g DG2

_g−1

, ∞

n=0

N_g,n^FLSq^n+g−1= DG2

_g−2

D²G2

=(g−1)⁻¹D DG2

_g−1 ,

whereDis the operatorq(d/dq)andG2is the Eisenstein series; that is, G2(q)= − 1

24+ ∞ k=1





d|k

d



q^k.

Note that the right-hand sides are quasi-modular forms (the set of quasi-modular forms is an algebra generated by modular forms,G2, andD, cf. [5]), and the left-hand

Received 8 September 1998. Revision received 22 October 1998.

1991 Mathematics Subject Classification. Primary 14J, 14K; Secondary 11F11.

Bryan’s work supported by a grant from the Ford Foundation and National Science Foundation grant number DMS-9802612.

Leung’s work supported by National Science Foundation grant number DMS-9626689.

311

sides are reminiscent of theta series since the power ofqis(C²/2)=n+g−1.

The surprising fact that quasi-modular forms appear in generating functions for counting curves on surfaces was first discovered by Yau and Zaslow [11], who gave a formula for the number of rational curves onK3 surfaces (cf. Beauville [1]). This formula was subsequently generalized by Göttsche to a conjectural formula that ap- plies for all genus and all surfaces (see [5]). The formula forN_g,n^FLS in Theorem 1.1 is a special case of the general Göttsche-Yau-Zaslow formula thus proving it (given our definition ofN_g,n^FLS) for Abelian surfaces. Göttsche was able to prove the genus-2 case for Abelian surfaces (see also [3]), and the case of K3 surfaces (for arbitrary genus) was proved in [2].

In general, the Göttsche-Yau-Zaslow formula provides a conjectural generating function for the number of curves withnnodes in anyn-dimensional sublinear system of|C|for any sufficiently ample divisorC on any surfaceS. The formulas involve only c2(S),C·C,C·K, K·K, and universal functions (K is the canonical class).

For surfaces with numerically trivial canonical class, the formula reduces to quasi- modular forms.

For concreteness, we expand the equations of the theorem to write ∞

n=0

Ng,nq^n+g−1=g



^∞

k=1

k





d|k

d



q^k





g−1

,

∞ n=0

N_g,n^FLSq^n+g−1=



^∞

k=1

k





d|k

d



q^k





g−2

^∞

k=1

k²





d|k

d



q^k



.

So, for example, for small values of n and g, N_g,n^FLS and Ng,n are given in the following tables.

N_g,n^FLS n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

g=2 1 12 36 112 150 432 392 960

g=3 1 18 120 500 1620 4116 9920 19440

g=4 1 24 240 1464 6594 23808 73008 198480

g=5 1 30 396 3220 18960 88452 344960 1169520

Ng,n n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

g=2 2 12 24 56 60 144 112 240

g=3 3 36 180 600 1620 3528 7440 12960

g=4 4 72 576 2928 11304 35712 97344 238176

g=5 5 120 1320 9200 47400 196560 689920 2126400

Gromov-Witten invariants have been remarkably effective in answering many ques- tions in enumerative geometry for certain varieties such as Pⁿ; however, the ordinary Gromov-Witten invariants are not very effective for counting curves on most surfaces.

One basic reason is that the moduli space of stable maps often fails to be a good model for a linear system (and the corresponding Severi varieties) for dimensional reasons.

For a bundleLsuch thatL−K is ample, the dimension of the Severi varietyVg(L) (the closure of the set of geometric genus-gcurves in the complete linear system|L|) is

dimCVg(L)= −K·L+g−1+pg−q,

whereKis the canonical class,pg=dimH⁰(X,K), andq=dimH¹(X,ᏻ). On the other hand, the virtual dimension of the moduli space _ᏹg,L(X)of stable maps of genus-gin the class dual toc1(L)is

virdimCᏹg,L(X)= −K·L+g−1.

The discrepancy pg−q arises from two sources. Since the image of maps in ᏹg,L(X)are divisors not only in |L|but also potentially in every linear system in Pic^c1(L)(X), one would expect dim_ᏹg,L(X) to exceed dimVg(L) by q = dim Pic^c1(L)(X). (We use Pic^c(X)to denote the component of Pic(X) with Chern classc.) We show that this discrepancy can be accounted for within the framework of the usual Gromov-Witten invariants (see Theorem 2.1).

However, even if we consider_ᏹg,Las a model for the parameterized Severi varieties Vg

c1(L)

≡

L∈Pic^c1(L)(X)

Vg L

, there is still apg-dimensional discrepancy (see also [4]).

The reason is the following. The virtual dimension of_ᏹg,L(X)is the dimension of the space of curves that persist as pseudoholomorphic curves when we perturb the Kähler structure to a generic almost Kähler structure. The difference of pg in the dimensions of_ᏹg,L andVg(c1(L))means that only a codimensionpg subspace of Vg(c1(L)) persists as pseudoholomorphic curves when we perturbthe Kähler structure. One way to rectify this situation is to find a compact pg-dimensional¹ family of almost Kähler structures that has the property that the only almost Kähler structure in the family that supports pseudoholomorphic curves in the classc1(L)is the original Kähler structure. IfT is such a family, then the moduli spaceᏹg,L(X,T ) of stable maps for the familyT is a better model for the spaceVg(c1(L))in the sense that its dimension is stable under generic perturbations of the familyT →T.

It is straightforward to extend the notion of the ordinary Gromov-Witten invariants to invariants for families of almost Kähler structures. The invariants will only depend

1By this we mean a real 2pg-dimensional family. Note that the parameter space for the family need not have an almost complex structure.

on the deformation class of the underlying family of symplectic structures. Given the existence of apg-dimensional family as described above, these invariants can be used to answer enumerative geometry questions for the corresponding surface and linear system.

In general, it is not clear when such a family will exist; however, ifXhas a hyper- Kähler metricg(i.e.,Xis an Abelian orK3 surface), then there is a natural candidate forT, namely, the hyper-Kähler family of Kähler structures. We call this the twistor family associated to the metric g, and we denote it byTg. It is parameterized by a 2-sphere, and so dimRTg=2=2pg as it should. Furthermore, the property that all the curves in_ᏹg,L(X,Tg)are holomorphic for the original complex structure can be proved with Hodge theory (of course, this need no longer be the case for a perturbation ofTg to a generic family of almost Kähler structures).

We will define the numbers N_g,n^FLS and Ng,n as certain Gromov-Witten invariants for the twistor familyTgassociated to a hyper-Kähler metric on an Abelian surfaceX (see our previous paper [2] for theK3 case). We show that the invariants only depend ong andn(notXorC), and they count each irreducible geometric genus-gcurve with positive integral multiplicity. Furthermore, the multiplicity is 1 if, additionally, the curve is nodal. There are additional Gromov-Witten invariants for the twistor family that we also compute. These invariants are easier to describe with the notation of the Gromov-Witten invariants, so we postpone the statement of the result until Section 3 (Definition 3.3 and Theorem 4.1). These additional invariants correspond to the enumerative problem of counting curves that pass throughg−1 points and lie in a certain one-dimensional family of linear systems.

Section 2 reviews Gromov-Witten invariants for families and formulates our result that equates the subset of _ᏹg,L(X)consisting of maps whose images have a fixed divisor class with a cycle more familiar in ordinary Gromov-Witten theory (see The- orem 2.1). In Section 3, we discuss properties of the twistor family, defineN_g,n^FLS and Ng,n, and prove that they have the enumerative properties discussed above. In Sec- tion 4, we compute the invariants to complete the proof of our main theorem and its generalization. We conclude with an appendix containing the proof of Theorem 2.1, which was postponed in the main exposition.

The authors would like to thank O. DeBarre, A. Givental, L. Göttsche, K. Kedlaya, and C. Taubes for helpful discussions and correspondence.

2. Gromov-Witten invariants for families. We review Gromov-Witten invari- ants for families, and we refer the reader to [2] for details.

Let(X,ω)be any compact symplectic manifold with an almost Kähler structure.

Recall that ann-marked, genus-gstable map of degreeC∈H²(X,Z)is a (pseudo)- holomorphic map f : → X from an n-marked nodal curve (,x1,...,xn) of geometric genusgtoXwithf∗([])=Cthat has no infinitesimal automorphisms.

Two stable mapsf :→Xandf:→Xare equivalent if there is a biholomor- phismh:→such thatf =f◦h. We write_ᏹg,n,C(X,ω)for the moduli space

of equivalence classes of genus-g,n-marked, stable maps of degreeCtoX. We often drop theωorXfrom the notation if they are understood, and we sometimes drop the nfrom the notation when it is zero. IfB is a family of almost Kähler structures, then we denote a parameterized version of the moduli space by

ᏹg,n,C(X,B)=

t∈B

ᏹg,n,C(X,ωt).

If B is a compact, connected, oriented manifold, then ᏹg,n,C(X,B) has a fidu- ciary cycle [ᏹg,n,C(X,B)]^vir called the virtual fundamental cycle (see [2] and the fundamental papers of Li and Tian [8], [7], [9]). The dimension of the cycle is

dimR

ᏹg,n,C(X,B)vir

=2c1(X)(C)+

6−dimRX

(g−1)+2n+dimRB.

The invariants are defined by evaluating cohomology classes of ᏹg,n,C on the virtual fundamental cycle. The cohomology classes are defined via incidence relations of the maps with cycles inX. The framework is as follows. There are maps

ᏹg,1,C ev //

ft

X

ᏹg,C

called the evaluation and forgetful maps, defined by ev({f :(,x1)→X})=f (x1) and ft({f :(,x1)→X})= {f :→X}.²The diagram should be regarded as the universal map overᏹg,C.

Given geometric cycles α1,...,αl in X representing classes [α1],...,[αl] ∈ H∗(X,Z) with Poincaré duals [α1]^∨,...,[αl]^∨, we can define the Gromov-Witten invariant

"^(X,B)_g,C

α1,...,αl

=

[ᏹg,C(X,B)]^virft_∗ev^∗ [α1]^∨

∪···∪ft∗ev^∗ [αl]^∨

.

"^(X,B)_g,C (α1,...,αl)counts the number of genus-g, degree Cmaps that are pseudo- holomorphic with respect to some almost Kähler structure in B and such that the image of the map intersects each of the cycles α¹,...,αl. (The integral is defined to be zero if the integrand is not a class of the correct degree.) The Gromov-Witten invariants are multilinear in theα’s, and they are symmetric forα’s of even degree and skew-symmetric forα’s of odd degree. If p¹,...,pk are points in a path-connected X, we use the shorthand

"^(X,B)_g,C

pt.^k,αk+1,...,αl

:="^(X,B)_g,C

p1,...,pk,αk+1,...,αl .

2There is some subtlety to making this definition rigorous since forgetting the point may make a stable map unstable, but it can be done.

Now suppose thatXis a Kähler surface. In order to count curves in a fixed linear system|L| with c1(L)= [C]^∨, one would like to restrict the above integral to the cycle defined by#⁻₀¹(0), where#0 is the map

#0:ᏹg,C(X,ω)→Pic⁰(X)

given by f →ᏻ(Im(f )−0), where0 ∈ |L| is a fixed divisor. Dually, one can add the pullback by#0of the volume form on Pic⁰(X)to the integrand defining the invariant, as follows:

[ᏹ_g,C(X)]^vir#^∗₀ [pt.]^∨

∪ft_∗ev^∗ [α1]^∨

∪···∪ft_∗ev^∗ [αl]^∨

.

We show that#^∗₀([pt.]^∨)can be expressed in the usual Gromov-Witten framework.

Theorem2.1. Let X be a Kähler surface, let [γ] ∈ H1(X,Z), and let ˜γ be the corresponding class inH¹(Pic⁰(X),Z)induced by the identification Pic⁰(X)∼= H¹(X,R)/H¹(X,Z). Then

#^∗₀(γ )˜ =ft_∗ev^∗ [γ]^∨

.

Corollary2.2. Let[γ1],...,[γb1] be an oriented integral basis for H1(X;Z). Then#^∗₀([pt.]^∨)=ft_∗ev^∗([γ1]^∨)∪···∪ft_∗ev^∗([γb1]^∨).

We defer the proof of Theorem 2.1 to the appendix. The corollary follows imme- diately.

The upshot is that we count curves in a fixed linear system using the usual con- straints from Gromov-Witten theory. Namely, the invariant

"g,C

γ1,...,γb1,α1,...,αl

counts the number of genus-g maps whose images lie in a fixed linear system |L|

withc1(L)= [C]^∨ and hit the cyclesα1,...,αl.

3. The twistor family and the definition ofNg,nandN_g,n^FLS. The discussion in this section is very similar to the corresponding discussion for K3 surfaces given in Section 3 of [2]. We show that there is a unique family T (up to deformation) corresponding to the twistor family. We define N_g,n^FLS andNg,nusing a suitable set of Gromov-Witten invariants for the familyT. We show that the invariants solve the enumerative problems in which we are interested. We use this framework to prove that the invariantsN_g,n^FLS andNg,nare universal numbers independent of the choice of the Abelian surfaceXand the linear system|L|.

Let(X,ω)be an Abelian surface and letgbe the unique hyper-Kähler metric given by Yau’s theorem (see [10]). DefineTgto be the family of Kähler structures given by the unit sphere in_Ᏼ²_+,g, the space of self-dual, harmonic 2-forms.

Proposition3.1. For any two hyper-Kähler metricsgandg, the corresponding twistor families Tg and T_g are deformation equivalent. There is therefore a well- defined deformation class, which we denote byT.

Proof. The moduli space of complex structures on the 4-torus is connected, and the space of hyper-Kähler structures for a fixed complex torus is contractible (it is the Kähler cone). Therefore, the space parameterizing hyper-Kähler 4-tori is connected (in fact, it is just the space of flat metrics). We can thus find a path of hyper-Kähler metrics connectinggtog, and, by associating the twistor family to each metric, we obtain a continuous deformation ofTg toTg.

An important observation concerning the twistor family is the following corollary.

Corollary3.2. Letf :X→Xbe an orientation-preserving diffeomorphism.

Thenf^∗(T )is deformation equivalent toT.

Proof. LetTg be the twistor family for a hyper-Kähler metricg. Thenf^∗(Tg)= Tf^∗(g), wheref^∗(g)is the pullback metric which is also hyper-Kähler.

The corollary has the consequence that for any orientation-preserving diffeomor- phismf, the Gromov-Witten invariants for the twistor family satisfy

"^X,T_g,C

α1,...,αl

="^X,T_g,f∗(C)

f∗α1,...,f∗αl .

There is an orientation-preserving diffeomorphism of the 4-torus for every element ofSl4(Z)given by the descent of the linear action on the universal cover R⁴. It follows from the elementary divisor theorem that for any two classesCandCinH²(X,Z) with the same divisibility and square, there is a linear diffeomorphism f such that f∗(C)=C (see [6, p. 47]). This means that there is a lot of symmetry among the Gromov-Witten invariants for the twistor family so that (for primitive classes) they can be encompassed byNg,n,N_g,n^FLS, and the other invariants, which we define below.

Definition 3.3. Let γ1,...,γ4 be loops in X representing an oriented basis for H1(X,Z), and letC∈H2(X,Z)be a primitive class withC·C=2g−2+2n. We define

Ng,n="^{(X,T )}_g,C pt.^g

, N_g,n^FLS="^{(X,T )}_g,C

γ1,γ2,γ3,γ4,pt.^g−2 . Now letC= [γ¹]∧[γ²]+(n+g−1)[γ³]∧[γ⁴]and define³

Ng,n^ij ="^{(X,T )}_g,C

γi,γj,pt.^g−1 fori < j.

3Here we use the fact that on any torus there is a natural identification )²H1(X;Z)∼=H2(X,Z).

The invariantsNg,n,N_g,n^FLS, andNg,n^ij encompass all possible Gromov-Witten invari- ants for the twistor family and primitive homology classes. SinceXhas real dimension 4, the only nontrivial constraints come from the point class andH1(X;Z). Since the point class is invariant under orientation-preserving diffeomorphisms, "^{(X,T )}_g,C (pt.^g) only depends on the square (and divisibility) of C. Similarly, since " is skew- symmetric in the classes γi, the only possibilities with four γ-constraints are the invariants "^{(X,T )}_g,C (γ1,γ2,γ3,γ4,pt.^g−2), which also only depend on the square (and divisibility) ofC.⁴

For dimensional reasons, the invariants with one or threeγ’s are zero. The invariants Ng,n^ij encompass the remaining invariants since for any primitiveCwithC²=2g− 2+2n, we can first moveCto[γ1]∧[γ2]+(n+g−1)[γ3]∧[γ4]by an orientation- preserving diffeomorphism.

We now wish to show thatNg,n,N_g,n^FLS, andNg,n^ij enumerate holomorphic curves as we choose.

Lemma3.4. Suppose thatXis an Abelian surface with a hyper-Kähler metricg and suppose thatC⊂Xis a holomorphic curve. Then the only Kähler structure in Tg that has a holomorphic curve in the class[C] ∈H2(X;Z)is the original Kähler structure for whichCis holomorphic.

Proof. Necessary conditions for the class[C]to contain holomorphic curves are [C]^∨ ∈ H^1,1(X,R) and [C] pairs positively with the Kähler form. Since classes orthogonal toᏴ²_+,gare always of type(1,1), we just need to see when the projection of[C]^∨to_Ᏼ²_+,g is type(1,1).

We may assume [C]² is nonnegative since X has no rational curves and so the projection of[C]^∨onto_Ᏼ²_+,gis nonzero. By definition, the complex structure associ- ated to a formω∈Tg⊂Ᏼ²_+,g defines the orthogonal splittingᏴ²_+,g∼=ωR⊕(H^2,0⊕ H^0,2)R. Therefore, the onlyω∈Tgfor which[C]^∨is type(1,1)and pairs positively with ωis whenω is a positive multiple of the projection of[C]^∨ toᏴ²_+,g. This is unique and so must be the original Kähler structure for whichC⊂Xis holomorphic.

In general, Gromov-Witten-type invariants may give a different count from the corresponding enumerative problem because Gromov-Witten invariants count maps instead of curves and the image of a map may have geometric genus smaller than its domain. We show that, in the case at hand, this does not happen.

4This invariant only depends on the square and divisibility ofCsince, for any orientation-preserving diffeomorphismf, we have

"^{(X,T )}_g,C (γ1,...,γ4,pt.^g−2)="^{(X,T )}_g,f∗(C)(f∗γ1,...,f∗γ4,pt.^g−2)

=det(f∗:H1→H1)"^{(X,T )}_g,f∗(C)(γ1,...,γ4,pt.^g−2)

="^{(X,T )}_g,f∗(C)(γ1,...,γ4,pt.^g−2).

Lemma3.5. Suppose X is generic among those Abelian surfaces admitting a curve in the class ofC. Then the invariantsNg,n,N_g,n^FLS, andNg,n^ij count only maps whose images have geometric genusg, and they are counted with positive integral multiplicity.

Proof. The assumption guarantees that C generates Pic(X)/Pic⁰(X)so that all the curves in the class[C]are reduced and irreducible. Then a dimension count shows that maps with contracting components of genus 1 or greater are not counted. The details are the same as in the proof of Theorem 3.5 in [2]. The argument shows that all the maps are birational onto their image and the corresponding moduli scheme is zero-dimensional. The invariant is then just the length of this scheme, and so all the curves are counted with positive integral multiplicities. Moreover, it is easy to see that if the image of a stable map has only nodal singularities, then the map is a smooth point of the moduli scheme.

We summarize the results of this section in the following theorem.

Theorem3.6. LetC⊂Xbe a holomorphic curveCin an Abelian surfaceXwith C²=2g−2+2n. Then the invariantsNg,n,N_g,n^FLS, andNg,n^ij defined in Definition 3.3 count the number of genus-gcurves in the class[C]that

(1) pass throughggeneric points (Ng,n);

(2) pass through g−2 generic points and are confined to the linear system |C|

(N_g,n^FLS);

(3) pass though g−1 generic points and are confined to lie in a certain one- dimensional family of linear systems determined byij∈{12,13,14,23,24,34} (Ng,n^ij ).

Furthermore, the invariants do not depend on the particular Abelian surfaceX or the curveC; and the invariants count all irreducible, reduced curves with positive multiplicity that is 1 for nodal curves.

4. Computing the invariants. In this section, we compute the invariantsNg,n, N_g,n^FLS, andNg,n^ij by using a particular choice forXandC. Namely, we chooseXto be a product of elliptic curves andC to be a section together with a multiple of the fiber. The computations give our main theorem as well as the generalization to the Ng,n^ij case. For convienence, we state the results below.

Theorem4.1. The invariants Ng,n, N_g,n^FLS, andNg,n^ij (defined in Definition 3.3) are given by the following generating functions:

∞ n=0

Ng,nq^n+g−1=g DG2

_g−1

, (1)

∞ n=0

N_g,n^FLSq^n+g−1= DG2

_g−2

D²G2=(g−1)⁻¹D DG2

_g−1 , (2)

∞ n=0

N_g,n¹²q^n+g−1=D DG2

_g−1 , (3)

∞ n=0

N_g,n³⁴q^n+g−1= DG2

_g−1

, (4)

∞ n=0

N_g,n¹³q^n+g−1=0, (5)

∞ n=0

N_g,n¹⁴q^n+g−1=0, ∞

n=0

N_g,n²³q^n+g−1=0, ∞

n=0

N_g,n²⁴q^n+g−1=0, whereD=q(d/dq)andG2= −(1/24)+_∞

k=1(

d|kd)q^k.

From Theorem 3.6, we are free to compute the invariants for any choice ofXand C. LetXbe the product of two generic elliptic curvesS×F, and letCbe the primitive homology classS+(g+n−1)F ∈H2(X,Z). We writeX=S¹×S¹×S¹×S¹ by choosing a diffeomorphism betweenS¹×S¹with S and also one betweenS¹×S¹ withF. Next we choose representatives for the loops and points onX. We consider the following four loops generatingH1(X,Z):

γ¹:S¹→X, γ¹ e^it

=

e^it,b¹,c¹,d¹ , γ2:S¹→X, γ2

e^it

=

a2,e^it,c2,d2

, γ3:S¹→X, γ3

e^it

=

a3,b3,e^it,d3

, γ4:S¹→X, γ4

e^it

=

a4,b4,c4,e^it ,

where theai’s,bi’s,ci’s, anddi’s are distinct points onS¹.We also chooseggeneric pointsp1=(s1,f1),...,pg=(sg,fg)onX=S×F.For anyf ∈F ands∈S, we callSf =S×{f} ⊂Xa section curve andFs= {s}×F a fiber curve ofX.

Proof of equation (1). We suppose thatφ:D→Xis a stable map from a genus-g curveDwithgmarked points toXrepresenting the classC=S+(g+n−1)F and sending corresponding marked points to thepi’s. First we observe that the image ofφ consists of one section curve and some fiber curves. This is because the projection of any irreducible component of Im(φ)toSis of degree zero except for one component which has degree-1. On the other hand, the projection of the degree-1 component to F must have degree zero because there is no nontrivial morphism between two generic elliptic curvesS andF. Therefore, Im(φ)consists of a single section curve and a number of fibers.

In order for φto pass through the g generic pointsp1,...,pg onX,we need at least(g−1)fiber curves in the image ofφ. On the other hand, the geometric genus of Disg, and covering each section or fiber curve will take up at least one genus ofD. Therefore, Im(φ)consists of exactlyg−1 fiber curves (possibly with multiplicity) and one section curve.

In fact,D has to have precisely g irreducible componentsD1,...,Dg, and each component is a genus-1 curve and contains one marked point. (Since an Abelian surface has no nonconstant rational curves, stability implies that every rational com- ponent ofDmust meet at least three other components. It is easy to rule this possibility out.) The restriction ofφto eachDiis either (i) a covering of some fiber curve con- taining one of thepi’s or (ii) an isomorphism to a section curve containing one of the pi’s.

We can assume thatφrestricted toDg is an isomorphism onto one of the section curves containing somepi=(si,fi). There aregchoices of suchpi’s. Without loss of generality, we assume φ(Dg)containspg or, equivalently,φ(Dg)=Sfg.In this case, the marked point onDgmust be the unique pointφ⁻¹(pg).

Fori < g,we label the component ofDcoveringFsiasDi.Thenφ:Di→Fsiis an unbranched cover by the Hurwitz theorem because bothDiandFs_iare genus-1 curves.

We denoteki=deg(φ:Di→Fsi) >0; then we have_g−1

i=1ki=g+n−1 sinceφ represents the homology class ofS+(g+n−1)F. The number of elliptic curves that admit a degree-khomomorphism to a fixed elliptic curve is classically known to be d|kd. We fix the origin of the elliptic curveDi(andFsi) to be its intersection point withDg (andSfg, respectively), so thatφ:Di→Fsi is a homomorphism and thus the number of choices of(φ:Di→Fsi)is given by

d|kid. Since the marked point on Di could be any one of theki points inφ⁻¹(pi), there are a total of ki

d|kid choices for each marked curveDi.

We denote the(g−1)-tuplek1,...,kg−1 by k, and we write|k|for

iki. From the preceding discussion, the number of stable mapsφ of geometric genusg andg marked points toXin the class ofS+(g+n−1)F is given by

g

k:|k|=n+g−1 g−1 i=1

ki





d|k_i

d



. (6)

It is not difficult to see that each such stable mapφcontributes+1 to the Gromov- Witten invariant for the twistor family (cf. [1]). The analysis is essentially the same as in Section 5 of [2], where the analogous results are proved in theK3 case, and we refer the reader there for full details. We remark however that the situation here is much more elementary than in theK3 case: Each map is an isolated smooth point, as can be seen by examining the infinitesimal deformations of maps (cf. Section 5.2 of [2]). As in theK3 case, the infinitesimal obstruction to deforming the map f in the direction of the twistor family cancels withH¹(S,NS), where NS is the normal bundle ofS inX. The spaceH¹(S,NS)is essentially the obstruction to extending a

deformation of the complex structure of the section component of the domain to a deformation of the mapf.

The formula is now proved by summing equation (6) overnand rearranging:

∞ n=0

Ng,nq^n+g−1= ∞ n=0



g

|k|=n+g−1 g−1 i=1

ki





d|ki

d







q^n+g−1

=g ∞ n=0





|k|=n+g−1 g−1 i=1



ki

d|ki

dq^ki









=g



^∞

k=1

k

d|k

dq^k





g−1

=g DG2

_g−1

.

Proof of equations (3), (4), and (5). For these computations, we count the number of stable maps φ:D→Xof genusg withg+1 marked points that represent the homology classC=S+(g+n−1)F. The maps are constrained by requiring that the firstg−1 marked points are mapped to the pointsp¹,...,pg−1and that the image of the remaining two points must lie onγiandγj, respectively. We argue as before that Dhasgirreducible componentsD1,...,Dg, where eachDiis a genus-1 curve and it contains one marked point exceptDg, which contains two marked points. Moreover, the image of the two marked points on Dg must lie on the two loops γi andγj. It is easy to check that when the points pi,ai,...,di are in general positions, then no single section or fiber curve can pass through both one of the loopsγiand one of the pointspi. We can also verify directly the following lemma.

Lemma4.2. No fiber or section curve can pass through two differentγi’s unless they are(γ1,γ2)or(γ3,γ4).

No section curve can pass through bothγ1 andγ2. The only fiber curve passing throughγ1 andγ2isF12=F(a2,b1). Moreover,F12∩γ1=(a2,b1,c1,d1)andF12∩ γ2=(a2,b1,c2,d2).

Similarly, no fiber curve can pass through bothγ3andγ4.The only section curve passing throughγ3andγ4isS34=S(c4,d3). Moreover,S34∩γ3=(a3,b3,c4,d3)and S34∩γ4=(a4,b4,c4,d3).

From the lemma, the moduli space of stable maps is empty in all cases except (γi,γj)=(γ1,γ2)or(γ3,γ4)up to permutations. Therefore the corresponding family Gromov-Witten invariants vanish (proving equation (5)):

N_g,n¹³ =N_g,n¹⁴ =N_g,n²³ =N_g,n²⁴ =0.

Next we computeN_g,n³⁴.Letφbe any stable map in the corresponding moduli space.

We haveφ(Dg)=S34by the above lemma. Moreover, the restriction ofφtoDg is

an isomorphism andφmust send the two marked points onDg to(a3,b3,c4,d3)and (a4,b4,c4,d3). It is not difficult to check that the orientation of the moduli space is compatible with the one induced from(γ3,γ4). The calculations for the contribution from the otherDi’s are identical with the earlier computation, and we obtain

∞ n=0

N_g,n³⁴q^n+g−1=



^∞

k=1

k

d|k

dq^k





g−1

= DG2

_g−1

, thus proving equation (4).

Now we computeN_g,n¹². From the lemma again we haveφ(Dg)=F12for any stable mapφin the corresponding moduli space. Let us denote the degree ofφrestricted to Dgbyk0.Then the number of possible(φ:Dg→F12)is given by

d|k0das before.

The image of the two marked points onDg are(a2,b1,c1,d1)and(a2,b1,c2,d2). Sinceφis an unbranched covering map, there arek⁰choices for the location of each marked point. There are then a total ofk₀²

d|k0d different choices associated with the componentDg.

There are g−1 choices for which curve D¹,...,Dg−1 is mapped to a section.

After making such a choice, we may assume without loss of generality thatDg−1is mapped to the section curve passing throughpg−1(since we can relabel the points).

Then, sinceφ is an isomorphism onDg−1, the location of the marked point on it is determined. The analysis for the other fibers is the same as before, and the issue of the orientation is the same as above. In conclusion, we have

N_g,n¹² =(g−1)

k:|k|=n+g−1



k0²

d|k0

d

g−2 i=1

ki

d|k_i

d



,

where the summation is over (g−1)-tuples k = {k0,...,kg−2} with ki > 0 and

|k| =_g−2

i=0ki=g+n−1. Summing overn, we prove equation (3):

∞ n=0

N_g,n¹²q^n+g−1=(g−1)



^∞

k=1

k²

d|k

dq^k







^∞

k=1

k

d|k

dq^k





g−2

=(g−1) D²G²

DG²_g−2

=D DG2

_g−1 .

Proof of equation (2). Suppose that φ:D→Xis a stable map of genusgwith g+2 marked points, and suppose that it represents the homology class C=S+ (g+n−1)F. We argue as before thatDhas girreducible componentsD1,...,Dg. The curves D¹,...,Dg−2 each have one marked point mapping to p¹,...,pg−2, respectively. The curves Dg−1 and Dg contain the remaining four marked points which must lie on the four loops γ1,...,γ4. The image of eachDi must either be

a section curve or a fiber curve. By the previous lemma, the only fiber curve that intersects two γi’s must be F12, and it intersects γ1 and γ2 at (a2,b1,c1,d1) and (a2,b1,c2,d2), respectively. Similarly, the only section that intersects twoγ’s isS34, and it must intersect γ3 and γ4 at (a3,b3,c4,d3) and (a4,b4,c4,d3), respectively.

Hence the images ofDg−1andDg must beF¹²andS³⁴, respectively. Fori≤g−2, we haveφ(Di)=Fsi.Let the degree ofφ onDi beki, where 1≤i≤g−1.Then ki =g+n−1. The number of choices of (φ : Di →Fsi) is again given by d|k_id.For i < g−1, there are ki choices of the location of the marked point on Di.Fori=g−1,there are two marked points onDg−1and each haskg−1possible locations. Again it is not difficult to check that such maps each contribute+1 to the Gromov-Witten invariants for the twistor family (cf. [1]). In conclusion, we have

N_g,n^FLS=

k:|k|=n+g−1



k²_g−1

d|k_g−1

d



^g−

2 i=1



ki

d|ki

d



,

and so, summing overn, we have ∞

n=0

N_g,n^FLSq^n+g−1=



^∞

k=1

k

d|k

dq^k





g−2

^∞

k=1

k²

d|k

dq^k





= DG2

_g−1 D²G2

,

proving equation (2).

Appendix

Proof of Theorem 2.1. First we determine a more explicit formulation of the map

#0:ᏹg,C→Pic⁰(X)= H¹(X,R) H¹(X,Z).

Throughout this section, we abuse notation and refer to a stable mapf :→X by its image so that when we say ∈ᏹg,C, we mean the curve⊂Xgiven by the image of the mapf :→X. By definition,#0()=ᏻ(−0). We wish to get a 1-form representative forᏻ(−0). First, choose aC^∞trivialization of the bundleᏻ(−0). Lets0 ands be defining sections ofᏻ(0)andᏻ(). Then, via the trivialization,h=s0/s is a nonvanishingC^∞ function on X^o=X− {0∪}. Define an integrable real 1-formahby

ah= 1 2πi

∂h¯ h −∂h

h

.

The form ah defines a real 1-current, and let a₀ be the 1-form representing the harmonic projection of ah. We wish to define a map to H¹(X,R)/H¹(X,Z)using a₀, and we need the following lemma.

LemmaA.1. The harmonic 1-forma₀is independent of the choices ofs0,s, and the trivialization up to translation by an integral harmonic 1-form.

Proof. Let h=s₀/s be defined using a possibly different trivialization. Then h=ghfor someC^∞mapg:X→C^∗, and so

ah−ah = 1 2πi

∂g¯ g −∂g

g

=ag

is a smooth 1-form. We need to show that the harmonic projection ofag is integral.

Writeg=e^tθ, wheret:X→R andθ:X→S¹⊂C areC^∞. A direct computation shows that

ag=d^ct+ 1

2πiθ⁻¹dθ=d^∗(tω)+ 1

2πiθ⁻¹dθ,

where the last equality uses the Kähler identities, and so we see that the harmonic projection ofag is the integral class(1/2πi)θ⁻¹dθ.

The lemma shows that

→a₀ determines a well-defined map

ᏹg,C→H¹(X,R) H¹(X,Z). This is the same map as#0since:

(1) ifis linearly equivalent to⁰, thena₀≡0; and

(2) the sum of divisors corresponds to the sum of forms, that is,a₂⁺₀ =a₀+a₀. The first property holds since, if∼0, thenhcan be chosen to be a meromorphic function and soah≡0. The second property follows fromahh =ah+ah.

To prove Theorem 2.1, we need to show that#^∗₀(γ )˜ =ft_∗ev^∗([γ]^∨). Recall thatγ is a loop inX,[γ]^∨is the Poincaré dual of the 1-cycle[γ], and ˜γ∈H¹(Pic⁰(X),Z)is the natural class associated to [γ] arising from the identification Pic⁰(X) ∼= H¹(X,R)/H¹(X,Z).

We use the general correspondence between elements ofH¹(M,Z)and homotopy classes of mapsM →S¹. One can get a circle-valued function onM from a class φ∈H¹(M;Z)by choosing a base pointx0∈Mand definingfφ:M→R/Z by

fφ(x)=

4_x0^x φmod Z,

where4_x^x₀is a path fromx0tox. Sinceφis an integral class,fφdoes not depend on the choice of the path (mod Z).