Local GW Invariants of Elliptic Multiple Fibers

We use simple geometric arguments to calculate the dimension zero local Gromov-Witten invariants of elliptic multiple fibers. This completes the calculation of all dimension zero GW invariants of elliptic surfaces with pg > 0. Let X be a Kähler surface with pg > 0. By the Enriques-Kodaira classification (cf. [BHPV]), its minimal model is a K3 or Abelian surface, a surface of general type or an elliptic surface. Each holomorphic 2-form α on X defines an almost complex structure Jα = (Id+ JKα) J (Id+ JKα). (0.1) Here, J is the complex structure on X and the endomorphism Kα of TX is defined by the formula 〈u,Kαv〉 = α(u, v) where 〈 , 〉 is the Kähler metric. This Jα satisfies : Lemma 0.1 ([L]). If f is a Jα-holomorphic map that represents a nontrivial (1,1) class then its image lies in the support of the zero divisor Dα of α and f is, in fact, holomorphic. The Gromov-Witten invariant GWg,n(X,A) is a (virtual) count of holomorphic maps representing the class A. In particular, the invariant GWg,n(X,A) vanishes unless A is a (1,1) class since every holomorphic map represents (1,1) class. Note that each canonical divisor D of X is a zero divisor of a holomorphic 2-form. Lemma 0.1 thus shows that the GW invariant is a sum GWg,n(X,A) = ∑ GW loc g,n(Dk, Ak) over the connected components Dk of the canonical divisor D of local invariants that counts the contribution of maps whose image lies in Dk. It follows that the GW invariants of minimal K3 or Abelian surfaces are trivial except possibly for the trivial homology class because their canonical divisors are trivial. The local GW invariants have a universal property. If X is a minimal surface of general type with a smooth canonical divisor D then the local invariants associated with D, and hence GW invariants, are determined by the normal bundle of D — in fact, there exists a universal function of c1 and c2 that gives the GW invariants of X (cf. Section 7 of [LP]). If π : X → C is a minimal elliptic surface with pg > 0, after suitable deformation, we can assume X has a canonical divisor of the form ∑