A structure theorem for the Gromov-Witten invariants of Käler surfaces

Abstract

We prove a structure theorem for the Gromov-Witten invariants of compact Kahler surfaces with geometric genus pg > 0. Under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components, the theorem shows that the GW invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic Euler characteristic of the surface. We compute special cases of these universal functions. Much of the work on the Gromov-Witten invariants of Kahler surfaces has focused on rational and ruled surfaces, which have geometric genus pg = 0. This paper focuses on surfaces with pg > 0, a class that includes most elliptic surfaces and most surfaces of general type. In this context we prove a general “structure theorem” that shows (with one technical assumption) how the GW invariants are completely determined by the local geometry of a generic canonical divisor. The structure theorem is a consequence of a simple fact: the “Image Localization Lemma” of Section 3. Given a Kahler surface X and a canonical divisor D ∈ |KX |, this lemma shows that the complex structure J on X can be perturbed to a non-integrable almost complex structure JD with the property the image of all JD-holomorphic maps lies in the support of D. This immediately gives some striking vanishing theorems for the GW invariants of Kahler surfaces (see Section 3). More importantly, it implies that the Gromov-Witten invariant of X for genus g and n marked points is a sum GWg,n(X,A) = ∑ GW loc g,n(Dk, Ak) over the connected components Dk of D of “local invariants” that count the contribution of maps whose image lies in or (after perturbing to a generic moduli space) near Dk. These local invariants have not been previously defined. The proof of their existence relies on using non-integrable structures and geometric analysis techniques. Our structure theorem characterizes the local invariants by expressing them in terms of usual GW invariants of certain standard surfaces. For this we make the mild assumption that one can deform the Kahler structure on X and choose a canonical divisor D so that all components of D are smooth. With this assumption, the restriction of KX to each component Dk of multiplicity mk is the normal bundle Nk to Dk, and Nk k = KDk , ∗partially supported by the N.S.F.