Family Gromov-Witten invariants for Kähler surfaces

Abstract

We prove a structure theorem for the Gromov-Witten invariants of compact Kähler surfaces with geometric genus p g > 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 Kähler surfaces has focused on rational and ruled surfaces, which have geometric genus p g = 0. This paper focuses on surfaces with p g > 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 Kähler surface X and a canonical divisor D ∈ |K X |, this lemma shows that the complex structure J on X can be perturbed to a non-integrable almost complex structure J D with the property the image of all J D -holomorphic maps lies in the support of D. This immediately gives some striking vanishing theorems for the GW invariants of Kähler 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 GW g,n (X, A) = GW loc g,n (D k , A k ) over the connected components D k of D of "local invariants" that count the contribution of maps whose image lies in or (after perturbing to a generic moduli space) near D k . 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 Kähler structure on X and choose a canonical divisor D so that all components of D are smooth. With this assumption, the restriction of K X to each component D k of multiplicity m k is the normal bundle N k to D k , and N m k +1 k = K D k , * partially supported by the N.S.F.

Low degree GW invariants of surfaces II

“…In [4], using twisted family of K3 surfaces Bryan-Leung introduced the family GW-invariants of a K3 surface. Later, by deforming almost complex structures of surfaces Lee introduced the family GWinvariants of surfaces with p g > 0 [12]. For the case of K3 surfaces, Lee showed that the two versions of family GW-invariants coincide.…”

Sum formulas for local Gromov-Witten invariants of spin curves

“…Proof. This proof is similar to that of Lemma 3.2 in [LP1]. Use the same notation α for the real part of the holomorphic 2-form α on N D .…”

Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

“…The index relations between simple J -holomorphic curves and their multiple covers make the following conjecture plausible: 1 CONJECTURE 1.1. On any closed symplectic manifold .M; !/ of real dimension at least 4, there exists a Baire subset J reg in the space of smooth !-tame almost complex structures such that for all J 2 J reg , every closed, connected, and simple J -holomorphic curve with deformation index 0 is super-rigid.Some special cases of this conjecture have been proved previously by Lee and Parker [16,17] and Eftekhary [5]. The techniques used in the present paper are related to those of [16,17], which also play a role in the announced solution by Ionel and Parker to the Gopakumar-Vafa conjecture [13].For an unbranched cover of a simple curve, the super-rigidity condition is equivalent to the usual notion of Fredholm regularity, and our main result (stated as Theorem 1.3 below) is that this can always be achieved by choosing J generically.…”