Counting Curves in Elliptic Surfaces by Symplectic Methods

Abstract

We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from [IP3]. In particular, as in [BL1], these calculations also confirm the well-known Yau-Zaslow Conjecture [YZ] for primitive classes in K3 surfaces. In [L] we introduced “family GW invariants” for Kahler surfaces with pg > 0. Since these invariants are defined by using non-compact family of almost Kahler structures, we can easily extend several existing techniques for calculating GW invariants to the family GW invariants. In particular, the ‘TRR formula’ applies to the family invariants, and at least some special cases of the symplectic sum formula [IP3] apply, with appropriate minor modifications to the formula. Those formulas enable us to enumerate the curves in the elliptic surfaces E(n) for the class A= section plus multiples of the fiber. Theorem 0.1 Let E(n) → P1 be a standard elliptic surface with a section of self-intersection −n. Denote by S and F the homology class of the section and the fiber. Then the genus g family GW invariants for the classes S + dF are given by the generating function ∑ d≥0 GWH S+dF,g(E(n)) ( pt ) t = ( tG′(t) )g ∏ d≥1 ( 1 1− td )12n (0.1) where G(t) = ∑ d≥1 σ(d) td and σ(d) = ∑ k|d k . Bryan and Leung ([BL1],[BL2]) defined family invariants for K3 and Abelian surfaces by using the Twistor family. They used algebraic methods to show (0.1) for GW invariants of the rational elliptic surface E(1) and for family invariants of E(2) = K3 surfaces. For K3, that confirms the famous Yau-Zaslow Conjecture [YZ] for those cases when the homology class A is primitive. They also pointed out that one can define family invariants of E(n) for n ≥ 3 using compact