Gromov–Witten Theory of K3 × P1 and Quasi-Jacobi Forms.

Let S be a K3 surface with primitive curve class β⁠. We solve the relative Gromov–Witten theory of S × P¹ in classes (β,1) and (β,2)⁠. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus 0 Gromov–Witten invariants on the Hilbert scheme of points of S⁠. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let E be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of S × E in classes (β,1) and (β,2) are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type (1,1,d)⁠. [ABSTRACT FROM AUTHOR]