Gromov–Witten Theory of $\text{K3} \times {\mathbb{P}}^1$ and Quasi-Jacobi Forms

Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov–Witten theory of $S \times {\mathbb{P}}^1$ in classes $(\beta,1)$ and $(\beta,2)$. 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 $S \times E$ in classes $(\beta,1)$ and $(\beta,2)$ 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 $(1,1,d)$.