Gromov-Witten theory of $\mathrm{K3} \times \mathbb{P}^1$ and quasi-Jacobi forms

TL;DR

This paper computes the Gromov-Witten invariants of K3 surfaces times projective lines, revealing their structure as quasi-Jacobi forms and connecting them to Hilbert schemes and modular forms, thus advancing understanding in enumerative geometry.

Contribution

It explicitly solves the relative Gromov-Witten theory for K3 surfaces times P^1 in specific classes, proving a case of a conjecture and linking invariants to modular forms.

Findings

Generating series are quasi-Jacobi forms.

Gromov-Witten invariants relate to Hilbert scheme invariants.

Invariants of K3 surfaces times elliptic curves involve the Igusa cusp form.

Abstract

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 $(1,1,d)$.