Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface

Abstract

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let $S$ be a K3 surface and let $\mathsf{Hilb}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3 surfaces $S \to \mathbb{P}^1$, we calculate genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $\mathsf{Hilb}^d(S) \to \mathbb{P}^d$. The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form. We also prove results for genus $0$ Gromov-Witten invariants of $\mathsf{Hilb}^d(S)$ for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of $2$ points of $\mathbb{P}^1 \times E$, where $E$ is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on $\mathsf{Hilb}^d(S)$ with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of $S$. We prove the conjecture in the first non-trivial case $\mathsf{Hilb}^2(S)$. As a corollary, we find that the full genus $0$ Gromov-Witten theory of $\mathsf{Hilb}^2(S)$ in primitive classes is governed by Jacobi forms. We present two applications. A conjecture relating genus $1$ invariants of $\mathsf{Hilb}^d(S)$ to the Igusa cusp form was proposed in joint work with R. Pandharipande in \cite{K3xE}. Our results prove the conjecture in case $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through $2$ general points.