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 ${\mathsf{Hilb}}^d\left(S\right)$ be the Hilbert scheme of $d$ points of $S$. In the case of elliptically fibered K3 surfaces $S\to {\mathbb{P}}^1$, we calculate genus-0 Gromov–Witten invariants of ${\mathsf{Hilb}}^d\left(S\right)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration ${\mathsf{Hilb}}^d\left(S\right)\to {\mathbb{P}}^d$. 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 ${\mathsf{Hilb}}^d\left(S\right)$ 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 two 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 ${\mathsf{Hilb}}^d\left(S\right)$ 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 nontrivial case ${\mathsf{Hilb}}^2\left(S\right)$. As a corollary, we find that the full genus-0 Gromov–Witten theory of ${\mathsf{Hilb}}^2\left(S\right)$ in primitive classes is governed by Jacobi forms.

We present two applications. A conjecture relating genus-1 invariants of ${\mathsf{Hilb}}^d\left(S\right)$ to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.