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.

Fully revised version. New material: A second proof of main result using a Kummer K3, and a conjecture for hyperelliptic curve counts on K3 surfaces. 117 pages. Compiling problem in last version

References in corpus (1)

• The Katz-Klemm-Vafa conjecture for K3 surfaces

Cited by in corpus (9)

• Holomorphic anomaly equations and the Igusa cusp form conjecture
• Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
• Quasimaps to moduli spaces of sheaves on a $K3$ surface
• Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
• Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds
• Elliptic genus of singular algebraic varieties and quotients
• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Quasimaps to moduli spaces of sheaves
• Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence