The enumerative geometry 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 Hilb(S) be the Hilbert scheme of d points of S. In case of elliptically fibered K3 surfaces S → P1, we calculate genus 0 Gromov-Witten invariants of Hilb(S), which count rational curves incident to two generic fibers of the induced Lagrangian fibration Hilb(S)→ Pd. The generating series of these invariants is the Fourier expansion of a product of Jacobi theta functions and modular forms, hence of a Jacobi form. The result is a generalization of the classical Yau-Zaslow formula which relates the number of rational curves on a K3 surface to the modular discriminant. We also prove results for genus 0 Gromov-Witten invariants of Hilb(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 P1 × E, where E is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on Hilb(S) with respect to primitive 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 Hilb(S). As a corollary, the full genus 0 Gromov-Witten theory of Hilb(S) in primitive classes is governed by Jacobi forms. We state three applications of our results. First, in joint work with R. Pandharipande, a conjecture counting the number of maps from a fixed elliptic curve to Hilb(S) is presented. The result, summed over all d, is expressed in terms of the reciprocal of a Siegel modular form, the Igusa cusp form χ10. Second, we give a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through 2 general points. Third, we discuss a relationship between the Jacobi forms appearing in curve counting on Hilb(S) and the moduli space of holomorphic symplectic varieties.