## Geometry & Topology

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

Georg Oberdieck

#### 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 $Hilbd(S)$ be the Hilbert scheme of $d$ points of  $S$. In the case of elliptically fibered K3 surfaces $S→ℙ1$, we calculate genus-0 Gromov–Witten invariants of $Hilbd(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $Hilbd(S)→ℙd$. 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 $Hilbd(S)$ 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 two 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 $Hilbd(S)$ 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 nontrivial case $Hilb2(S)$. As a corollary, we find that the full genus-0 Gromov–Witten theory of $Hilb2(S)$ in primitive classes is governed by Jacobi forms.

We present two applications. A conjecture relating genus-1 invariants of $Hilbd(S)$ 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.

#### Article information

Source
Geom. Topol., Volume 22, Number 1 (2018), 323-437.

Dates
Revised: 1 March 2017
Accepted: 30 March 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513774915

Digital Object Identifier
doi:10.2140/gt.2018.22.323

Mathematical Reviews number (MathSciNet)
MR3720346

Zentralblatt MATH identifier
06805081

#### Citation

Oberdieck, Georg. Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface. Geom. Topol. 22 (2018), no. 1, 323--437. doi:10.2140/gt.2018.22.323. https://projecteuclid.org/euclid.gt/1513774915

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