Geometry & Topology
- Geom. Topol.
- Volume 22, Number 1 (2018), 323-437.
Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface
We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.
Let be a K3 surface and let be the Hilbert scheme of points of . In the case of elliptically fibered K3 surfaces , we calculate genus-0 Gromov–Witten invariants of , which count rational curves incident to two generic fibers of the induced Lagrangian fibration . 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 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 , where is an elliptic curve.
Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of . We prove the conjecture in the first nontrivial case . As a corollary, we find that the full genus-0 Gromov–Witten theory of in primitive classes is governed by Jacobi forms.
We present two applications. A conjecture relating genus-1 invariants of to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when . Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.
Geom. Topol., Volume 22, Number 1 (2018), 323-437.
Received: 11 November 2015
Revised: 1 March 2017
Accepted: 30 March 2017
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 11F50: Jacobi forms
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