Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

Oberdieck, Georg¹

¹ Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

Geometry & topology, Tome 28 (2024) no. 8, pp. 3779-3868

Résumé

We conjecture that the generating series of Gromov–Witten invariants of the Hilbert schemes of n points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus 0 and for at most three markings — for all Hilbert schemes and for arbitrary curve classes. In particular, for fixed n, the reduced quantum cohomologies of all hyperkähler varieties of K3 ⁡ [n]–type are determined up to finitely many coefficients.

As an application we show that the generating series of 2–point Gromov–Witten classes are vector-valued Jacobi forms of weight − 10, and that the fiberwise Donaldson–Thomas partition functions of an order-2 CHL Calabi–Yau threefold are Jacobi forms.

Détail

DOI : 10.2140/gt.2024.28.3779

Keywords: Gromov–Witten theory, K3 surfaces, Jacobi forms, Hilbert schemes of points, holomorphic anomaly equations

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