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

Oberdieck, Georg

doi:10.2140/gt.2024.28.3779

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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

Georg Oberdieck

Geometry & Topology 28 (2024) 3779–3868

DOI: 10.2140/gt.2024.28.3779

Abstract

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.

Keywords

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

Mathematical Subject Classification

Primary: 11F50, 14J28, 14J42, 14N35

References

Publication

Received: 27 May 2022

Revised: 12 July 2023

Accepted: 18 August 2023

Published: 20 December 2024

Proposed: Jim Bryan

Seconded: Richard P Thomas, Mark Gross

Authors

Georg Oberdieck
Department of Mathematics KTH Royal Institute of Technology Stockholm Sweden

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Volume 28, issue 8 (2024)