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Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface.(English)Zbl 07960664

Summary: 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 \(\operatorname{K3}^{[n]}\)-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.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties
11F50 Jacobi forms

Cite

References:

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