G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

Abstract

For an Abelian surface A with a symplectic action by a finite group G, one can define the partition function for G-invariant Hilbert schemes $$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$We prove the reciprocal \(Z_{A,G}^{-1}\) is a modular form of weight \(\frac{1}{2}e(A/G)\) for the congruence subgroup \(\Gamma _{0}(|G|)\) and give explicit expressions in terms of eta products. Refined formulas for the \(\chi _{y}\)-genera of \(\text {Hilb}(A)^{G}\) are also given. For the group generated by the standard involution \(\tau : A \rightarrow A\), our formulas arise from the enumerative geometry of the orbifold Kummer surface \([A/\tau ]\). We prove that a virtual count of curves in the stack is governed by \(\chi _{y}(\text {Hilb}(A)^{\tau })\). Moreover, the coefficients of \(Z_{A, \tau }\) are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.