$G$-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 \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] 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 $\tau : A \to A$, 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.