Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions

TL;DR

This paper studies the action of finite groups on the Hodge structures of Hilbert schemes of points on smooth projective surfaces, revealing connections to modular forms especially for K3 and abelian surfaces.

Contribution

It describes how finite group actions influence the Hodge numbers of Hilbert schemes and generalized Kummer varieties, linking geometric symmetries to modular forms.

Findings

Group actions induce trace formulas on Hodge structures.

Generating functions relate to modular forms for K3 and abelian surfaces.

Provides explicit formulas for Hodge number transformations under group actions.

Abstract

G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge pieces via point counting. Each element of $G$ gives a trace on $\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. In the case that $S$ is a K3 surface or an abelian surface, the resulting generating functions give some interesting modular forms when $G$ acts faithfully and symplectically on $S$.