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

Göttsche 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 $\sum^\infty_{n=0} \sum^\infty_{i=0} (-1)^i H^i (S^{[n]}, \mathbb{C} ) q^n$ 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 $S$.

Keywords

smooth projective surfaces, group representations, Hilbert schemes of points, generalized Kummer varieties

2010 Mathematics Subject Classification

14G17, 14J15, 14J50

Full Text (PDF format)

Received 3 February 2022

Received revised 1 October 2022

Accepted 24 February 2023

Published 13 November 2023