Arkiv för Matematik

Volume 61 (2023)

Number 2

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

Pages: 475 – 493



Sailun Zhan (Department of Mathematical Sciences, Binghamton University, Binghamton, New York, U.S.A.)


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 $S$.


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

2010 Mathematics Subject Classification

14G17, 14J15, 14J50

Received 3 February 2022

Received revised 1 October 2022

Accepted 24 February 2023

Published 13 November 2023