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# 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

DOI: https://dx.doi.org/10.4310/ARKIV.2023.v61.n2.a9

#### Author

#### Abstract

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

Received 3 February 2022

Received revised 1 October 2022

Accepted 24 February 2023

Published 13 November 2023