Counting rational curves on K3 surfaces with finite group actions

TL;DR

This paper investigates the action of finite groups on the cohomology of Hilbert schemes of points on K3 surfaces, linking group actions to modular forms and providing conditions for contributions of certain curve orbits.

Contribution

It describes how finite group actions influence the Hodge structure of Hilbert schemes on K3 surfaces and relates these to cusp forms and Jacobians, offering new insights into symplectic group actions.

Findings

Group actions induce trace formulas involving cusp forms.

The Hodge structure of Hilbert schemes relates to Jacobians under group actions.

A criterion is provided for certain curve orbits to not contribute to the representation.

Abstract

G\"ottsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.