# About Chow groups of certain hyperk\"ahler varieties with non-symplectic automorphisms

## TL;DR

This paper investigates the action of non-symplectic automorphisms on the Chow groups of hyperk"ahler varieties, providing partial proofs of conjectures and implications for quotient varieties.

## Contribution

It proves the predicted behavior of certain Chow group pieces under automorphisms for Hilbert schemes of K3 surfaces, supporting Beauville's and Bloch-Beilinson's conjectures.

## Key findings

- Confirmed invariance of specific Chow group pieces under automorphisms.
- Established that certain Chow groups contain no non-trivial invariant cycles.
- Derived consequences for the Chow ring of quotient varieties.

## Abstract

Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms. This has consequences for the Chow ring of the quotient $X/G$.