Algebraic cycles on some special hyperk\"ahler varieties

TL;DR

This paper provides examples of hyperk"ahler varieties with non-symplectic automorphisms where the group action on Chow groups aligns with Bloch's conjecture, revealing interesting Chow-theoretic properties of their quotients.

Contribution

It introduces new examples of hyperk"ahler varieties with specific automorphism groups and analyzes their Chow groups and quotient properties, supporting conjectures in algebraic geometry.

Findings

Group actions on Chow groups match Bloch's conjecture predictions.

Quotients are Calabi-Yau varieties with special Chow-theoretic properties.

Supports a strong version of the Beauville-Voisin conjecture.

Abstract

This note contains some examples of hyperk\"ahler varieties $X$ having a group $G$ of non-symplectic automorphisms, and such that the action of $G$ on certain Chow groups of $X$ is as predicted by Bloch's conjecture. The examples range in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ is a Calabi-Yau variety which has interesting Chow-theoretic properties; in particular, the variety $Y$ satisfies (part of) a strong version of the Beauville-Voisin conjecture.