# On the Chow ring of certain Lehn-Lehn-Sorger-van Straten eightfolds

## TL;DR

This paper investigates the Chow ring structure of certain hyperk"ahler eightfolds with automorphisms, confirming predictions of the Bloch-Beilinson conjectures and revealing implications for intersection theory.

## Contribution

It demonstrates the automorphism actions on Chow groups align with conjectural predictions using finite-dimensional motives theory.

## Key findings

- Automorphism action on 0-cycle Chow groups matches conjectural predictions
- Anti-symplectic involution exhibits similar Chow group behavior
- Results impact understanding of intersection products in these hyperk"ahler varieties

## Abstract

We consider a $10$-dimensional family of Lehn-Lehn-Sorger-van Straten hyperk\"ahler eightfolds which have a non-symplectic automorphism of order $3$. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of $0$-cycles is as predicted by the Bloch-Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product in the Chow ring of these varieties.