# A remark on the Chow ring of some hyperk\"ahler fourfolds

## TL;DR

This paper investigates Voisin's conjecture on the Chow ring of hyperk"ahler varieties, focusing on the Fano variety of lines on a very general cubic fourfold, and explores whether certain subvarieties' classes inject into cohomology.

## Contribution

The paper provides new insights into Voisin's conjecture by analyzing the Chow ring of the Fano variety of lines on a very general cubic fourfold.

## Key findings

- Evidence supporting the conjecture for specific hyperk"ahler fourfolds
- Identification of subvarieties whose classes may lie in the conjectured subring
- Analysis of the injectivity of the cycle class map for these subvarieties

## Abstract

Let $X$ be a hyperk\"ahler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.