On the Chow groups of certain EPW sextics

TL;DR

This paper investigates the action of a specific involution on the Chow groups of a Hilbert scheme of a K3 surface and explores implications for the Chow ring of associated EPW sextics, supporting Bloch's conjecture.

Contribution

It demonstrates that the involution acts as expected on Chow groups of the Hilbert scheme, with consequences for the Chow ring of related EPW sextics.

Findings

Involution acts as predicted on certain Chow groups.

Results support Bloch's conjecture for these varieties.

Implications for the Chow ring of EPW sextics.

Abstract

This note is about the Hilbert square $X=S^{[2]}$, where $S$ is a general $K3$ surface of degree $10$, and the anti-symplectic birational involution $\iota$ of $X$ constructed by O'Grady. The main result is that the action of $\iota$ on certain pieces of the Chow groups of $X$ is as expected by Bloch's conjecture. Since $X$ is birational to a double EPW sextic $X^\prime$, this has consequences for the Chow ring of the EPW sextic $Y\subset{\mathbb{P}}^5$ associated to $X^\prime$.