Algebraic cycles and EPW cubes

TL;DR

This paper investigates the action of an anti-symplectic involution on the Chow groups of certain hyperk"ahler sixfolds called double EPW cubes, providing evidence for Beauville's splitting conjecture and analyzing the Chow ring of their quotients.

Contribution

It verifies part of the Bloch-Beilinson conjecture for a family of hyperk"ahler sixfolds known as double EPW cubes, linking involution actions to Chow group decompositions.

Findings

Partial verification of Beauville's splitting property for double EPW cubes

Description of the Chow ring structure of the quotient EPW cubes

Implications for the Bloch-Beilinson conjectures in this setting

Abstract

Let $X$ be a hyperk\"ahler variety with an anti-symplectic involution $\iota$. According to Beauville's conjectural "splitting property", the Chow groups of $X$ should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how $\iota$ should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a $19$-dimensional family of hyperk\"ahler sixfolds that are "double EPW cubes" (in the sense of Iliev-Kapustka-Kapustka-Ranestad). This has interesting consequences for the Chow ring of the quotient $X/\iota$, which is an "EPW cube" (in the sense of Iliev-Kapustka-Kapustka-Ranestad).