Algebraic cycles and Lehn-Lehn-Sorger-van Straten eightfolds

TL;DR

This paper studies Lehn-Lehn-Sorger-van Straten eightfolds, providing an explicit formula for an anti-symplectic involution's action on zero-cycles, with implications for the Chow ring and Bloch-Beilinson conjectures.

Contribution

It offers a new explicit formula for the involution on Chow groups of these eightfolds when birational to Hilbert schemes of K3 surfaces.

Findings

Formula matches Bloch-Beilinson conjectures

Implications for Chow ring of the quotient

Provides explicit involution action on zero-cycles

Abstract

This article is about Lehn-Lehn-Sorger-van Straten eightfolds $Z$, and their anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement with the Bloch-Beilinson conjectures, and has some non-trivial consequences for the Chow ring of the quotient.