K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration

TL;DR

This paper investigates the relationship between K3 categories and 0-cycles on holomorphic symplectic varieties, proposing a conjecture and verifying it for certain rational curves in cubic fourfolds.

Contribution

It introduces a filtration on the CH_1 group of cubic 4-folds, conjectures a sheaf/cycle correspondence for associated K3 categories, and constructs Beauville-Voisin filtration and coisotropic subvarieties.

Findings

Verified the conjecture for sheaves supported on low degree rational curves.

Provided systematic constructions of Beauville-Voisin filtration on CH_0.

Constructed algebraically coisotropic subvarieties in moduli spaces of stable objects.

Abstract

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on the $\mathrm{CH}_1$-group of a cubic 4-fold $Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$ category $\mathcal{A}_Y$. This is a noncommutative analog of O'Grady's conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sheaves supported on low degree rational curves. Our method provides systematic constructions of (a) the Beauville-Voisin filtration on the $\mathrm{CH}_0$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli sace of stable objects in $\mathcal{A}_Y$.