CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION

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$\text{CH}_{1}$-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$\text{CH}_{0}$-group and (b) algebraically coisotropic subvarieties of a holomorphic symplectic variety which is a moduli space of stable objects in${\mathcal{A}}_{Y}$.