K3 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 ${\mathrm{CH}}_1$-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 ${\mathcal{A}}_Y$.

29 pages. Comments are welcome!

References in corpus (2)

• Semiorthogonal decompositions in algebraic geometry
• Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case

Cited by in corpus (1)

• Chow groups of surfaces of lines in cubic fourfolds