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$K3$ CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 5 / September 2020
- Published online by Cambridge University Press:
- 05 November 2018, pp. 1601-1627
- Print publication:
- September 2020
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- Article
- Export citation
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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 
${\mathcal{A}}_{Y}$.
