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  • Junliang Shen (a1) and Qizheng Yin (a2)


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 ${\mathcal{A}}_{Y}$ .



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