Topology Seminar: Classical and noncommutative Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds

In a seminal paper, Voevodsky introduced the smash-nilpotence equivalence relation on the group of algebraic cycles on a smooth projective variety. He also conjectured that the nilpotence equivalence corresponds to the classical numerical equivalence on cycles. More recently, Bernardara, Marcolli and Tabuada defined a noncommutative version of this conjecture for smooth and proper dg categories. They proved the equivalence between the classical conjecture and their noncommutative version for the unique enhancement of the derived category of perfect complexes on a smooth projective k-scheme.

The aim of this talk is to prove Voevodsky's conjecture for cubic fourfolds and generic Gushel-Mukai fourfolds. Then, we deduce the noncommutative version of this conjecture for the K3 subcategory appearing in the semiorthogonal decomposition of the derived category of perfect complexes on a cubic fourfold and on a generic GM fourfold, introduced by Kuznetsov and Kuznetsov-Perry. Finally, we apply this result to deduce Voevodsky's conjecture for special classes of GM fourfolds. This is a joint work with Mattia Ornaghi.