Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

Abstract

In the first part of this paper we will prove the Voevodsky’s nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel–Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky’s nilpotence conjecture for generic Gushel–Mukai fourfolds containing a $\tau$-plane Gr(2, 3) and for ordinary Gushel–Mukai fourfolds containing a quintic del Pezzo surface.