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

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 $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.