A refinement of some previous results of Bernardara-Marcolli-Tabuada and Ornaghi-Pertusi

Abstract: The Voevodsky nilpotence conjecture was proved by Bernardara-Marcolli-Tabuada for certain quadric fibrations, intersections of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, and Moishezon manifolds, and later by Ornaghi-Pertusi for certain cubic fourfolds and Gushel-Mukai fourfolds. In this paper we refine these results by using the algebraic equivalence relation instead of the nilpotence equivalence relation. Along the way, we address also certain cases of K\"uchle fourfolds, families of Sextic del Pezzo surfaces and families of Fano fourfolds of K3 type.