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

TL;DR

This paper refines previous results on the Voevodsky nilpotence conjecture by employing algebraic equivalence, expanding the scope to include K"uchle fourfolds, Sextic del Pezzo surfaces, and Fano fourfolds of K3 type.

Contribution

It advances the understanding of the Voevodsky nilpotence conjecture by replacing the nilpotence relation with algebraic equivalence in various geometric contexts.

Findings

Refined results for quadric fibrations and intersections of quadrics.

Extended the conjecture to K"uchle fourfolds and Fano fourfolds of K3 type.

Demonstrated the applicability of algebraic equivalence in these cases.

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.