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

## TL;DR

This paper proves Voevodsky's nilpotence conjecture for certain classes of cubic fourfolds and Gushel-Mukai fourfolds, utilizing noncommutative motives to extend the results to more specific cases.

## Contribution

It establishes the conjecture for broad classes of fourfolds and introduces noncommutative motive techniques for these proofs.

## Key findings

- Voevodsky's conjecture proven for smooth cubic fourfolds.
- Voevodsky's conjecture proven for generic Gushel-Mukai fourfolds.
- Extension of results to special Gushel-Mukai fourfolds containing specific 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 $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.