The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$

TL;DR

This paper proves the Tate conjecture for certain Gushel-Mukai varieties of dimensions 4 and 6 over fields of characteristic p ≥ 5, extending known results and employing methods analogous to those used for K3 surfaces.

Contribution

It establishes the Tate conjecture for GM fourfolds and sixfolds in characteristic p ≥ 5, using reduction techniques and properties of generalized partners.

Findings

Proves Tate conjecture for GM sixfolds in characteristic p ≥ 5.

Reduces Tate conjecture for GM fourfolds to the sixfold case.

Shows GM sixfolds have no nonzero global vector fields.

Abstract

We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic $p$. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic $p\geq 5$. In the case of GM sixfolds, we follow the method used by Madapusi Pera in his proof of the Tate conjecture for K3 surfaces. As input for this, we prove a number of basic results about GM sixfolds, such as the fact that there are no nonzero global vector fields. For GM fourfolds, we prove the Tate conjecture by reducing it to the case of GM sixfolds by making use of the notion of generalised partners plus the fact that generalised partners in characteristic 0 have isomorphic Chow motives in the middle degree. Several steps in the proofs rely on results in characteristic 0 that are proven our paper "Algebraic cycles on Gushel-Mukai varieties", \'Epijournal G\'eom\'etrie Alg\'ebrique, Volume sp\'ecial en l'honneur de Claire Voisin, 2024.