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

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 ≥ 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 middle degree. Several steps in the proofs rely on results in characteristic 0 that are proven in our paper [19].

On the Tate Conjecture in Codimension One for Varieties with $h^{2, 0} = 1$ over Finite Fields

“…We remark that the theorem above is only nontrivial when i = n. Recently, the theorem above was also obtained by Fu-Moonen [25] for p ≥ 5 but for all k finitely generated over its prime field. Their methods are very different from ours, and we refer the reader to §10 for a comparison.…”

“…Finally, we discuss how our approach is compared to that of the recent paper [25] by Fu and Moonen. The key difference is that they do solve the "local Schottky problem".…”

Some motivic properties of Gushel–Mukai sixfolds

“…It is worth mentioning the recent papers [22,23], which have appeared on arXiv some months after ours, that continue the program of studying the Chow motives and algebraic cycles on Gushel-Mukai varieties, with some remarkable new results.…”