HPD-invariance of the Tate conjecture(s)

Abstract

We prove that the Tate conjecture (and its variants) is invariant under homological projective duality. As an application, we obtain a proof, resp. an alternative proof, of the Tate conjecture (and of its variants) in the new case of linear sections of determinantal varieties, resp. in the old cases of Pfaffian cubic fourfolds and complete intersections of quadrics. In addition, we generalize the Tate conjecture (and its variants) from schemes to stacks and prove this generalized conjecture(s) for low-dimensional root stacks and low-dimensional (twisted) orbifolds.