A Variational Tate Conjecture in crystalline cohomology

Abstract

Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends cohomologically to the entire family. This is a characteristic $p$ analogue of Grothendieck’s Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension; the former result can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.