A Variational Tate Conjecture in crystalline cohomology

TL;DR

This paper formulates and proves a characteristic p analogue of Grothendieck's Variational Hodge Conjecture within crystalline cohomology, providing new tools to relate cycle extension problems to the Tate conjecture.

Contribution

It introduces a Variational Tate Conjecture in crystalline cohomology, proves it for divisors, and extends the approach to higher codimension cycles, linking to the Tate conjecture over finite fields.

Findings

Proved the conjecture for divisors.

Established an infinitesimal variant for higher codimension cycles.

Reduced the Tate conjecture for divisors to the case of surfaces.

Abstract

Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate 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. This can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.