A Variational Tate Conjecture in crystalline cohomology
Matthew Morrow
Source 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.
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