A Variational Tate Conjecture in crystalline cohomology
Matthew MorrowSorbonne Université, Paris, France
Given a smooth, proper family of varieties in characteristic , and a cycle on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of , whether extends cohomologically to the entire family. This is a characteristic 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 -adic Tate conjecture for divisors over finite fields to the case of surfaces.
Cite this article
Matthew Morrow, A Variational Tate Conjecture in crystalline cohomology. J. Eur. Math. Soc. 21 (2019), no. 11, pp. 3467–3511DOI 10.4171/JEMS/907