A Variational Tate Conjecture in crystalline cohomology

2014/08/28 by Morrow, Matthew · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1408.6783

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 ℓ-adic Tate conjecture for divisors over finite fields to the case of surfaces.

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• F-isocrystals of Higher Direct Images of p-Divisible Groups
• An approach to the Tate conjecture for surfaces over a finite field

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