A Variational Tate Conjecture in crystalline cohomology

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 char...

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Bibliographic Details
Published inarXiv.org
Main Author Morrow, Matthew
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.03.2015
Subjects
Crystal structure
Crystallinity
Fields (mathematics)
Homology
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Summary: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.
ISSN:2331-8422