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.
- Pub Date:
- August 2014
- Mathematics - Algebraic Geometry;
- Mathematics - K-Theory and Homology
- The former Section 3.3, containing a sketched treatment of line bundles with Q_p-coefficients, has been removed as an error was found. This affects the validity of none of the main results, but necessitates giving a different proof of the application to the Tate Conjecture. Other minor changes