Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz (1,1) theorem

摘要

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for k a perfect field of characteristic p, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over k[t] lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with Q(p)-coefficients.