Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

Abstract

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$k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$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$\mathbb{Q}_{p}$-coefficients.