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

## TL;DR

This paper proves a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that certain line bundles lift if and only if their Chern classes do, with implications for algebraicity over function fields.

## Contribution

It establishes a semistable variational Tate conjecture for divisors using elementary methods and explores its consequences and limitations.

## Key findings

- Proves a semistable variational Tate conjecture for divisors.
- Shows line bundle lifting is equivalent to Chern class lifting.
- Provides a counterexample for the conjecture with dic coefficients.

## Abstract

In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that 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 $\mathbb{Q}_p$-coefficients.