Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz $(1, 1)$ theorem. (May 2019)

• Record Type: Journal Article
• Authors: Lazda, Christopher; Pál, Ambrus
• Is Part Of: Compositio mathematica. Volume 155:Number 5(2019)
• Page Start: 1025
• Page End: 1045
• Publication Date: 2019-05
• Subjects: 14C22 (primary), -- 14F30, -- 11G25 (secondary); Picard groups, -- crystalline cohomology, -- semistable reduction, -- Tate conjecture; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X19007164
• Languages: English
• ISSNs: 0010-437X

Abstract

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.