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

Part of: Cycles and subschemes (Co)homology theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 May 2019

Christopher Lazda  and
Ambrus Pál
Christopher Lazda
Affiliation:
Korteweg-de Vries Institute, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands email [email protected]
Ambrus Pál
Affiliation:
Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7 2AZ, UK email [email protected]
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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.

Keywords

Picard groupscrystalline cohomologysemistable reductionTate conjecture

MSC classification

Primary: 14C22: Picard groups
Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 11G25: Varieties over finite and local fields
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 1025 - 1045
Copyright
© The Authors 2019 

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