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Non-algebraic geometrically trivial cohomology classes over finite fields

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Federico Scavia
  • Fumiaki Suzuki

Research Organisations

External Research Organisations

  • University of Paris Sorbonne North

Details

Original languageEnglish
Article number109964
JournalAdvances in Mathematics
Volume458
Issue numberA
Early online date7 Oct 2024
Publication statusPublished - Dec 2024

Abstract

We give the first examples of smooth projective varieties $X$ over a finite field $\mathbb{F}$ admitting a non-algebraic torsion $\ell$-adic cohomology class of degree $4$ which vanishes over $\overline{\mathbb{F}}$. We use them to show that two versions of the integral Tate conjecture over $\mathbb{F}$ are not equivalent to one another and that a fundamental exact sequence of Colliot-Th\'el\`ene and Kahn does not necessarily split. Some of our examples have dimension $4$, and are the first known examples of fourfolds with non-vanishing $H^{3}_{\text{nr}}(X,\mathbb{Q}_{2}/\mathbb{Z}_{2}(2))$.

Keywords

    math.AG, 14C25, 14G15, 55R35, Algebraic cycles, Integral Tate conjecture, Unramified cohomology

ASJC Scopus subject areas

Cite this

Non-algebraic geometrically trivial cohomology classes over finite fields. / Scavia, Federico; Suzuki, Fumiaki.
In: Advances in Mathematics, Vol. 458, No. A, 109964, 12.2024.

Research output: Contribution to journal › Article › Research › peer review

Scavia F, Suzuki F. Non-algebraic geometrically trivial cohomology classes over finite fields. Advances in Mathematics. 2024 Dec;458(A):109964. Epub 2024 Oct 7. doi: 10.1016/j.aim.2024.109964, 10.48550/arXiv.2206.12732
Scavia, Federico ; Suzuki, Fumiaki. / Non-algebraic geometrically trivial cohomology classes over finite fields. In: Advances in Mathematics. 2024 ; Vol. 458, No. A.
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