TY - JOUR

T1 - Non-algebraic geometrically trivial cohomology classes over finite fields

AU - Scavia, Federico

AU - Suzuki, Fumiaki

N1 - Publisher Copyright: © 2024 The Author(s)

PY - 2024/12

Y1 - 2024/12

N2 - 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 $H^{3}_{\text{nr}}(X,\mathbb{Q}_{2}/\mathbb{Z}_{2}(2))$.

AB - 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 $H^{3}_{\text{nr}}(X,\mathbb{Q}_{2}/\mathbb{Z}_{2}(2))$.

KW - math.AG

KW - 14C25, 14G15, 55R35

KW - Algebraic cycles

KW - Integral Tate conjecture

KW - Unramified cohomology

UR - http://www.scopus.com/inward/record.url?scp=85205715200&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2024.109964

DO - 10.1016/j.aim.2024.109964

M3 - Article

VL - 458

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - A

M1 - 109964

ER -