TY - UNPB

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

AU - Scavia, Federico

AU - Suzuki, Fumiaki

N1 - 23 pages, comments welcome, v3. final version, to appear in Advances in Mathematics

PY - 2024/9/23

Y1 - 2024/9/23

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

M3 - Preprint

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

ER -