Details
| Originalsprache | Englisch |
|---|---|
| Publikationsstatus | Elektronisch veröffentlicht (E-Pub) - 23 Sept. 2024 |
Abstract
When the base field F is finite, we show that the equality of the two filtrations over the algebraic closure F¯¯¯ has some consequences for algebraic equivalence for codimension-2 cycles over F. As an application, we prove that the third unramified cohomology group H3nr(X,Qℓ/Zℓ) vanishes for a large class of rationally chain connected threefolds X over F, confirming a conjecture of Colliot-Thélène and Kahn.
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2024.
Publikation: Arbeitspapier/Preprint › Preprint
}
TY - UNPB
T1 - Two coniveau filtrations and algebraic equivalence over finite fields
AU - Scavia, Federico
AU - Suzuki, Fumiaki
N1 - 31 pages, comments are welcome, v2. final version, to appear in Algebraic Geometry. arXiv admin note: substantial text overlap with arXiv:2206.12732
PY - 2024/9/23
Y1 - 2024/9/23
N2 - We extend the basic theory of the coniveau and strong coniveau filtrations to the $\ell$-adic setting. By adapting the examples of Benoist--Ottem to the $\ell$-adic context, we show that the two filtrations differ over any algebraically closed field of characteristic not $2$. When the base field $\mathbb{F}$ is finite, we show that the equality of the two filtrations over the algebraic closure $\overline{\mathbb{F}}$ has some consequences for algebraic equivalence for codimension-$2$ cycles over $\mathbb{F}$. As an application, we prove that the third unramified cohomology group $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ vanishes for a large class of rationally chain connected threefolds $X$ over $\mathbb{F}$, confirming a conjecture of Colliot-Th\'el\`ene and Kahn.
AB - We extend the basic theory of the coniveau and strong coniveau filtrations to the $\ell$-adic setting. By adapting the examples of Benoist--Ottem to the $\ell$-adic context, we show that the two filtrations differ over any algebraically closed field of characteristic not $2$. When the base field $\mathbb{F}$ is finite, we show that the equality of the two filtrations over the algebraic closure $\overline{\mathbb{F}}$ has some consequences for algebraic equivalence for codimension-$2$ cycles over $\mathbb{F}$. As an application, we prove that the third unramified cohomology group $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ vanishes for a large class of rationally chain connected threefolds $X$ over $\mathbb{F}$, confirming a conjecture of Colliot-Th\'el\`ene and Kahn.
KW - math.AG
KW - 14C25, 14G15, 55R35
U2 - 10.48550/arXiv.2304.08560
DO - 10.48550/arXiv.2304.08560
M3 - Preprint
BT - Two coniveau filtrations and algebraic equivalence over finite fields
ER -