Two coniveau filtrations and algebraic equivalence over finite fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916406415589376 |
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| author | Scavia, Federico Suzuki, Fumiaki |
| author_facet | Scavia, Federico Suzuki, Fumiaki |
| contents | 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-Thélène and Kahn. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08560 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Two coniveau filtrations and algebraic equivalence over finite fields Scavia, Federico Suzuki, Fumiaki Algebraic Geometry 14C25, 14G15, 55R35 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-Thélène and Kahn. |
| title | Two coniveau filtrations and algebraic equivalence over finite fields |
| topic | Algebraic Geometry 14C25, 14G15, 55R35 |
| url | https://arxiv.org/abs/2304.08560 |