{
  "id": "2304.08560",
  "version": "v1",
  "published": "2023-04-17T18:58:12.000Z",
  "updated": "2023-04-17T18:58:12.000Z",
  "title": "Two coniveau filtrations and algebraic equivalence over finite fields",
  "authors": [
    "Federico Scavia",
    "Fumiaki Suzuki"
  ],
  "comment": "30 pages, comments are welcome. This is an expanded version of the second part of arXiv:2206.12732v1",
  "categories": [
    "math.AG"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-17T18:58:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "14G15",
      "55R35"
    ],
    "keywords": [
      "algebraic equivalence",
      "finite fields",
      "strong coniveau filtrations",
      "third unramified cohomology group",
      "large class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}