{
  "id": "1112.1080",
  "version": "v3",
  "published": "2011-12-05T21:01:08.000Z",
  "updated": "2015-07-27T12:48:55.000Z",
  "title": "Remarks on motives of abelian type",
  "authors": [
    "Charles Vial"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "A motive over a field $k$ is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over $k$. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension $K/k$ and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow--Kuenneth decompositions for varieties that become isomorphic to an abelian variety after some field extension. Second, let $\\Omega$ be a universal domain containing $k$. We show that Murre's conjectures for motives of abelian type over $k$ reduce to Murre's conjecture (D) for products of curves over $\\Omega$. In particular, we show that Murre's conjecture (D) for products of curves over $\\Omega$ implies Beauville's vanishing conjecture on abelian varieties over $k$. Finally, we give criteria on Chow groups for a motive to be of abelian type. For instance, we show that $M$ is of abelian type if and only if the total Chow group of algebraically trivial cycles $CH_*(M_\\Omega)_{alg}$ is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives $f: N \\to M$, with $N$ Kimura finite-dimensional, which induces a surjection $f_* : CH_*(N_\\Omega)_{alg} \\to CH_*(M_\\Omega)_{alg}$ also induces a surjection $f_* : CH_*(N_\\Omega)_{hom} \\to CH_*(M_\\Omega)_{hom}$ on homologically trivial cycles.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-12-04T21:06:14.000Z",
      "comment": "22 pages. Title changed, largely expands the previous version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-07-27T12:48:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "14C15",
      "14K15"
    ],
    "keywords": [
      "abelian type",
      "abelian variety",
      "murres conjecture",
      "trivial cycles",
      "finite-dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.1080V"
    }
  }
}