{
  "id": "1110.4264",
  "version": "v2",
  "published": "2011-10-19T12:38:28.000Z",
  "updated": "2012-10-05T13:02:24.000Z",
  "title": "On the Chow motive of an abelian scheme with non-trivial endomorphisms",
  "authors": [
    "Ben Moonen"
  ],
  "comment": "32 pages. This is a completely revised and greatly expanded version of the first submission with this number",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let X be an abelian scheme over a base variety S with endomorphism algebra D. We prove that the relative Chow motive R(X/S) has a canonical decomposition as a direct sum of motives R^(\\xi)$ where \\xi runs over an explicitly determined finite set of irreducible representations of the group D^{opp,*}, such that R^(\\xi), seen as a functor from Chow motives to D^{\\opp,*}-representations, is \\xi-isotypic. Our decomposition refines the motivic decomposition of Deninger and Murre, as well as Beauville's decomposition of the Chow group. The second main result is that we construct a canonical generalized motivic Lefschetz decomposition. Inspired by work of Looijenga and Lunts, the role of sl_2 in the classical theory is here replaced by a larger Lie algebra, generated by all Lefschetz and Lambda operators associated to non-degenerate line bundles. We construct an action of this larger Lie algebra on R(X/S) and deduce from this a generalized Lefschetz decomposition. We also give a precise structure result for the Lefschetz components that arise. As an application of our techniques, we provide a positive answer to a question of Claire Voisin concerning the analogue for abelian varieties of a conjecture of Beauville.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-05T13:02:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chow motive",
      "abelian scheme",
      "non-trivial endomorphisms",
      "larger lie algebra",
      "precise structure result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.4264M"
    }
  }
}