{
  "id": "1710.05753",
  "version": "v1",
  "published": "2017-10-13T07:21:09.000Z",
  "updated": "2017-10-13T07:21:09.000Z",
  "title": "Rationality questions and motives of cubic fourfolds",
  "authors": [
    "Michele Bolognesi",
    "Claudio Pedrini"
  ],
  "comment": "46 pages. arXiv admin note: text overlap with arXiv:1701.05743",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this note we propose an approach to some questions about the birational geometry of smooth cubic fourfolds through the theory of Chow motives. Let $X \\subset \\mathbb{P}^5$ be such a cubic hypersurface. We prove that, assuming the Hodge conjecture for the product $S \\times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family $f : \\mathcal{S} \\to B$ of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. We introduce the transcendental part $t(X)$ of the motive of $X$ and prove that it is isomorphic to the (twisted) transcendental part $h^{tr}_2(F(X))$ in a suitable Chow-K\\\"unneth decomposition for the motive of the Fano variety of lines $F(X)$. If $X$ is special, in the sense of B.Hassett, and $F(X) \\simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we show that $t(X)\\simeq t_2(S)(1)$. Then we relate the existence of an isomorphism between the transcendental motive $t(X)$ and the (twisted) transcendental motive of a K3 surface with the conjectures by Hasset and Kuznetsov on the rationality of a special cubic fourfold. Finally we give examples of cubic fourfolds such that the motive $t(X)$ is finite dimensional and of abelian type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-13T07:21:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rationality questions",
      "k3 surface",
      "transcendental part",
      "conjecture",
      "decomposable integral polarized hodge structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}