{
  "id": "0806.1461",
  "version": "v5",
  "published": "2008-06-09T13:47:41.000Z",
  "updated": "2008-09-29T15:20:32.000Z",
  "title": "Generalized Thomas hyperplane sections and relations between vanishing cycles",
  "authors": [
    "Morihiko Saito"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between the vanishing cycles of type (0,0) with unipotent monodromy around a singular hyperplane section defines a primitive Hodge class such that this singular hyperplane section is a generalized Thomas hyperplane section if and only if the pairing between a given primitive Hodge class and some of the constructed primitive Hodge classes does not vanish. This is a generalization of a construction by P. Griffiths.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2008-09-29T15:20:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S40"
    ],
    "keywords": [
      "generalized thomas hyperplane section",
      "vanishing cycles",
      "primitive hodge class",
      "singular hyperplane section defines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1461S"
    }
  }
}