{
  "id": "math/0311180",
  "version": "v1",
  "published": "2003-11-12T05:40:25.000Z",
  "updated": "2003-11-12T05:40:25.000Z",
  "title": "Vanishing cycles, the generalized Hodge Conjecture and Gröbner bases",
  "authors": [
    "Ichiro Shimada"
  ],
  "comment": "30pages, 2 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in $X$, we prove that the vanishing cycles in the middle homology group of $X$ are represented by topological cycles whose support is contained in a proper Zariski closed subset $T\\subset X$ of certain codimension. In some cases, we can find such a Zariski closed subset $T$ with codimension equal to the upper bound obtained from the Hodge structure of the middle cohomology group of $X$ by means of Gr\\\"obner bases. Hence a consequence of the generalized Hodge conjecture is verified in these cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-11-12T05:40:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C30",
      "14M10"
    ],
    "keywords": [
      "generalized hodge conjecture",
      "vanishing cycles",
      "gröbner bases",
      "general complete intersection",
      "proper zariski closed subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11180S"
    }
  }
}