{
  "id": "1404.7305",
  "version": "v1",
  "published": "2014-04-29T10:45:20.000Z",
  "updated": "2014-04-29T10:45:20.000Z",
  "title": "Algebraic versus homological equivalence for singular varieties",
  "authors": [
    "Vincenzo Di Gennaro",
    "Davide Franco",
    "Giambattista Marini"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $ Y \\subseteq \\Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that $\\dim Y=n+h$ and $ \\dim Y_{sing} \\le \\min\\{ d+h-1 , n-1 \\} $. Let $Z$ be an algebraic cycle on $Y$ of dimension $d+h$, whose homology class in $H_{2(d+h)}(Y; \\Bbb Q)$ is non-zero. In the present paper we prove that the restriction of $Z$ to $X$ is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case $Y$ is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-29T10:45:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "14C15",
      "14C25",
      "14F17",
      "14F43",
      "14F45",
      "14J17",
      "14M10"
    ],
    "keywords": [
      "singular varieties",
      "homological equivalence",
      "explicit examples",
      "complex numbers",
      "possibly singular projective variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7305D"
    }
  }
}