{
  "id": "1704.01357",
  "version": "v1",
  "published": "2017-04-05T10:36:58.000Z",
  "updated": "2017-04-05T10:36:58.000Z",
  "title": "On the topology of a resolution of isolated singularities",
  "authors": [
    "Vincenzo Di Gennaro",
    "Davide Franco"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $\\pi:X\\to Y$ a resolution of singularities, $G:=\\pi^{-1}{\\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)\\to H^k(Y,Y\\backslash {\\rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $\\pi$. A consequence is a short proof of the Decomposition Theorem for $\\pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $\\pi$ is the blowing-up of $Y$ along ${\\rm{Sing}}(Y)$ with smooth and connected fibres, or when $\\pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)\\to H^k(Y,Y\\backslash {\\rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $\\pi^*_k:H^k(Y)\\to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-05T10:36:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B05",
      "14E15",
      "14F05",
      "14F43",
      "14F45",
      "32S20",
      "32S60",
      "58K15"
    ],
    "keywords": [
      "isolated singularities",
      "decomposition theorem",
      "resolution",
      "short proof",
      "natural gysin morphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}