{
  "id": "2001.02266",
  "version": "v1",
  "published": "2020-01-07T20:04:54.000Z",
  "updated": "2020-01-07T20:04:54.000Z",
  "title": "On the topology of a resolution of isolated singularities, II",
  "authors": [
    "Vincenzo Di Gennaro",
    "Davide Franco"
  ],
  "comment": "7 pages, no figures",
  "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, $G:=\\pi^{-1}\\left(\\rm{Sing}(Y)\\right)$ the exceptional locus. From the Decomposition Theorem one knows that the map $H^{k-1}(G)\\to H^k(Y,Y\\backslash {\\rm{Sing}}(Y))$ vanishes for $k>n$. It is also known that, conversely, assuming this vanishing one can prove the Decomposition Theorem for $\\pi$ in few pages. The purpose of the present paper is to exhibit a direct proof of the vanishing. As a consequence, it follows a complete and short proof of the Decomposition Theorem for $\\pi$, involving only ordinary cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T20:04:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isolated singularities",
      "decomposition theorem",
      "resolution",
      "exceptional locus",
      "complex projective variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}