{
  "id": "1412.4746",
  "version": "v1",
  "published": "2014-12-15T20:08:19.000Z",
  "updated": "2014-12-15T20:08:19.000Z",
  "title": "Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension",
  "authors": [
    "Lucian Badescu"
  ],
  "comment": "To appear in Bulletin Mathematique de la Societe des Sciences Mathematiques de la Romanie",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a submanifold of dimension $n$ of the complex projective space $\\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\\geq\\frac{N+3}{2}\\geq 4$ we prove a geometric criterion for the existence of an extension of $E$ to a vector bundle on the first order infinitesimal neighborhood of $X$ in $\\mathbb P^N$ in terms of the splitting of the normal bundle sequence of $Y\\subset X\\subset\\mathbb P^N$, where $Y$ is the zero locus of a general section of a high twist of $E$. In the last section we show that the universal quotient vector bundle on the Grassmann variety $\\mathbb G(k,m)$ of $k$-dimensional linear subspaces of $\\mathbb P^m$, with $m\\geq 3$ and $1\\leq k\\leq m-2$ (i.e. with $\\mathbb G(k,m)$ not a projective space), embedded in any projective space $\\mathbb P^N$, does not extend to the first infinitesimal neighborhood of $\\mathbb G(k,m)$ in $\\mathbb P^N$ as a vector bundle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-15T20:08:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M07"
    ],
    "keywords": [
      "small codimension",
      "infinitesimal extensions",
      "submanifold",
      "projective space",
      "first order infinitesimal neighborhood"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}