{
  "id": "math/0601210",
  "version": "v1",
  "published": "2006-01-10T10:03:06.000Z",
  "updated": "2006-01-10T10:03:06.000Z",
  "title": "On the Brieskorn (a,b)-module of an hypersurface singularity",
  "authors": [
    "D. Barlet"
  ],
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We show in this note that for a germ $g$ of holomorphic function with an isolated singularity at the origin of $\\mathbb{C}^n$ there is a pole for the meromorphic extension of the distribution \\begin{equation*} \\frac{1}{\\Gamma(\\lambda)} \\int_X | g |^{2\\lambda}\\bar{g}^{-n} \\square \\tag{*} \\end{equation*} at $- n - \\alpha$ when $ \\alpha$ is the smallest root in its class modulo $\\mathbb{Z}$ of the reduce Bernstein-Sato polynomial of $g$. This is rather unexpected result comes from the fact that the self-duality of the Brieskorn (a,b)-module $E_g$ associated to $g$ exchanges the biggest simple pole sub-(a,b)-module of $E_g$ with the saturation of $E_g$ by $b^{-1}a$. In the first part of this note, we prove that the biggest simple pole sub-(a,b)-module of the Briekorn (a,b)-module $E$ of $g$ is \"geometric\" in the sense that it depends only on the hypersurface germ $\\{g = 0 \\}$ at the origin in $\\mathbb{C}^n$ and not on the precise choice of the reduced equation $g$, as the poles of (*). By duality, we deduce the same property for the saturation $\\tilde{E}$ of $E$. This duality gives also the relation between the \"dual\" Bernstein-Sato polynomial and the usual one, which is the key of the proof of the theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-10T10:03:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S05",
      "32S25",
      "32S40"
    ],
    "keywords": [
      "hypersurface singularity",
      "biggest simple pole",
      "reduce bernstein-sato polynomial",
      "result comes",
      "first part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1210B"
    }
  }
}