{
  "id": "2208.03277",
  "version": "v1",
  "published": "2022-08-05T16:52:01.000Z",
  "updated": "2022-08-05T16:52:01.000Z",
  "title": "V-filtrations and minimal exponents for locally complete intersection singularities",
  "authors": [
    "Qianyu Chen",
    "Bradley Dirks",
    "Mircea Mustaţă",
    "Sebastián Olano"
  ],
  "comment": "37 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We define and study a notion of minimal exponent for a locally complete intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology $H^r_Z({\\mathcal O}_X)$, where $r$ is the codimension of $Z$ in $X$. We also study its relation to the Bernstein-Sato polynomial of $Z$. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension $1$ case. A key ingredient for our main result is a description of the Kashiwara-Malgrange $V$-filtration associated to any ideal $(f_1,\\ldots,f_r)$ in terms of the microlocal $V$-filtration associated to the hypersurface defined by $\\sum_{i=1}^rf_iy_i$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-05T16:52:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F10",
      "14B05",
      "14J17"
    ],
    "keywords": [
      "locally complete intersection singularities",
      "minimal exponent",
      "main result",
      "smooth complex algebraic variety",
      "v-filtrations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}