{
  "id": "2212.01898",
  "version": "v1",
  "published": "2022-12-04T19:17:11.000Z",
  "updated": "2022-12-04T19:17:11.000Z",
  "title": "The minimal exponent and $k$-rationality for locally complete intersections",
  "authors": [
    "Qianyu Chen",
    "Bradley Dirks",
    "Mircea Mustaţă"
  ],
  "comment": "21 pages. Comments are welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show that if $Z$ is a locally complete intersection subvariety of a smooth complex variety $X$, of pure codimension $r$, then $Z$ has $k$-rational singularities if and only if $\\widetilde{\\alpha}(Z)>k+r$, where $\\widetilde{\\alpha}(Z)$ is the minimal exponent of $Z$. We also characterize this condition in terms of the Hodge filtration on the intersection cohomology Hodge module of $Z$. Furthermore, we show that if $Z$ has $k$-rational singularities, then the Hodge filtration on the local cohomology sheaf $\\mathcal{H}^r_Z(\\mathcal{O}_X)$ is generated at level $\\dim(X)-\\lceil \\widetilde{\\alpha}(Z)\\rceil-1$ and, assuming that $k\\geq 1$ and $Z$ is singular, of dimension $d$, that $\\mathcal{H}^k(\\underline{\\Omega}_Z^{d-k})\\neq 0$. All these results have been known for hypersurfaces in smooth varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-04T19:17:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F10",
      "14B05",
      "14J17",
      "32S35"
    ],
    "keywords": [
      "minimal exponent",
      "rationality",
      "intersection cohomology hodge module",
      "hodge filtration",
      "rational singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}