{
  "id": "2202.08425",
  "version": "v1",
  "published": "2022-02-17T03:08:42.000Z",
  "updated": "2022-02-17T03:08:42.000Z",
  "title": "The log canonical threshold and rational singularities",
  "authors": [
    "Raf Cluckers",
    "János Kollár",
    "Mircea Mustaţă"
  ],
  "comment": "26 pages. This supersedes arXiv:1901.08111",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show that if $f$ is a nonzero, noninvertible function on a smooth complex variety $X$ and $J_f$ is the Jacobian ideal of $f$, then ${\\rm lct}(f,J_f^2)>1$ if and only if the hypersurface defined by $f$ has rational singularities. Moreover, if it does not have rational singularities, then ${\\rm lct}(f,J_f^2)={\\rm lct}(f)$. We give two proofs, one relying on arc spaces and one that goes through the inequality $\\widetilde{\\alpha}(f)\\geq{\\rm lct}(f,J_f^2)$, where $\\widetilde{\\alpha}(f)$ is the minimal exponent of $f$. In the case of a polynomial over $\\overline{\\mathbf{Q}}$, we also prove an analogue of this latter inequality, with $\\widetilde{\\alpha}(f)$ replaced by the motivic oscillation index ${\\rm moi}(f)$. We also show a part of Igusa's strong monodromy conjecture, for poles larger than $-{\\rm lct}(f,J_f^2)$. We end with a discussion of lct-maximal ideals: these are ideals $I$ with the property that ${\\rm lct}(I)<{\\rm lct}(J)$ for every $J$ with $I\\subsetneq J$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-17T03:08:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B05",
      "14E18",
      "14J17",
      "11L07"
    ],
    "keywords": [
      "rational singularities",
      "log canonical threshold",
      "igusas strong monodromy conjecture",
      "motivic oscillation index",
      "smooth complex variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}