{
  "id": "2205.12326",
  "version": "v1",
  "published": "2022-05-24T19:12:35.000Z",
  "updated": "2022-05-24T19:12:35.000Z",
  "title": "On the boundedness of singularities via normalized volume",
  "authors": [
    "Joaquín Moraga",
    "Hendrik Süß"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on hypersurface singularities and singularities with torus action of complexity one. Given a positive integer n and a real value v>0, we prove that the class of n-dimensional K-semistable hypersurface singularities with normalized volume at least v forms a bounded family. Analogously, we prove that the class of n-dimensional K-semistable singularities with torus action of complexity one and with normalized volume at least v forms a bounded family. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with complexity one torus action. We exhibit a 3-dimensional example which shows that this last statement is optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-24T19:12:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B05",
      "14M25",
      "53C25"
    ],
    "keywords": [
      "torus action",
      "boundedness",
      "n-dimensional k-semistable hypersurface singularities",
      "complexity",
      "study conjectures regarding normalized volume"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}