{
  "id": "0811.2965",
  "version": "v1",
  "published": "2008-11-18T18:40:43.000Z",
  "updated": "2008-11-18T18:40:43.000Z",
  "title": "A new geometric approach to problems in birational geometry",
  "authors": [
    "Chen-Yu Chi",
    "Shing-Tung Yau"
  ],
  "comment": "13 pages, to appear in PNAS",
  "doi": "10.1073/pnas.0809030105",
  "categories": [
    "math.AG",
    "math.DG"
  ],
  "abstract": "A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-18T18:40:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "birational geometry",
      "geometric approach",
      "first varietys pseudonormed spaces",
      "vector spaces admits",
      "typical metric structure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Proceedings of the National Academy of Science",
      "year": 2008,
      "month": "Dec",
      "volume": 105,
      "number": 48,
      "pages": 18696
    },
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008PNAS..10518696C"
    }
  }
}