{
  "id": "math/0006026",
  "version": "v2",
  "published": "2000-06-05T04:50:57.000Z",
  "updated": "2000-09-18T09:06:16.000Z",
  "title": "Deformation of Okamoto-Painlevé Pairs and Painlevé equations",
  "authors": [
    "Masa-Hiko Saito",
    "Taro Takebe",
    "Hitomi Terajima"
  ],
  "comment": "38 pages, 4 figures, minor corrections, typos",
  "journal": "J. Algebraic Geom. 11 (2002), 311-362.",
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper, we introduce the notion of generalized rational Okamoto-Painlev\\'e pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlev\\'e equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlev\\'e differential equations from the deformation of Okamoto-Painlev\\'e pairs by using the local cohomology groups. Moreover the reason why the Painlev\\'e equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlev\\'e pairs of type $\\tilde{E}_7 (= P_{II})$ and $\\tilde{D}_8 (= P_{III}^{\\tilde{D}_8})$ are calculated explicitly as examples of our theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2000-09-18T09:06:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D15",
      "34M55",
      "32G10",
      "14J26"
    ],
    "keywords": [
      "deformation",
      "painleve equations",
      "generalized rational okamoto-painleve pair",
      "painleve differential equations",
      "holomorphic symplectic structure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......6026S"
    }
  }
}