{
  "id": "math/0006027",
  "version": "v1",
  "published": "2000-06-05T05:09:48.000Z",
  "updated": "2000-06-05T05:09:48.000Z",
  "title": "Local cohomology of generalized Okamoto-Painlevé pairs and Painlevé equations",
  "authors": [
    "Hitomi Terajima"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "In the theory of deformation of Okamoto-Painlev\\'e pair (S,Y), a local cohomology group $H^1_D(\\Theta_S(-\\log D))$ plays an important role. In this paper, we estimate the local cohomology group of pair (S,Y) for several types, and obtain the following results. For a pair (S,Y) corresponding to the space of initial conditions of the Painlev\\'e equations, we show that the local cohomology group $H^1_D(\\Theta_S(-\\log D))$ is at least 1 dimensional. This fact is the key to understand Painlev\\'e equation related to (S,Y). Moreover we show that, for the pairs (S,Y) of type $\\tilde{A_8}$, the local cohomology group $H^1_D(\\Theta_S(-\\log D))$ vanish. Therefore in this case, there is no differential equation on S-Y in the sense of the theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-06-05T05:09:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14D15",
      "14J26",
      "32G10",
      "34M55"
    ],
    "keywords": [
      "local cohomology group",
      "initial conditions",
      "okamoto-painleve pair",
      "understand painleve equation",
      "important role"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......6027T"
    }
  }
}