{
  "id": "1709.07140",
  "version": "v1",
  "published": "2017-09-21T02:58:36.000Z",
  "updated": "2017-09-21T02:58:36.000Z",
  "title": "Classification of rational 1-forms on the Riemann sphere up to PSL(2,C)",
  "authors": [
    "Julio C. Magaña-Cáceres"
  ],
  "comment": "17 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the family $\\Omega^1(-s)$ of rational 1--forms on the Riemann sphere, having exactly $-s \\leq -2$ simple poles. Three equivalent $(2s-1)$--dimensional complex parametrizations on $\\Omega^1(-s)$, using coefficients, residues--poles and zeros--poles of the 1--forms, are recognized. A rational 1--form is isochronous when all their residues are purely imaginary. We prove that the subfamily $\\mathcal{RI}\\Omega^1(-s)$ of isochronous 1--forms is a $(3s-1)$--dimensional real analytic submanifold in the complex manifold $\\Omega^1(-s)$. The complex Lie group $PSL(2,\\mathbb{C})$ acts holomorphically on $\\Omega^1(-s)$. For $s \\geq 3$, the $PSL(2,\\mathbb{C})$--action is proper on $\\Omega^1(-s)$ and $\\mathcal{RI}\\Omega^1(-s)$. Therefore, the quotients $\\Omega^1(-s)/PSL(2,\\mathbb{C})$ and $\\mathcal{RI}\\Omega^1(-s)/PSL(2,\\mathbb{C})$ admit a stratification by orbit types. Using an explicit set of $PSL(2,\\mathbb{C})$--invariant functions, we give parametrizations for the quotients $\\Omega^1(-s)/PSL(2,\\mathbb{C})$ and $\\mathcal{RI}\\Omega^1(-s)/PSL(2,\\mathbb{C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-21T02:58:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32M05",
      "30F30"
    ],
    "keywords": [
      "riemann sphere",
      "dimensional real analytic submanifold",
      "classification",
      "complex lie group",
      "dimensional complex parametrizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}