{
  "id": "1902.04830",
  "version": "v1",
  "published": "2019-02-13T10:06:06.000Z",
  "updated": "2019-02-13T10:06:06.000Z",
  "title": "Volume form on moduli spaces of d-differentials",
  "authors": [
    "Duc-Manh Nguyen"
  ],
  "comment": "preliminary version, comments welcome!",
  "categories": [
    "math.GT",
    "math.AG",
    "math.DG"
  ],
  "abstract": "Given $d\\in \\mathbb{N}$, $g\\in \\mathbb{N} \\cup\\{0\\}$, and an integral vector $\\kappa=(k_1,\\dots,k_n)$ such that $k_i>-d$ and $k_1+\\dots+k_n=d(2g-2)$, let $\\Omega^d\\mathcal{M}_{g,n}(\\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\\kappa$. We show that $\\Omega^d\\mathcal{M}_{g,n}(\\kappa)$ carries a volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\\mathbb{P}\\Omega^d\\mathcal{M}_{g,n}(\\kappa)=\\Omega^d\\mathcal{M}_{g,n}/\\mathbb{C}^*$ with respect to the measure induced by this volume form is finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-13T10:06:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51H25",
      "51M05"
    ],
    "keywords": [
      "volume form",
      "moduli space",
      "d-differentials",
      "affine complex manifold structure",
      "integral vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}