{
  "id": "1312.6932",
  "version": "v2",
  "published": "2013-12-25T06:42:49.000Z",
  "updated": "2015-05-02T15:41:15.000Z",
  "title": "Curvatures of moduli space of curves and applications",
  "authors": [
    "Kefeng Liu",
    "Xiaofeng Sun",
    "Xiaokui Yang",
    "Shing-Tung Yau"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, \\omega_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient. We also show that, on a Kodaira (deformation) surface, there is no K\\\"ahler metric with non-positive Riemannain sectional curvature although it possesses K\\\"ahler metrics with strictly negative holomorphic bisectional curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-25T06:42:49.000Z",
      "abstract": "In this paper, we investigate the curvature relations between K\\\"ahler metrics and background Riemannian metrics. We prove that Siu's strongly non-positivity (resp. non-negativity) is equivalent to the non-positivity (resp. non-negativity) of the complex sectional curvature of the background Riemannain metric; the semi-dual-Nakano-positivity (resp. the semi-dual-Nakano-negativity) implies the non-negativity (resp. non-positivity) of the Riemannian curvature operator. As applications, we show that the moduli space $(\\sM_g, \\omega_{WP})$ of curves with genus $g>1$ has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. We also prove that any submanifold in $\\sM_g$ which is totally geodesic in $\\sA_g$ with finite volume must be a ball quotient.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-05-02T15:41:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C55",
      "32C10",
      "14K10"
    ],
    "keywords": [
      "moduli space",
      "applications",
      "direct image sheaves",
      "non-positive complex sectional curvature",
      "strictly negative holomorphic bisectional curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6932L"
    }
  }
}