{
  "id": "1706.01518",
  "version": "v1",
  "published": "2017-06-05T19:56:10.000Z",
  "updated": "2017-06-05T19:56:10.000Z",
  "title": "Degeneration of Kahler-Einstein manifolds of negative scalar curvature",
  "authors": [
    "Jian Song"
  ],
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "Let $\\pi: \\mathcal{X}^* \\rightarrow B^*$ be an algebraic family of compact K\\\"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\\in \\mathbb{C}$. Let $g_t$ be the unique K\\\"ahler-Einstein metric on $\\mathcal{X}_t= \\pi^{-1}(t)$. We show that as $t\\rightarrow 0$, $(\\mathcal{X}_t, g_t)$ converges in pointed Gromov-Hausdorff topology to a unique finite disjoint union of complete metric length spaces $\\coprod_{\\alpha=1}^\\mathcal{A} (Y_\\alpha, d_\\alpha)$ without loss of volume. Each $(Y_\\alpha, d_\\alpha)$ is a smooth open K\\\"ahler-Einstein manifold of complex dimension n outside its closed singular set of Hausdorff dimension no greater than $2n-4$. Furthermore, $\\coprod_{\\alpha=1}^\\mathcal{A} Y_\\alpha$ is a quasi-projective variety isomorphic to $\\mathcal{X}_0 \\setminus LCS(\\mathcal{X}_0)$, where $\\mathcal{X}_0$ is a projective semi-log canonical model and $LCS(\\mathcal{X}_0)$ is the non-log terminal locus of $\\mathcal{X}_0$. This is the first step of our approach toward compactification of the analytic geometric moduli space of K\\\"ahler-Einstein manifolds of negative scalar curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-05T19:56:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "negative scalar curvature",
      "kahler-einstein manifolds",
      "degeneration",
      "complex dimension",
      "complete metric length spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}