{
  "id": "0806.2457",
  "version": "v1",
  "published": "2008-06-15T16:05:39.000Z",
  "updated": "2008-06-15T16:05:39.000Z",
  "title": "On the simply connectedness of non-negatively curved Kähler manifolds and applications",
  "authors": [
    "Albert Chau",
    "Luen-Fai Tam"
  ],
  "comment": "24 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "We study complete noncompact long time solutions $(M, g(t))$ to the K\\\"ahler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\\ijb \\ge cRg_\\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient K\\\"ahler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \\to \\infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for K\\\"ahler Ricci flow \\cite{Cao04} and techniques of Ni-Tam \\cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \\in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete K\\\"ahler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient K\\\"ahler Ricci solitons.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-15T16:05:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J37",
      "35B35"
    ],
    "keywords": [
      "non-negatively curved kähler manifolds",
      "noncompact long time solutions",
      "holomorphic fiber bundle structure",
      "nonnegative holomorphic bisectional curvature",
      "simply connectedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2457C"
    }
  }
}