{
  "id": "1308.5667",
  "version": "v2",
  "published": "2013-08-26T19:52:28.000Z",
  "updated": "2013-09-26T08:31:26.000Z",
  "title": "Kobayashi pseudometric on hyperkahler manifolds",
  "authors": [
    "Ljudmila Kamenova",
    "Steven Lu",
    "Misha Verbitsky"
  ],
  "comment": "21 pages, some proofs simplified (minor), a few corrections",
  "categories": [
    "math.AG",
    "math.DG"
  ],
  "abstract": "The Kobayashi pseudometric on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkaehler manifold if it admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. The SYZ conjecture claims that any parabolic nef line bundle on a deformation of a given hyperkaehler manifold is semi-ample. We prove that the Kobayashi pseudometric vanishes for all hyperkaehler manifolds satisfying the SYZ property. This proves the Kobayashi conjecture for K3 surfaces and their Hilbert schemes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-26T08:31:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperkahler manifolds",
      "hyperkaehler manifold",
      "parabolic nef line bundle",
      "syz conjecture claims",
      "kobayashi pseudometric vanishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.5667K"
    }
  }
}