{
  "id": "1408.3892",
  "version": "v2",
  "published": "2014-08-18T03:30:12.000Z",
  "updated": "2014-10-26T17:10:43.000Z",
  "title": "Morrison-Kawamata cone conjecture for hyperkahler manifolds",
  "authors": [
    "Ekaterina Amerik",
    "Misha Verbitsky"
  ],
  "comment": "23 pages, added a section about ample cones and polyhedral fundamental domains",
  "categories": [
    "math.AG",
    "math.DG",
    "math.DS"
  ],
  "abstract": "Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits, whenever $b_2(M)\\neq 5$. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and $\\{S_i\\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-18T03:30:12.000Z",
      "abstract": "Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and $\\{S_i\\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.",
      "comment": "21 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-26T17:10:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C26",
      "32G13"
    ],
    "keywords": [
      "morrison-kawamata cone conjecture",
      "hyperkahler manifolds",
      "simple holomorphically symplectic manifold",
      "complete riemannian orbifold",
      "infinite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3892A"
    }
  }
}