{
  "id": "1704.02731",
  "version": "v1",
  "published": "2017-04-10T07:04:36.000Z",
  "updated": "2017-04-10T07:04:36.000Z",
  "title": "Remarks on degenerations of hyper-Kähler manifolds",
  "authors": [
    "János Kollár",
    "Radu Laza",
    "Giulia Saccà",
    "Claire Voisin"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-K\\\"ahler setting, we can then deduce a finiteness statement for monodromy acting on $H^2$, once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the geometry of hyper-K\\\"ahler manifolds, we prove that a finite monodromy projective degeneration of hyper-K\\\"ahler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain geometric constructions of hyper-K\\\"ahler manifolds (e.g. Debarre--Voisin or Laza--Sacc\\`a--Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-K\\\"ahler manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-10T07:04:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyper-kähler manifolds",
      "rational homology type",
      "kulikov type form",
      "minimal model program",
      "singular central fibers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}