REMARKS ON DEGENERATIONS OF HYPER-KÄHLER MANIFOLDS

János Kollár, Radu Laza, Giulia Saccà, Claire Voisin

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

Using the Minimal model program, any degeneration of Ktrivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for the monodromy acting on H2, once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the theory of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler 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 explicit models of projective hyper-Kähler manifolds. 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-Kähler manifolds.

Original languageEnglish (US)
Pages (from-to)2837-2882
Number of pages46
JournalAnnales de l'Institut Fourier
Volume68
Issue number7
StatePublished - 2018

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Geometry and Topology

Keywords

  • Hyper-Kähler manifold
  • Torelli theorem
  • deformations
  • degeneration

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