Remarks on degenerations of hyper-K\"ahler manifolds

By J. Kollár, R. Laza, Giulia Saccà, Claire Voisin. 2017

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1fb19645e7d5f7c6a624976c719c8cbba7a15c1d

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In Geometry and Topology

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