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Remarks on degenerations of hyper-Kähler manifolds. (Remarques sur les dégénérescences des variétés hyper-kählériennes.) (English. French summary) Zbl 1435.14038

Summary: 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-Kähler setting, we can then deduce a finiteness statement for the 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 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.

MSC:

14J42 Holomorphic symplectic varieties, hyper-Kähler varieties
14B05 Singularities in algebraic geometry
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14C34 Torelli problem
14E30 Minimal model program (Mori theory, extremal rays)
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References:

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