Remarks on degenerations of hyper-Kähler manifolds

Kollár, János; Laza, Radu; Saccà, Giulia; Voisin, Claire

Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2837-2882.

Comme conséquence du Programme du modèle minimal, toute dégénérescence de variétés projectives à fibré canonique trivial admet une forme de Kulikov, c’est-à-dire que les singularités de la fibre centrale sont modérées et le fibré canonique relatif est trivial. Dans le cas hyper-kählérien, on en déduit un résultat de finitude pour l’action de monodromie sur $H^2$, dès qu’on sait qu’une composante de la fibre centrale n’est pas uniréglée. Nous montrons par ailleurs, en utilisant des résultats puissants de la théorie des variétés hyper-kählériennes, qu’une dégénérescence de variétés hyper-kählériennes à monodromie finie sur $H^2$ admet un remplissage lisse, c’est-à-dire, après changement de base, un modèle birationnel à fibre centrale lisse. Combinant ces deux résultats, nous obtenons une version du théorème de Huybrechts sur l’équivalence birationnelle et le type de déformations, valable pour les familles à fibre centrale singulière. Ce résultat nous permet de retrouver de façon simple le type de déformations de la plupart des modèles projectifs connus de variétés hyper-kählériennes. Dans une direction différente, nous établissons des résultats basiques (dimension et type d’homotopie rationnelle) concernant le complexe dual de la dégénérescence de Kulikov d’une variété hyper-kählérienne.

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.

Classification : 14B05, 14D05, 14J32, 14E99
Keywords: Hyper-Kähler manifold, degeneration, deformations, Torelli theorem
Mot clés : Variété hyper-kählérienne, dégénérescence, déformations, théorème de Torelli

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