Hybrid convergence of Kähler–Einstein measures

Pille-Schneider, Léonard

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 587-615.

Nous calculons la limite hybride (au sens de Boucksom–Jonsson) de la famille des formes volumes de Kähler–Einstein sur une dégénérescence de variétés canoniquement polarisées. La mesure limite est une somme pondérée de masses de Dirac en des valuations divisorielles, déterminées par la limite algébro-géométrique naturelle de la famille. Nous formulons aussi des remarques sur l’opérateur de Monge–Ampère non-archimédien et la continuité hybride des potentiels de Kähler–Einstein dans ce contexte.

We compute the hybrid limit (in the sense of Boucksom–Jonsson) of the family of Kähler–Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make some remarks on the non-archimedean Monge–Ampère operator and hybrid continuity of Kähler–Einstein potentials in this context.

Classification : 32Q20, 32P05, 14T05
Keywords: Kähler–Einstein manifolds, degenerations, Berkovich spaces
Mot clés : Variétés de Kähler–Einstein, dégénérescences, espaces de Berkovich

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