Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 287-336.

Let K be a non-archimedean local field, X a smooth and proper K-scheme, and fix a pluricanonical form on X. For every finite extension K of K, the pluricanonical form induces a measure on the K -analytic manifold X(K ). We prove that, when K runs through all finite tame extensions of K, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of X converge to a Lebesgue-type measure on the temperate part of the Kontsevich–Soibelman skeleton, assuming the existence of a strict normal crossings model for X. We also prove a similar result for all finite extensions K under the assumption that X has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi–Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields.

Soient K un corps local non-archimédien et X un K-schéma lisse et propre, et fixons une forme pluricanonique sur X. Pour chaque extension finie K de K, la forme pluricanonique induit une mesure sur la K -variété analytique X(K ). Nous démontrons que, lorsque K parcourt toutes les extensions finies modérément ramifiées de K, les normalisations appropriées des images directes de ces mesures sur l’analytifié de X au sens de Berkovich convergent vers une mesure de type Lebesgue sur la partie tempérée du squelette de Kontsevich-Soibelman, en supposant l’existence d’un modèle à croisements normaux stricts de X. Nous démontrons également un résultat similaire pour toutes les extensions finies K en supposant que X admet un modèle log lisse. Il s’agit d’une version non-archimédienne de résultats analogues pour les formes de volumes sur les familles dégénérées de variétés complexes de Calabi–Yau dus à Boucksom et au premier auteur. En cours de route, nous développons une théorie générale des mesures de Lebesgue sur les squelette de Berkovich sur des corps à valuation discrète.

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DOI: 10.5802/jep.118
Classification: 14G22, 14J32, 32P05, 14T05
Keywords: Volume forms, local fields, Berkovich spaces
Mot clés : Formes volumes, corps locaux, espaces de Berkovich
Jonsson, Mattias 1; Nicaise, Johannes 2

1 Dept of Mathematics, University of Michigan Ann Arbor, MI 48109-1043, USA
2 Imperial College, Department of Mathematics South Kensington Campus, London SW72AZ, UK and KU Leuven, Department of Mathematics Celestijnenlaan 200B, 3001 Heverlee, Belgium
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     title = {Convergence of $p$-adic pluricanonical measures to {Lebesgue} measures on skeleta {in~Berkovich} spaces},
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Jonsson, Mattias; Nicaise, Johannes. Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 287-336. doi : 10.5802/jep.118. http://archive.numdam.org/articles/10.5802/jep.118/

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