Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

Jonsson, Mattias; Nicaise, Johannes

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 287-336.

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 $K^{\prime }$ de $K$, la forme pluricanonique induit une mesure sur la $K^{\prime }$-variété analytique $X\left(K^{\prime }\right)$. Nous démontrons que, lorsque $K^{\prime }$ 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 $K^{\prime }$ 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.

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 $K^{\prime }$ of $K$, the pluricanonical form induces a measure on the $K^{\prime }$-analytic manifold $X\left(K^{\prime }\right)$. We prove that, when $K^{\prime }$ 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 $K^{\prime }$ 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.

Classification : 14G22, 14J32, 32P05, 14T05
Keywords: Volume forms, local fields, Berkovich spaces
Mot clés : Formes volumes, corps locaux, espaces de Berkovich

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