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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 theBerkovich analytification of $X$ converge to a Lebesgue-type measure on thetemperate part of the Kontsevich--Soibelman skeleton, assuming the existence ofa strict normal crossings model for $X$. We also prove a similar result for allfinite extensions $K'$ under the assumption that $X$ has a log smooth model.This is a non-archimedean counterpart of analogous results for volume forms ondegenerating complex Calabi--Yau manifolds by Boucksom and the first-namedauthor. Along the way, we develop a general theory of Lebesgue measures onBerkovich skeleta over discretely valued fields.

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