[Convergence des mesures pluricanoniques p-adiques vers des mesures de Lebesgue sur des squelettes dans les espaces de Berkovich]
Soient un corps local non-archimédien et un -schéma lisse et propre, et fixons une forme pluricanonique sur . Pour chaque extension finie de , la forme pluricanonique induit une mesure sur la -variété analytique . Nous démontrons que, lorsque parcourt toutes les extensions finies modérément ramifiées de , les normalisations appropriées des images directes de ces mesures sur l’analytifié de 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 . Nous démontrons également un résultat similaire pour toutes les extensions finies en supposant que 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 be a non-archimedean local field, a smooth and proper -scheme, and fix a pluricanonical form on . For every finite extension of , the pluricanonical form induces a measure on the -analytic manifold . We prove that, when runs through all finite tame extensions of , suitable normalizations of the pushforwards of these measures to the Berkovich analytification of 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 . We also prove a similar result for all finite extensions under the assumption that 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.
Accepté le :
Publié le :
DOI : 10.5802/jep.118
Keywords: Volume forms, local fields, Berkovich spaces
Mot clés : Formes volumes, corps locaux, espaces de Berkovich
@article{JEP_2020__7__287_0, author = {Jonsson, Mattias and Nicaise, Johannes}, title = {Convergence of $p$-adic pluricanonical measures to {Lebesgue} measures on skeleta {in~Berkovich} spaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {287--336}, publisher = {Ecole polytechnique}, volume = {7}, year = {2020}, doi = {10.5802/jep.118}, zbl = {1430.14056}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.118/} }
TY - JOUR AU - Jonsson, Mattias AU - Nicaise, Johannes TI - Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 287 EP - 336 VL - 7 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.118/ DO - 10.5802/jep.118 LA - en ID - JEP_2020__7__287_0 ER -
%0 Journal Article %A Jonsson, Mattias %A Nicaise, Johannes %T Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 287-336 %V 7 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.118/ %R 10.5802/jep.118 %G en %F JEP_2020__7__287_0
Jonsson, Mattias; Nicaise, Johannes. Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 287-336. doi : 10.5802/jep.118. http://www.numdam.org/articles/10.5802/jep.118/
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