Monge–Ampère measures for toric metrics on abelian varieties

Gubler, Walter; Stadlöder, Stefan

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2023), pp. 49-84

Toric metrics on a line bundle of an abelian variety $A$ are the invariant metrics under the natural torus action coming from Raynaud’s uniformization theory. We compute here the associated Monge–Ampère measures for the restriction to any closed subvariety of $A$. This generalizes the computation of canonical measures done by the first author from canonical metrics to toric metrics and from discrete valuations to arbitrary non-archimedean fields.

Les métriques toriques sur un fibré en droites sur une variété abélienne $A$ sont les métriques invariantes sous l’action naturelle du tore issue de la théorie de l’uniformisation de Raynaud. Nous calculons les mesures de Monge–Ampère associées pour les restrictions à toutes les sous-variétés fermées de $A$. Ceci généralise des travaux du premier auteur sur le calcul des mesures canoniques pour des valuations discrètes au cas des métriques toriques pour des corps non archimédiens arbitraires.

Classification : 14G40, 11G10, 14G22
Keywords: Berkovich analytic spaces, formal geometry, abelian varieties, canonical measures

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