Monge-Amp\`ere measures for toric metrics on abelian varieties

TL;DR

This paper computes Monge-Ampère measures for toric metrics on abelian varieties, extending previous work from canonical metrics and discrete valuations to more general non-archimedean fields, providing new insights into invariant measures.

Contribution

It generalizes the computation of canonical measures to toric metrics on abelian varieties over arbitrary non-archimedean fields.

Findings

Computed Monge-Ampère measures for toric metrics on subvarieties of abelian varieties.

Extended previous canonical measure computations to non-archimedean settings.

Provided explicit formulas for measures invariant under torus actions.

Abstract

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-Amp\`ere 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.