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

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.