Monge-Ampère measures for toric metrics on abelian varieties

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-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 nonarchimedean fields. Contents 1. Introduction 1 2. Notation and preliminaries 4 3. Piecewise linear approximation 6 4. Toric metrics 11 5. Strictly polystable alterations 15 6. Monge-Ampère measures of toric metrics 18 7. The canonical subset 19 Appendix A. Differential forms, currents and positivity 24 References 27