Pluripotential theory for tropical toric varieties and non-Archimedean Monge–Ampère equations

Abstract

Tropical toric varieties are partial compactifications of finite dimensional real vector spaces associated with rational polyhedral fans. We introduce plurisubharmonic functions and a Bedford–Taylor product for Lagerberg currents on open subsets of a tropical toric variety. The resulting tropical toric pluripotential theory provides the link to give a canonical correspondence between complex and non-Archimedean pluripotential theories of invariant plurisubharmonic functions on toric varieties. We will apply this correspondence to solve invariant non-Archimedean Monge–Ampère equations on toric and abelian varieties over arbitrary non-Archimedean fields.