On solutions to tropical Monge-Ampère equations

Abstract: This paper develops convex analysis on tropical varieties and investigates the properties of the tropical Monge-Ampère equations. We start by introducing notions of plurisubharmonic functions and Kähler forms on arbitrary tropical varieties. We show that many classical results of pluripotential theory extend to this setting, including an analogue of Demailly's regularization theorem. Next, we study the solutions to the tropical Monge-Ampère equations. While we establish their uniqueness, we show that the existence is not guaranteed in general. More precisely, we prove that the tropical analogue of the Calabi-Yau theorem fails for any pair $(X, θ)$ consisting of a compact tropical variety $X$ of dimension at least two satisfying a mild genericity assumption, and a tropical Kähler form $θ$ on $X$. In particular, there exist measures compactly supported in the open part of $X$ that admit no solution. In contrast, for tropical varieties that are regular in codimension one, we show the existence of solutions. Finally, via strongly faithful tropicalizations whose existence is established in this paper, we relate the tropical solutions, when they exist, to the solutions of non-Archimedean Monge-Ampère equations on Berkovich spaces.