A comparison of the real and non-archimedean Monge-Amp\`ere operator

Abstract

Let $X$ be a proper algebraic variety over a non-archimedean, non-trivially valued field. We show that the non-archimedean Monge-Amp\`ere measure of a metric arising from a convex function on an open face of some skeleton of $X^{\text{an}}$ is equal to the real Monge-Amp\`ere measure of that function up to multiplication by a constant. As a consequence we obtain a regularity result for solutions of the non-archimedean Monge-Amp\`ere problem on curves.