A comparison of the real and non-archimedean Monge–Ampère operator

Abstract

Let X be a proper algebraic variety over a non-archimedean, non-trivially valued field. We show that the non-archimedean Monge–Ampère measure of a metric arising from a convex function on an open face of some skeleton of Xan is equal to the real Monge–Ampère 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–Ampère problem on curves.