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

TL;DR

This paper establishes a precise relationship between non-archimedean and real Monge-Ampère measures on algebraic varieties, leading to regularity results for solutions on curves.

Contribution

It demonstrates that non-archimedean Monge-Ampère measures of certain metrics correspond to real Monge-Ampère measures, bridging two geometric frameworks.

Findings

Non-archimedean Monge-Ampère measure equals scaled real Monge-Ampère measure for metrics from convex functions.

Derived regularity results for solutions of the non-archimedean Monge-Ampère problem on curves.

Established a fundamental link between non-archimedean and real pluripotential theory.

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.