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

Christian Vilsmeier

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arXiv · 1912.06220

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

Christian Vilsmeier

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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.

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Christian Vilsmeier. 2019-12-12. A comparison of the real and non-archimedean Monge-Amp\`ere operator. https://arxiv.org/abs/1912.06220

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