Addendum to the article `Global pluripotential theory over a trivially valued field'

TL;DR

This paper extends previous work on global pluripotential theory over trivially valued fields by proving the envelope property for smooth varieties and relating Monge–Ampère energy to regularized envelopes in certain cases.

Contribution

It establishes the envelope property for smooth varieties and links Monge–Ampère energy to usc regularized envelopes for possibly singular varieties with ample classes.

Findings

Envelope property holds for smooth varieties.

Monge–Ampère energy equals the energy of the usc regularized envelope.

Results apply to varieties over characteristic zero fields or of dimension at most two.

Abstract

This note is an addendum to the paper `Global pluripotential theory over a trivially valued field' by the present authors, in which we prove two results. Let $X$ be an irreducible projective variety over an algebraically closed field field $k$, and assume that $k$ has characteristic zero, or that $X$ has dimension at most two. We first prove that when $X$ is smooth, the envelope property holds for any numerical class on $X$. Then we prove that for $X$ possibly singular and for an ample numerical class, the Monge--Amp\`ere energy of a bounded function is equal to the energy of its usc regularized plurisubharmonic envelope.