Addendum to the article “Global pluripotential theory over a trivially valued field”

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> be an irreducible projective variety over an algebraically closed field field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> , and assume that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> has characteristic zero, or that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> has dimension at most two. We first prove that when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is smooth, the envelope property holds for any numerical class on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . Then we prove that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> possibly singular and for an ample numerical class, the Monge–Ampère energy of a bounded function is equal to the energy of its usc regularized plurisubharmonic envelope.