Appendix to the paper by W. Gubler, Ph. Jell, K. Künnemann and F. Martin, Continuity of plurisubharmonic envelopes in non-archimedean geometry and test ideals

Abstract

Let L be an ample line bundle on a smooth projective variety $X$ over a non-archimedean field $K$. For a continuous metric on $L^{\text {an }},$ we show In the following two cases that the semipositive envelope is a continuous semipositive metric on $L^{\text {an }}$ and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that $X$ is a surface defined geometrically over the function field of a curve over a perfect field $k$ of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over $k .$ The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.