Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by Jos\'e Ignacio Burgos Gil and Mart\'in Sombra)

TL;DR

This paper investigates the continuity of plurisubharmonic envelopes and solutions to the non-archimedean Monge-Ampère equation on algebraic varieties over non-archimedean fields, using test ideals and providing counterexamples.

Contribution

It extends the understanding of semipositive envelopes and Monge-Ampère solutions in non-archimedean geometry, replacing multiplier ideals with test ideals and providing new examples.

Findings

Semipositive envelopes are continuous in certain cases.

Solutions to the non-archimedean Monge-Ampère equation exist under specified conditions.

Counterexample showing retraction may not preserve semipositivity.

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, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L and that the non-archimedean Monge-Amp\`ere 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.