On the non-archimedean Monge-Amp\`ere equation in mixed characteristic

TL;DR

This paper advances the understanding of non-archimedean Monge-Ampère equations in mixed characteristic by establishing solutions using a variational approach and novel test ideals, extending techniques from equicharacteristic zero.

Contribution

It introduces a method to solve non-archimedean Monge-Ampère equations in mixed characteristic using perturbation friendly test ideals, replacing multiplier ideals.

Findings

Proves continuity of the plurisubharmonic envelope on X.

Develops a framework for solving Monge-Ampère equations in mixed characteristic.

Extends variational methods to a new setting.

Abstract

Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-archimedean Monge-Amp\`ere equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.