Envelopes of positive metrics with prescribed singularities

TL;DR

This paper studies envelopes of positive metrics with specific singularities, proving regularity, analyzing their Monge-Ampere measures, and exploring their connections to Bergman functions, Legendre transforms, and geometric structures like Okounkov bodies.

Contribution

It generalizes Berman's work to new singularity settings, establishing regularity and linking these envelopes to various geometric and analytic concepts.

Findings

Envelopes are proven to be C^{1,1} regular.

Monge-Ampere measures are supported on equilibrium sets.

Connections are established between envelopes, Bergman functions, and geometric structures.

Abstract

We investigate envelopes of positive metrics with a prescribed singularity type. First we generalise work of Berman to this setting, proving C^{1,1} regularity of such envelopes, showing their Monge-Ampere measure is supported on a certain "equilibrium set" and connecting with the asymptotics of the partial Bergman functions coming from multiplier ideals. We investigate how these envelopes behave on certain products, and how they relate to the Legendre transform of a test curve of singularity types in the context of geodesic rays in the space of K\"ahler potentials. Finally we consider the associated exhaustion function of these equilibrium sets, connecting it both to the Legendre transform and to the geometry of the Okounkov body.