Analytic test configurations and geodesic rays

TL;DR

This paper develops a method to construct weak geodesic rays in the space of metrics on a line bundle using Legendre transforms, linking algebraic test configurations to geometric structures.

Contribution

It introduces a new approach to generate weak geodesic rays from singularity data and filtrations, recovering known results for algebraic test configurations.

Findings

Constructs weak geodesic rays from singularity data.

Links filtrations of section algebras to geodesic rays.

Recovers Phong-Sturm geodesic rays for algebraic test configurations.

Abstract

Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle L over a compact manifold. Using this we associate weak geodesics to suitable filtrations of the algebra of sections of L. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how in this case we recover the weak geodesic ray of Phong-Sturm.