Analytic test configurations and geodesic rays

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 this recovers the weak geodesic ray of Phong-Sturm.