Geometry at the infinity of the space of positive metrics: test configurations, geodesic rays and chordal distances

From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the Monge-Amp\`ere equation on an equivariant resolution of singularities of the test configuration. We prove that the Mabuchi chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at the infinity of the space of positive metrics, viewed - as it is usually done for spaces of negative curvature - through geodesic rays.