Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations

Abstract

Abstract The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential 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 infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays.