A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations

For any polarized variety (X,L), we show that test configurations and, more generally, R-test configurations (defined as finitely generated filtrations of the section ring) can be analyzed in terms of Fubini-Study functions on the Berkovich analytification of X with respect to the trivial absolute value on the ground field. Building on non-Archimedean pluripotential theory, we describe the (Hausdorff) completion of the space of test configurations, with respect to two natural pseudo-metrics, in terms of plurisubharmonic functions and measures of finite energy on the Berkovich space. We also describe the Hausdorff quotient of the space of all filtrations, and establish a 1--1 correspondence between divisorial norms and divisorial measures, both being determined in terms of finitely many divisorial valuations.