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 compare various induced metrics on the space of valuations of linear growth.