Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs

Abstract

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We rst provide a detailed analysis of Duistermaat-Heckman measures in the context of test congurations, characterizing in par- ticular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kahler geometry. These functionals are dened on the space of test congurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and sin- gularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.