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

The purpose of the present paper is to set up a formalism inspired from\nnon-Archimedean geometry to study K-stability. We first provide a detailed\nanalysis of Duistermaat-Heckman measures in the context of test configurations,\ncharacterizing in particular the trivial case. For any normal polarized variety\n(or, more generally, polarized pair in the sense of the Minimal Model Program),\nwe introduce and study the non-Archimedean analogues of certain classical\nfunctionals in K\\"ahler geometry. These functionals are defined on the space of\ntest configurations, and the Donaldson-Futaki invariant is in particular\ninterpreted as the non-Archimedean version of the Mabuchi functional, up to an\nexplicit error term. Finally, we study in detail the relation between uniform\nK-stability and singularities of pairs, reproving and strengthening Y. Odaka's\nresults in our formalism. This provides various examples of uniformly K-stable\nvarieties.\n