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

TL;DR

This paper develops a non-Archimedean geometric framework to analyze K-stability, linking it to singularities of pairs and providing new insights and examples of uniformly K-stable varieties.

Contribution

It introduces non-Archimedean analogues of classical Kähler functionals and relates uniform K-stability to singularities of pairs, extending Odaka's results.

Findings

Characterization of Duistermaat-Heckman measures in test configurations

Non-Archimedean interpretation of the Mabuchi functional and Donaldson-Futaki invariant

Examples of varieties that are uniformly K-stable

Abstract

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular 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 K\"ahler geometry. These functionals are defined on the space of test configurations, 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 singularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.