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

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.Comment: Small changes. To appear in Ann. Inst. Fourie

Uniform K-stability and asymptotics of energy functionals in K\"ahler geometry

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability.Comment: New version with errata (an error was found in the proof of Theorem 5.6). The affected parts of the paper are marked in re

Asymptotic analysis of Bergman kernels for linear series and its application to Kähler ｇeometry

報告番号: ; 学位授与年月日: 2013.3.25 ; 学位の種別: 課程博士 ; 学位の種類: 博士(数理科学) ; 学位記番号: ; 研究科・専攻: 数理科学研究科数理科学専