Kähler–Einstein metrics, canonical random point processes and birational geometry

Abstract

In the present paper and the companion paper (Berman, 2017) a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical measure, coinciding with the canonical measure of Song-Tian and Tsuji. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) Kahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a Kahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.