Kahler-Einstein metrics, canonical random point processes and birational geometry

TL;DR

This paper introduces a probabilistic approach to canonical metrics on complex algebraic varieties, showing convergence of random point processes to canonical measures and linking stability notions to Kähler-Einstein metrics.

Contribution

It develops a new probabilistic framework for studying canonical metrics, defining canonical random point processes, and establishing their convergence to canonical measures, with implications for stability and Kähler-Einstein metrics.

Findings

Canonical random point processes converge to canonical measures.

Kähler-Einstein metrics are limits of Bergman type metrics.

Gibbs stability relates to existence of Kähler-Einstein metrics.

Abstract

In the present paper and the companion paper [8] 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 deterministic measure on X, coinciding with the canonical measure of Song-Tian and Tsuji. The proof is based on new large deviation principle for Gibbs measures with singular Hamiltonians which relies on an asymptotic submean inequality in large dimensions, proved in a companion paper. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) K\"ahler-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 K\"ahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.