Large deviations for Gibbs measures with singular Hamiltonians and emergence of Kahler-Einstein metrics

TL;DR

This paper develops a large deviation principle for Gibbs measures with singular Hamiltonians and demonstrates how Kahler-Einstein metrics with negative Ricci curvature emerge from many-particle limits of determinantal point processes on algebraic varieties.

Contribution

It introduces a probabilistic approach using large deviations for singular Hamiltonians to construct canonical metrics on complex algebraic varieties.

Findings

Large deviation principle established for Gibbs measures with singular Hamiltonians.

Kahler-Einstein metrics with negative Ricci curvature emerge from many-particle limits.

Framework extends to varieties with positive Kodaira dimension in companion work.

Abstract

In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a complex algebraic varieties X is introduced, by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kahler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper [9] the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kahler-Einstein metrics with positive Ricci curvature is developed.