The probabilistic vs the quantization approach to K\"ahler-Einstein geometry

TL;DR

This paper compares probabilistic and quantization methods for constructing K"ahler-Einstein metrics, providing new bounds and proofs that connect stability conditions with metric existence.

Contribution

It introduces a new quantitative bound on the partition function and offers a direct analytic proof linking Gibbs stability to the existence of K"ahler-Einstein metrics.

Findings

New bound on the partition function

Analytic proof of existence under Gibbs stability

Connection between probabilistic and quantization approaches

Abstract

In the probabilistic construction of K\"ahler-Einstein metrics on a complex projective algebraic manifold X - involving random point processes on X - a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a K\"ahler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to K\"ahler-Einstein geometry.