The probabilistic vs the quantization approach to Kähler–Einstein geometry

Abstract

In the probabilistic construction of Kähler–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 Kähler–Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler–Einstein geometry.