K\"ahler-Einstein metrics and Archimedean zeta functions

TL;DR

This paper explores the connection between K"ahler-Einstein metrics, probabilistic models, and Archimedean zeta functions, proposing conjectures on zero-free properties and their implications for complex geometry and number theory.

Contribution

It introduces a probabilistic construction of K"ahler-Einstein metrics and links conjectural zero-free properties of Archimedean zeta functions to geometric and arithmetic problems.

Findings

Convergence results for log Fano curves using Selberg integrals

Proposed zero-free conjectures for Archimedean zeta functions

Relations to automorphic L-functions and potential p-adic extensions

Abstract

While the existence of a unique K\"ahler-Einstein metric on a canonically polarized manifold X was established by Aubin and Yau already in the 70s there are only a few explicit formulas available. In previous work a probabilistic construction of the K\"ahler-Einstein metric was introduced - involving canonical random point processes on X - which yields canonical approximations of the K\"ahler-Einstein metric, expressed as explicit period integrals over a large number of products of X. Here it is shown that the conjectural extension to the case when X is a Fano variety suggests a zero-free property of the Archimedean zeta functions defined by the partition functions of the probabilistic model. A weaker zero-free property is also shown to be relevant for the Calabi-Yau equation. The convergence in the case of log Fano curves is settled, exploiting relations to the complex Selberg integral in the orbifold case. Some intriguing relations to the zero-free property of the local automorphic L-functions appearing in the Langlands program and arithmetic geometry are also pointed out. These relations also suggest a natural p-adic extension of the probabilistic approach.