# Kähler–Einstein metrics and Archimedean zeta functions

### Robert J. Berman

Chalmers University of Technology and the University of Gothenburg, Sweden

This book chapter is published *open access.*

## Abstract

While the existence of a unique Kähler–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 a previous work, a probabilistic construction of the Kähler–Einstein metric
was introduced – involving canonical random point processes on *X* – which yields canonical
approximations of the Kähler–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.