Surveys in Differential Geometry

Volume 23 (2018)

An invitation to Kähler–Einstein metrics and random point processes

Pages: 35 – 87

DOI: https://dx.doi.org/10.4310/SDG.2018.v23.n1.a2

Author

Robert J. Berman (Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Göteborg, Sweden)

Abstract

This is an invitation to the probabilistic approach for constructing Kähler–Einstein metrics on complex projective algebraic manifolds $X$. The metrics in question emerge in the large $N$-limit from a canonical way of sampling $N$ points on $X$, i.e. from random point processes on $X$, defined in terms of algebro-geometric data. The proof of the convergence towards Kähler–Einstein metrics with negative Ricci curvature is explained. In the case of positive Ricci curvature a variational approach is introduced to prove the conjectural convergence, which can be viewed as a probabilistic constructive analog of the Yau–Tian–Donaldson conjecture. The variational approach reveals, in particular, that the convergence holds under the hypothesis that there is no phase transition, which—from the algebro-geometric point of view—amounts to an analytic property of a certain Archimedean zeta function.

Published 5 May 2020