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# 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

#### 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