Ressource pédagogique : R. Berman - Canonical metrics, random point processes and tropicalization

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In this talk I will present a survey of the connections between canonical metrics and random point processes on a complex algebraic variety X. When the variety X has positive Kodaira dimension, this leads to a probabilistic construction of the canonical metric on X introduced by Tsuji and Song-Tian (coinciding with the Kähler-Einstien metric when X is of general type). In the opposite setting of Fano varieties this suggests a probalistic analog of the Yau-Tian-Donaldson conjecture. The probabilistic version of the conjecture is open, in general. But, as shown in a recent joint work with Magnus Onnheim, for toric X the “tropicalized” version of the conjecture does hold and involves discrete optimal transport theory.

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