2013/07/13 by Berman, Robert J.
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1307.3634
In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and shown to converge in probability towards a canonical deterministic measure on X, coinciding with the canonical measure of Song-Tian and Tsuji. The proof is based on new large deviation principle for Gibbs measures with singular Hamiltonians which relies on an asymptotic submean inequality in large dimensions, proved in a companion paper. In the case of a variety X of general type we obtain as a corollary that the (possibly singular) Kähler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. In the opposite setting of a Fano variety X we relate the canonical point processes to a new notion of stability, that we call Gibbs stability, which admits a natural algebro-geometric formulation and which we conjecture is equivalent to the existence of a Kähler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.