1307.3634 (Robert J. Berman)

Kahler-Einstein metrics, canonical random point processes and birational
geometry    [PDF]

Robert J. Berman

A new 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. More generally, the convergence is shown to hold in the setting of log canonical pairs. In the case of variety X of general type we obtain as a corollary that the (possibly singular) Kahler-Einstein metric on X with negative Ricci curvature is the limit of a canonical sequence of quasi-explicit Bergman type metrics. When X is defined over the integers the partition functions of the point processes define height type arithmetic invariants which, in the case of certain Shimura varieties, are shown to converge to a logarithmic derivative of the corresponding Dedekind zeta function. Finally, in the opposite setting of a Fano variety 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 Kahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.

View original: http://arxiv.org/abs/1307.3634