Kahler-Einstein metrics, canonical random point processes and birational geometry
Paper in proceedings, 2018

In the present paper and the companion paper (Berman, 2017) 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 measure, coinciding with the canonical measure of Song-Tian and Tsuji. In the case of a 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. 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 Kahler-Einstein metric on X and hence to K-stability as in the Yau-Tian-Donaldson conjecture.

Author

Robert Berman

Chalmers, Mathematical Sciences, Algebra and geometry

ALGEBRAIC GEOMETRY: SALT LAKE CITY 2015, PT 1

2324-707X (ISSN)

Vol. 97 29-73

American-Mathematical-Society Summer Research Institute on Algebraic Geometry
Salt Lake City, USA,

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1090/pspum/097.1/01669

More information

Latest update

9/12/2018