Kähler-Einstein metrics, random point processes and variational principles (RANDOM-KAHLER)

Research Project, 2013 – 2017

In broad terms the aim of this proposal is to introduce a new probabilistic approach to the study of Kähler-Einstein (K-E) metrics on complex manifolds. A precise procedure, based on a blend of Statistical Mechanics, Pluripotential theory and Kähler Geometry, will be used to show that when a K-E metric exists on a complex manifold X it can be obtained from the large N limit of certain canonical random point processes on X with N particles.The canonical point processes are directly defined in terms of algebro-geometric data and the thrust of this approach is thus that it gives a new link between algebraic geometry on one and hand and complex differential (Kähler) geometry on the other. A major motivation for this project comes from the fundamental Yau-Tian-Donaldson conjecture in Kähler geometry, which aims at characterizing the obstructions to the existence of a K-E metric on a Fano manifold in terms of a suitable notion of algebro-geometric stability, notably K-Stability. In this project a new probabilistic/statistical mechanical version of stability will be introduced referred to as Gibbs stability, which also has an interesting purely algebro-geometric definition in the spirit of the Minimal Model Program in current algebraic geometry and another specific aim of this project is to prove or at least make substantial progress towards proving. There is a (unique) K-E metric on a Fano manifold X precisely when X is asymptotically Gibbs stable. The canonical random point processes will be defined as certain beta-deformations of determinantal point processes and share certain properties with the ones appearing in Random Matrix Theory and in the study of quantum chaos and zeroes of random polynomials (and random holomorphic sections). But a crucial new feature here is that the processes are independent of any back-ground data, such as a potential or a metric.