An invitation to Kähler-Einstein metrics and random point processes

2020/03/25 by Berman, Robert J.
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2003.11358

Abstract

This is an invitation to the probabilistic approach for constructing Kähler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X, i.e. from random point processes on X, defined in terms of algebro-geometric data. The proof of the convergence towards Kähler-Einstein metrics with negative Ricci curvature is explained. In the case of positive Ricci curvature a variational approach is introduced to prove the conjectural convergence, which can be viewed as a probabilistic constructive analog of the Yau-Tian-Donaldson conjecture. The variational approach reveals, in particular, that the convergence holds under the hypothesis that there is no phase transition, which - from the algebro-geometric point of view - amounts to an analytic property of a certain Archimedean zeta function.

Related

• Kähler-Einstein metrics and Archimedean zeta functions
• Kahler-Einstein metrics, canonical random point processes and birational geometry
• Emergent complex geometry
• The probabilistic vs the quantization approach to Kähler-Einstein geometry
• Kähler-Einstein metrics arising from micro-canonical measures and Hamiltonian dynamics