Emergent complex geometry

2021/09/01 by Robert J. Berman, Berman, Robert J.
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2109.00307

openalex publication_date 2021/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a double exposure of the probabilistic construction of Kahler-Einstein metrics on a complex projective algebraic variety X - where the Kahler-Einstein metric emerges from a canonical random point process on X - and the variational approach to the Yau-Tian-Donaldson conjecture, highlighting their connections. The final section is a report on joint work in progress with Sébastien Boucksom and Mattias Jonsson on how the non-Archimedean geometry of X (with respect to the trivial absolute value) also emerges from the probabilistic framework.

Citations

• A generalization of Ross-Thomas' slope theory
• A non-Archimedean approach to K-stability

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