Emergent complex geometry

TL;DR

This paper explores how complex geometry, specifically Kahler-Einstein metrics, can be derived from probabilistic methods and connects this approach to the variational perspective on the Yau-Tian-Donaldson conjecture, including non-Archimedean geometry insights.

Contribution

It introduces a novel probabilistic framework for understanding Kahler-Einstein metrics and links it to the variational approach and non-Archimedean geometry, advancing the theoretical understanding of complex geometry.

Findings

Kahler-Einstein metrics emerge from a canonical random point process.

Connections established between probabilistic constructions and the variational approach to the Yau-Tian-Donaldson conjecture.

Non-Archimedean geometry of the variety also arises from the probabilistic framework.

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 S\'ebastien 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.