From complex to non-Archimedean geometry: an approach to the YTD conjecture

TL;DR

This paper explores the connections between algebraic, analytic, and non-Archimedean geometry over complex numbers, and provides a sketch of a proof for a version of the Yau--Tian--Donaldson conjecture related to Kähler metrics.

Contribution

It offers a new approach linking complex and non-Archimedean geometries to address the Yau--Tian--Donaldson conjecture in Kähler geometry.

Findings

Sketches a proof of a version of the Yau--Tian--Donaldson conjecture

Highlights the relation between algebraic, analytic, and non-Archimedean geometry

Provides insights into constant scalar curvature Kähler metrics

Abstract

These notes expand on talks given by the authors at the 2025 Summer Research Institute in Algebraic Geometry in Fort Collins, Colorado. We discuss the relation between algebraic, analytic, and non-Archimedean geometry over the complex numbers, and sketch a proof of a version of the Yau--Tian--Donaldson conjecture for constant scalar curvature K\"ahler metrics.