A variational approach to the Yau-Tian-Donaldson conjecture

TL;DR

This paper provides a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, linking geometric stability to algebraic thresholds without relying on traditional analytic methods.

Contribution

It introduces a novel variational approach using pluripotential theory and valuations, avoiding the continuity method and Cheeger-Colding-Tian theory.

Findings

Expresses the greatest twisted Ricci lower bound via algebro-geometric stability.

Establishes a relationship between geodesic rays and non-Archimedean metrics.

Provides a new proof of a version of the Yau-Tian-Donaldson conjecture.

Abstract

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted K\"ahler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.