A variational approach to 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.Comment: Added Appendix B on a valuative analysis of singularities of plurisubharmonic functions. Various other small changes and improvements. To appear in Journal of the AM