A Yau-Tian-Donaldson correspondence for degenerations

Abstract

We prove an equivalence between the existence of a unique Kähler-Einstein metric on a degeneration of Fano varieties and uniform CM minimization. This proves a discretely valued analogue of the (uniform) Yau-Tian-Donaldson conjecture, in the sense of non-Archimedean geometry. As a main tool, we prove that Ding stability of a degeneration implies uniform Ding minimization, proving a converse to work of Blum-Xu. We also prove an analogous result in geometric invariant theory (GIT), stating that uniform GIT height minimization of a family of points is equivalent to GIT stability, proving a converse to work of Wang, Wang-Xu and the author.