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Contemporary Mathematics 

To any projective pair (X,B) equipped with an ample Q-line bundle L (or even any ample numerical class), we attach a new invariant $$\beta(\mu)$$, defined on convex combinations $$\mu$$ of divisorial valuations on X , viewed as point masses on the Berkovich analytification of X . The construction is based on non-Archimedean pluripotential theory, and extends the Dervan–Legendre invariant for a single valuation – itself specializing to Li and Fujita’s valuative invariant in the Fano case, which detects K-stability. Using our $$\beta$$-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies (X,B) is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization L. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of (X,L), as considered by Chi Li. 

We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–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 a continuity method or the 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.