Stability of Valuations: Higher Rational Rank

Given a klt singularity $x\in (X,D)$, we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function ${\widehat{\text{vol}}}_{(X,D),x}$, if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity $x\in X$ on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of $x\in X$, hence confirming a conjecture by Donaldson–Sun.