摘要
Given a klt singularity x∈(X,D),we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized vol-ume function 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-mono-mial valuations up to rescaling.As a consequence,we prove that for a klt singular-ity x∈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∈X,hence confirming a conjecture by Donaldson-Sun.
