Monomial

Algorithm

Combinatorics

Gravitational singularity

Mathematical analysis

Tangent

Local ring

Tangent cone

Singularity

Rank (graph theory)

Organic chemistry

Chemistry

Pure mathematics

Geometry

Ring (chemistry)

Conjecture

Cone (formal languages)

Mathematics

Logarithm

Algebraic number

Peking Mathematical Journal

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.

Stability of Valuations: Higher Rational Rank