Cone (formal languages)

Monomial

Algorithm

Algebraic number

Pure mathematics

Tangent cone

Statistics

Local ring

Gravitational singularity

Tangent

Rank (graph theory)

Stability (learning theory)

Singularity

Computer science

Organic chemistry

Mathematics

Econometrics

Mathematical analysis

Geometry

Combinatorics

Mathematical economics

Logarithm

Conjecture

Chemistry

Machine learning

Ring (chemistry)

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