Stability of Valuations: Higher Rational Rank
By Chi Li. Peking Mathematical Journal, 2017
81 citations recorded by the source corpus.
Given a klt singularity $$x\in (X, D)$$x∈(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 $${\widehat{\text{vol}}}_{(X,D),x}$$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 $$x\in X$$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 $$x\in X$$x∈X, hence confirming a conjecture by Donaldson–Sun.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/6aade43b71ff54dd3664a45480ec522ed7328c89
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Stability of valuations and Kollár components (2016, methodology): “mputed out in [Li15, Example 2.8] could be irrational, the result in this paper does not directly tell whether it is a minimizer in Val X;o. This irregular situation is studied in the following paper [LX17] (see also [LL16, section 6]). 4.A notion called weakly-exceptional singularity is introduced ”
- K-stability of Fano varieties via admissible flags (2020, unknown): “Moreover, graded rings associated to lc places of Q -complement as in the above statement are usually expected to be finitely generated; see, for example, [35, Conjecture 1.2].”
- Algebraicity of the metric tangent cones and equivariant K-stability (2018, methodology): “ ∩H 0(S,mL) = M p∈Z t−pa p. (15) Indeed, it is easy to verify that the map t−p+mkf →t−pf for any f ∈a p ∩H0(S,mL) is an isomorphism of the two algebras. As in the proof of Lemma 3.4 (see also [Li17], [LX17]), there is a rank 2 torus (C∗)2 acting on X(1) 0 , such that if we let ξ0 be the vector gener”