Stability of Valuations and Kollár Components

Abstract

We prove that among all Kollár components obtained by plt blow ups of a klt singularity $o\in (X,D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valuations. Conversely, we show any divisorial minimizer of the normalized volume function yields a K-semistable Kollár component. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by the normalized volumes of Kollár components.

44 pages. Fourth version: substantial improvement on various parts. Notably, Theorem D, Theorem 1.4 and Proposition 4.6. Final version to appear in JEMS

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• Kähler-Einstein metrics and volume minimization
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Cited by in corpus (26)

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• The normalized volume of a singularity is lower semicontinuous
• Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)
• Birational superrigidity and K-stability of singular Fano complete intersections
• On the Yau-Tian-Donaldson conjecture for singular Fano varieties
• A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties
• Uniqueness of K-polystable degenerations of Fano varieties
• Kähler-Einstein metrics and volume minimization
• Wall crossing for K-moduli spaces of plane curves
• Minimizing normalized volumes of valuations
• Calabi-Yau manifolds with isolated conical singularities
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• Stability of Valuations: Higher Rational Rank
• Equivariant K-stability under finite group action
• A Minimizing Valuation is Quasi-monomial
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• A Note on Equivariant K-stability
• On the semi-continuity problem of normalized volumes of singularities
• Minimizing CM degree and slope stability of projective varieties
• Interaction Between Singularity Theory and the Minimal Model Program
• Volume minimization and obstructions to solving some problems in Kähler geometry
• On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line