Stability of Valuations: Higher Rational Rank

Abstract

Given a klt singularity $x\in (X,D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function ${\widehat{\mathrm{vol}}}_{(X,D),x}$, 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 to its metric tangent cone is uniquely determined by the algebraic structure of $x\in X$, hence confirming a conjecture by Donaldson-Sun.

55 pages. Comments are welcome v2: the version accepted by Peking Math. J

References in corpus (2)

• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds

Cited by in corpus (7)

• 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
• Uniqueness of K-polystable degenerations of Fano varieties
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• On the semi-continuity problem of normalized volumes of singularities
• Interaction Between Singularity Theory and the Minimal Model Program
• Volume minimization and obstructions to solving some problems in Kähler geometry