Algebraicity of the Metric Tangent Cones and Equivariant K-stability

Abstract

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun's Conjecture which says that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kahler-Einstein Fano manifolds only depends on the algebraic structure of the singularity. The second result says that for any log Fano variety with a torus action, the K-polystability is equivalent to the equivariant K-polystability, that is, to check K-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

33 pages. To appear in J. Amer. Math. Soc

References in corpus (7)

• A generalization of Ross-Thomas' slope theory
• Sasaki-Einstein metrics and K-stability
• Optimal destabilizing centers and equivariant K-stability
• Existence of valuations with smallest normalized volume
• Normal singularities with torus actions
• Special test configurations and $K$-stability of Fano varieties
• Geometrical pluripotential theory on Sasaki manifolds

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• Equivariant K-stability under finite group action
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• On the sharpness of Tian's criterion for K-stability
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• Approximating delta invariants in the sense of complements of plt type
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• On K-polystability for log del Pezzo pairs of Maeda type