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 (3)

• Optimal destabilizing centers and equivariant K-stability
• Existence of valuations with smallest normalized volume
• Geometrical pluripotential theory on Sasaki manifolds

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