Algebraicity of the Metric Tangent Cones and Equivariant K-stability

TL;DR

This paper establishes new algebraic criteria for K-polystability of Fano varieties, proving a conjecture relating metric tangent cones to algebraic singularities, and simplifies stability checks using torus symmetries.

Contribution

It proves that K-semistable cones degenerate uniquely to K-polystable ones and confirms that K-polystability can be verified via equivariant test configurations for torus actions.

Findings

Proved unique degeneration of K-semistable cones to K-polystable cones.

Confirmed the algebraic dependence of metric tangent cones on singularities.

Established equivalence of K-polystability and equivariant K-polystability under torus actions.

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.