Algebraicity of the metric tangent cones and equivariant K-stability

By Chi Li, Xiaowei Wang. Journal of The American Mathematical Society, 2018

113 citations recorded by the source corpus.

We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K 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 point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e6e6f19e31fdb627df55f9a9d45dc3571b2d977e

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Citations in the corpus

Reductivity of the automorphism group of K-polystable Fano varieties (2019, methodology): “lenging. We now explain in more detail the proofof S-completeness. The ﬁrst extensive study of the geometry of K-semistable Q-Fano varieties belonging to the same S-equivalence class was completed in [LWX18]. In particular, it was shown that there is a unique object, namely a K-polystable Q-Fano var”
Openness of K-semistability for Fano varieties (2019, methodology): “ed a key role in proving results in [LX16,LWX18,BX18]. We also show the following theorem which gives a characterization of weakly special test conﬁgurations. It is obtained by combining arguments in [LWX18] and [Xu19], which uses the existence of bounded local complements. Theorem 1.6 (=Theorem A.2”