Algebraicity of the metric tangent cones and equivariant k-stability

Chi Li, Xiaowei Wang, Chenyang Xu

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


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 -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 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-polystability is equivalent to equivariant K-polystability, that is, to check K-polystability, it is sufficient to check special test configurations which are equivariant under the torus action..

Original languageEnglish (US)
Pages (from-to)1175-1214
Number of pages40
JournalJournal of the American Mathematical Society
Issue number4
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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