## Algebraicity of the metric tangent cones and equivariant K-stability

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Chi Li, Xiaowei Wang and Chenyang Xu
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## Abstract:

We prove two new results on the $K$-polystability of $\mathbb {Q}$-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 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.## References

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## Additional Information

**Chi Li**- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
- Address at time of publication: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
- MR Author ID: 929302
- ORCID: 0000-0001-8725-9389
- Email: chi.li@rutgers.edu
**Xiaowei Wang**- Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102-1222
- MR Author ID: 696391
- ORCID: 0000-0003-1935-1786
- Email: xiaowwan@rutgers.edu
**Chenyang Xu**- Affiliation: BICMR, Beijing 100871, People’s Republic of China; and MIT, Cambridge, Massachussetts 02139
- Address at time of publication: Princeton University, Princeton, New Jersey 08544
- MR Author ID: 788735
- ORCID: 0000-0001-6627-3069
- Email: chenyang@princeton.edu
- Received by editor(s): May 29, 2018
- Received by editor(s) in revised form: January 1, 2019, October 19, 2020, and December 19, 2020
- Published electronically: April 9, 2021
- Additional Notes: The first author was supported in part by NSF Grants DMS-1636488 and DMS-1810867, and an Alfred P. Sloan research fellowship.

The second author was supported in part by a Collaboration Grants for Mathematicians from Simons Foundation: 281299/631318 and NSF Grant DMS-1609335.

The third author was supported in part by ‘Chinese National Science Fund for Distinguished Young Scholars (11425101)’ and NSF Grant DMS-1901849. - © Copyright 2021 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**34**(2021), 1175-1214 - MSC (2020): Primary 14J17, 14J45
- DOI: https://doi.org/10.1090/jams/974
- MathSciNet review: 4301561