@article{f1e31b4f75654d4cb256dbf9f36f9c1a,

title = "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 -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{\textquoteright}s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of K{\"a}hler-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..",

author = "Chi Li and Xiaowei Wang and Chenyang Xu",

note = "Funding Information: Part of the work was done when the second author was visiting IHES and the third author was visiting Institut Henri Poincar{\'e}(partially sponsored by {\textquoteleft}the Poincar{\'e} Chair{\textquoteright}), to which they want to thank the inspiring research environment. Funding Information: Received by the editors May 29, 2018, and, in revised form, January 1, 2019, October 19, 2020, and December 19, 2020. 2020 Mathematics Subject Classification. Primary 14J17, 14J45 . 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 {\textquoteleft}Chinese National Science Fund for Distinguished Young Scholars (11425101){\textquoteright} and NSF Grant DMS-1901849. Publisher Copyright: {\textcopyright} 2021 American Mathematical Society.",

year = "2021",

doi = "10.1090/jams/974",

language = "English (US)",

volume = "34",

pages = "1175--1214",

journal = "Journal of the American Mathematical Society",

issn = "0894-0347",

publisher = "American Mathematical Society",

number = "4",

}