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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Algebraicity of the metric tangent cones and equivariant K-stability
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by Chi Li, Xiaowei Wang and Chenyang Xu HTML | PDF
J. Amer. Math. Soc. 34 (2021), 1175-1214 Request permission

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.
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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