## 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’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 language | English (US) |
---|---|

Pages (from-to) | 1175-1214 |

Number of pages | 40 |

Journal | Journal of the American Mathematical Society |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics