## Abstract

We prove that on any log Fano pair of dimension n whose stability threshold is less than (Formula Presented), any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of K-ahler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

Original language | English (US) |
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Pages (from-to) | 507-566 |

Number of pages | 60 |

Journal | Annals of Mathematics |

Volume | 196 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2022 |

### Funding

Acknowledgements. We would like to thank Harold Blum, Xiaowei Wang and Chuyu Zhou for helpful discussions. We also would like to thank the anonymous referees for many helpful comments. YL is partially supported by NSF Grant DMS-2148266 (formerly DMS-2001317). CX is partially supported by NSF Grant DMS-1901849 and DMS-1952531. ZZ is partially supported by NSF Grant DMS-2055531.

## Keywords

- Fano variety
- Higher rank finite generation
- K-ahler{einstein metric
- K-moduli
- K-stability

## ASJC Scopus subject areas

- Mathematics (miscellaneous)