## 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 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

## Keywords

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