## Abstract

We prove a version of Jonsson-Mustata's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with previous works by many people, we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space, whose geometric points parametrize K-polystable klt Fano varieties.

Original language | English (US) |
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Pages (from-to) | 1003-1030 |

Number of pages | 28 |

Journal | Annals of Mathematics |

Volume | 191 |

Issue number | 3 |

DOIs | |

State | Published - May 2020 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

## Keywords

- Complement
- K-moduli of fano varieties
- Local volume of klt singularities
- Quasi-monomial