Finite generation for valuations computing stability thresholds and applications to K-stability

• 成果类型: Article
• 署名作者: Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.196.2.2
• 发表日期: 2022
• 页码: 507-566
• 关键词: kahler-einstein metrics; log canonical thresholds; q-fano varieties; alpha-invariants; Boundedness; LIMITS; approximation; SINGULARITIES; MANIFOLDS; volumes

摘要

We prove that on any log Fano pair of dimension n whose stability threshold is less than n+1/n, 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 Kahler-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.