Finite generation for valuations computing stability thresholds and applications to K-stability
Finite generation for valuations computing stability thresholds and applications to K-stability
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DOI：
10.4007/annals.2022.196.2.2
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发表时间：
2021-02
影响因子：
4.9
通讯作者：
Yuchen Liu;Chenyang Xu;Ziquan Zhuang
中科院分区：
文献类型：
--
作者：
Yuchen Liu;Chenyang Xu;Ziquan Zhuang
我们证明了在任何n维对数Fano对的稳定性阈值小于$\frac{n+1}{n}$，任何赋值计算的稳定性阈值有一个n-生成的关联分次环。与以前的工作一起，这意味着：（a）log Fano对是一致K-稳定的（分别为）。当且仅当它是K-稳定的（分别为K-多稳态）;（B）K-模空间是真的且可投射的;结合先前已知的K\“ahler-Einstein度量的存在性与变分方法证明的约化一致K-稳定性之间的等价性，（C）Yau-Tian-唐纳森猜想对一般（可能是奇异的）log Fano对成立。
We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies: (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.