Thresholds, valuations, and K-stability
Thresholds, valuations, and K-stability
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DOI：
10.1016/j.aim.2020.107062
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发表时间：
2017-06
影响因子：
1.7
通讯作者：
Harold Blum;Mattias Jonsson
中科院分区：
文献类型：
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作者：
Harold Blum;Mattias Jonsson
设X是正规复射影簇，最坏有klt个奇点，且X上有La个大线丛.我们使用赋值来研究L的对数典型阈值，以及另一个不变量，稳定性阈值。后者推广了Fujita和Odaka的一个概念，并可用于刻画Q-Fano簇的K-半稳定性或一致K-稳定性.它也可以用来推广Fujita和Liu的体积界。这两个阈值可以写为X上赋值空间上某些泛函的下确界。当L充分时，我们证明了这些下确界是可达到的。在复曲面的情况下，复曲面估值实现这些infima，我们得到简单的表达式的两个阈值的时刻多面体的L。
LetXbe a normal complex projective variety with at worst klt singularities, andLa big line bundle onX. We use valuations to study the log canonical threshold ofL, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when aQ-Fano variety isK-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations onX. WhenLis ample, we prove that these infima are attained. In the toric case, toric valuations achieve these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope ofL.