Singularities of linear systems and boundedness of Fano varieties
Singularities of linear systems and boundedness of Fano varieties
复制标题
DOI：
10.4007/annals.2021.193.2.1
复制
发表时间：
2016-09
影响因子：
4.9
通讯作者：
C. Birkar
中科院分区：
文献类型：
--
作者：
C. Birkar
我们研究了$\mathbb{R}$-线性系统的对数正则阈值(也称为全局对数正则阈值或$\α$不变)。我们证明了不同环境下正下界的存在性，特别是证明了Ambro的一个猜想。然后，我们证明了Borisv-Alexeev-Borisv猜想成立，即给定一个自然数$d$和一个正实数$\epsilon$，具有$\epsilon$-对数正则奇点的维$d$的Fano簇构成一个有界族。这意味着由有理连通簇构成的双态自同构群是Jordan的，它特别回答了Serre的一个问题。接下来，我们证明了如果一个Fano簇的反正则系统的对数正则门限至多为1，那么它是由某个因子计算的，回答了田在这种情况下的一个问题。
We study log canonical thresholds (also called global log canonical threshold or $\alpha$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $\epsilon$, the set of Fano varieties of dimension $d$ with $\epsilon$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.