标题： Tians α-不变量的弗罗贝尼乌斯版本 显示英文标题

标题： A Frobenius Version of Tians Alpha-Invariant

摘要： 对于由一个在特征p>0的完美域上的射影簇X和X上的一个充分线丛L组成的对(X,L)，我们引入并研究了Tian提出的$\alpha$不变量的正特征类比，我们称之为$\alpha_F$不变量。我们利用正特征下的F奇异性的理论，我们的方法是用与之密切相关的全局F正则性的概念来替代klt奇点。我们证明，对(X,L)的$\alpha_F$不变量可以通过线性系统|mL|的全局弗罗贝尼乌斯分裂来理解，其中m>0。我们建立了$\alpha_F$不变量与F-签名之间的不等式，并利用此来证明所有全局F-正则射影簇（相对于X上的任何充分L）的$\alpha_F$不变量都是正的。当X是一个Fano簇且L是$-K_X$时，我们证明X的$\alpha_F$不变量总是被1/2上界，并与F-签名建立了更紧密的比较。 我们还证明对于toric Fano代数簇，$\alpha_F$不变量与通常的(复数)$\alpha$不变量相匹配。 最后，我们证明在全局F-正则的Fano代数簇族中，$\alpha_F$不变量是下半连续的。 显示英文摘要

摘要： For a pair (X,L) consisting of a projective variety X over a perfect field of characteristic p>0 and an ample line bundle L on X, we introduce and study a positive characteristic analog of the $\alpha$-invariant introduced by Tian, which we call the $\alpha_F$-invariant. We utilize the theory of F-singularities in positive characteristics, and our approach is based on replacing klt singularities with the closely related notion of global F-regularity. We show that the $\alpha_F$-invariant of a pair (X,L) can be understood in terms of the global Frobenius splittings of the linear systems |mL|, for m>0. We establish inequalities relating the $\alpha_F$-invariant with the F- signature, and use that to prove the positivity of the $\alpha_F$-invariant for all globally F-regular projective varieties (with respect to any ample L on X). When X is a Fano variety and L is $-K_X$, we prove that the $\alpha_F$-invariant of X is always bounded above by 1/2 and establish tighter comparisons with the F-signature. We also show that for toric Fano varieties, the $\alpha_F$-invariant matches with the usual (complex) $\alpha$-invariant. Finally, we prove that the $\alpha_F$-invariant is lower semicontinuous in a family of globally F-regular Fano varieties.