论文标题
KLT品种的还原盖
Reductive covers of klt varieties
论文作者
论文摘要
在本文中，我们研究了$ g $ covers of KLT品种，其中$ g $是一个还原的组。首先，我们展示了一个klt奇异性的示例，该奇点承认$ \ mathbb {p} {\ rm gl} _n（\ mathbb {k}）$ - 封面不为KLT类型。然后，我们将自己限制在$ g $ -Quasi-Torsors，这是一类特殊的$ g $ covers，其行为就像$ g $ - torsorsors在Codimension二的封闭子集之外。给定一个$ g $ -quasi-torsor $ x \ rightarrow y $，其中$ g $是torus $ \ mathbb {t} $的有限扩展，我们证明$ x $是klt类型的，并且仅当$ y $属于klt类型时。我们证明了$ \ mathbb {t} $ - 对普通品种的cox环的结构定理。作为一个应用程序，我们表明$ \ mathbb {t} $ - 与KLT类型奇点的各种序列的每个序列最终都是$ \ Mathbb {t} $ - torsors的序列。这是由于Greb-kebekus-peternell而引起的，这是关于KLT型奇异性品种有限的准标准。相反，我们表明，在任何维度上都存在一系列有限的准标准和$ \ mathbb {t} $ - klt型品种上的准tors子，因此无限的许多不是Torsors。我们表明，具有KLT型奇异性的每种品种都是具有规范阶乘奇异性的品种的商。我们证明，具有Zariski本地曲折的奇异性的变化确实是可解决的群体平滑品种的商。最后，在Stibitz的工作中，我们研究了先前结果所保持的最佳奇异性类别。
In this article, we study $G$-covers of klt varieties, where $G$ is a reductive group. First, we exhibit an example of a klt singularity admitting a $\mathbb{P}{\rm GL}_n(\mathbb{K})$-cover that is not of klt type. Then, we restrict ourselves to $G$-quasi-torsors, a special class of $G$-covers that behave like $G$-torsors outside closed subsets of codimension two. Given a $G$-quasi-torsor $X\rightarrow Y$, where $G$ is a finite extension of a torus $\mathbb{T}$, we show that $X$ is of klt type if and only if $Y$ is of klt type. We prove a structural theorem for $\mathbb{T}$-quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of $\mathbb{T}$-quasi-torsors over a variety with klt type singularities is eventually a sequence of $\mathbb{T}$-torsors. This is the torus version of a result due to Greb-Kebekus-Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and $\mathbb{T}$-quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.
