Reductive covers of klt varieties

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.