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 P{GL}ₙ(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→ Y , where G is a finite extension of a torus T , we show that X is of klt type if and only if Y is of klt type. We prove a structural theorem for T -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of T -quasi-torsors over a variety with klt type singularities is eventually a sequence of 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 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.