Resumen de Reductive covers of klt varieties
Lukas Braun, Joaquín Moraga
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 PGL 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.
