Reductive covers of klt varieties

  • Lukas Braun

    Leopold-Franzens-Universität Innsbruck, Innsbruck, Austria
  • Joaquín Moraga

    University of California, Los Angeles, USA
Reductive covers of klt varieties cover
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Abstract

In this article, we study -covers of klt varieties, where is a reductive group. First, we exhibit an example of a klt singularity admitting a -cover that is not of klt type. Then, we restrict ourselves to -quasi-torsors, a special class of -covers that behave like -torsors outside closed subsets of codimension two. Given a -quasi-torsor , where is a finite extension of a torus , we show that is of klt type if and only if is of klt type. We prove a structural theorem for -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of -quasi-torsors over a variety with klt type singularities is eventually a sequence of -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 -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.

Cite this article

Lukas Braun, Joaquín Moraga, Reductive covers of klt varieties. Rev. Mat. Iberoam. 40 (2024), no. 5, pp. 1701–1730

DOI 10.4171/RMI/1494