Reductive quotients of klt singularities

TL;DR

This paper proves that quotients of klt singularities by reductive groups preserve the klt property, impacting the understanding of moduli spaces, GIT quotients, and geometric structures in algebraic geometry.

Contribution

It establishes that taking quotients of klt singularities by reductive groups results in klt type spaces, extending the class of known klt spaces and applications.

Findings

Quotients of klt singularities by reductive groups are of klt type.

GIT-quotients of klt varieties are of klt type.

Moduli spaces of Fano type and related quotients have klt or Fano type singularities.

Abstract

We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ endowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian K\"ahler $G$-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$-dimensional K-polystable Fano manifolds of volume $v$ has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.