Reductive quotients of klt singularities (Q6585703)

Kawamata log terminal (klt for short) singularities are an important class of singularities in the minimal model program. A normal variety \(X\) is said to be of klt type if there exists an effective divisor \(B\) such that the pair \((X, B)\) is klt. A natural question is whether the condition of being klt is preserved under quotients by reductive group actions. The paper under review solves this problem for klt type singularities.\N\NSpecifically, let \(X\) be an affine variety of klt type and \(G\) be a reductive group acting on \(X\). Then, the authors show that the affine invariant theoretic quotient \(X// G:=\text{Spec } \mathbb K[X]^{G}\) is of klt type. Note that the boundary \(B\) making \(X\) klt is not assumed to be \(G\)-invariant. In particular, the authors obtain that reductive quotient singularities are of klt type. These results have consequences for complex spaces obtained as quotients of Hamiltonian Kähler \(G\)-manifolds, for collapsing of homogeneous vector bundles, and for good moduli spaces of smooth Artin stacks and so on.