Reductive quotient singularities

The study of quotients by reductive groups is an important topic in algebraic geometry. It manifests when studying moduli spaces, orbit spaces, and $G$-varieties. Many important classes of singularities, as rational singularities, are preserved under quotients by reductive groups. In this talk, we will show that the singularities of the MMP are preserved under reductive quotients. As an application, we show that many good moduli spaces, as the moduli of smoothable $K$-polystable varieties, have klt type singularities.