Complex-analytic quotients of algebraic G-varieties

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski’s class ${\mathcal{Q}}_G$ has a realisation as a good quotient, and that every complete algebraic variety in ${\mathcal{Q}}_G$ is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class ${\mathcal{Q}}_T$, where $T$ is an algebraic torus, is a toric variety.