At the beginning of the 20th century, it was known that any compact connected, simply connected Riemann surface is biholomorphic to the projective line.
Subsequently, several characterizations of projective spaces were established. For instance, Siu and Yau stated that projective spaces are the only Kähler manifolds with positive holomorphic bisectional curvature, and Mori proved that they are the only projective manifolds that have an ample tangent bundle. In a different direction, projective spaces are the only Kähler--Einstein manifolds with a positive constant satisfying the equality in the Miyaoka--Yau inequality. This result originating from uniformization theory was generalized in the singular setting by Greb, Kebekus, Peternell and Druel, Guenancia, Păun. More precisely, they characterize singular quotients of P^n by finite groups acting freely in codimension 1. The aim of this talk is to discuss a generalization of Greb--Kebekus--Peternell's result in order to characterize quotients of P^n by any group action.
