Sur un théorème de connexité de Mumford pour les espaces homogènes

We prove some consequences of an old unpublished connectivity result of Mumford. These mostly deal with the fundamental group of some (ramified) coverings of homogeneous spaces; for example, it is shown that the fundamental group of a complex projective irreducible normal variety which is a covering of a simple (in the sense of 2.2) homogeneous space V, of degree ≤dim (V), is isomorphic to π₁ (V).