We shall show that a smooth, quasi-projective variety X has a holomorphically convex universal covering eX when (i) 1(X) is residually nilpotent and (ii) there is an admissable variation of mixed Hodge structure over X whose monodromy representation has a nite kernel, and where in each case a corresponding period mapping is assumed to be proper.
I. Introduction and statements of results II. The nilpotent case III. The variation of Hodge structure and mixed Hodge structure cases IV. Further directions
