Shafarevich mappings and period mappings

Abstract

We shall show that a smooth, quasi-projective variety $X$ has a holomorphically convex universal covering $\wt X$ when (i) ${\pi }_1(X)$ is residually nilpotent and (ii) there is an admissable variation of \mhs\ over $X$ whose monodromy representation has a finite kernel, and where in each case a corresponding period mapping is assumed to be proper.