  We study toroidal compactifications of finite volume complex hyperbolic
manifolds. We obtain results on the existence or nonexistence of K\"ahler
metrics satisfying certain nonpositive curvature properties on these
compactifications. Starting from quotients of complex hyperbolic space by deep
enough non-uniform arithmetic lattices, we also verify the Shafarevich
conjecture for their compactifications, by showing that their universal covers
are Stein.
