  The goal of this paper is to study the geometry of cusped complex hyperbolic
manifolds through their compactifications. We characterize toroidal
compactifications with non-nef canonical divisor. We derive effective very
ampleness results for toroidal compactifications of finite volume complex
hyperbolic manifolds. We estimate the number of ends of such manifolds in terms
of their volume. We give effective bounds on the number of complex hyperbolic
manifolds with given upper bounds on the volume. Moreover, we give two sided
bounds on their Picard numbers in terms of the volume and the number of cusps.
