On the canonical divisor of smooth toroidal compactifications

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if n ≥ 3 we show that the numerical dimension of the canonical divisor of a smooth n-dimensional compactification is always bigger or equal to n - 1. We also show that up to a finite étale cover all such compactifications have ample canonical class, therefore refining a classical theorem of Mumford and Tai. Finally, we improve in all dimensions n ≥ 3 the cusp count for finite volume complex hyperbolic manifolds given in [DD15a].