On the geometry of smooth compactifications of complex hyperbolic manifolds

In 1984 Hirzebruch constructed the first examples of smooth compactifications of complex hyperbolic manifolds with non-nef canonical divisors. In this talk, we will show how such examples cannot exist if the dimension of the manifold is greater than or equal to three. Finally, we will explain how to use this result to prove effective versions of some classical theorems, such as boundedness of hyperbolic manifolds, Baily-Borel embedding and bounds on the number of cusps. We will conclude with some open problems.