Birational geometry of complex hyperbolic manifolds

In 1984 Hirzebruch constructed the first examples of smooth toroidal compactifications of ball quotients with non-nef canonical divisor. In this talk, I will show that if the dimension is greater or equal than three then such examples cannot exist. We will use this result to reprove and improve classical theorems, such as boundedness of hyperbolic manifolds, Baily-Borel embeddings and cusps count.