The Kodaira dimension of complex hyperbolic manifolds with cusps

Abstract

We prove a bound relating the volume of a curve near a cusp in a complex ball quotient to its multiplicity at the cusp. There are a number of consequences: we show that for an -dimensional toroidal compactification with boundary , is ample for , and in particular that is ample for . By an independent algebraic argument, we prove that every ball quotient of dimension is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green-Griffiths conjecture.