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$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$to its multiplicity at the cusp. There are a number of consequences: we show that for an$n$-dimensional toroidal compactification$\overline{X}$with boundary$D$,$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$is ample for$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$, and in particular that$K_{\overline{X}}$is ample for$n\geqslant 6$. By an independent algebraic argument, we prove that every ball quotient of dimension$n\geqslant 4$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.