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 hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for an $n$-dimensional toroidal compactification $\bar{X}$ with boundary $D$, $K_{\bar{X}}+(1-\frac{n+1}{2\pi })D$ is nef, and in particular that $K_{\bar{X}}$ is ample for $n\ge 6$. By an independent algebraic argument, we prove that every hyperbolic manifold of dimension $n\ge 3$ is of general type, and conclude that the phenomena famously exhibited by Hirzebruch in dimension 2 do 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.

Minor typos corrected. Comments welcome

References in corpus (1)

• Effective results for complex hyperbolic manifolds

Cited by in corpus (3)

• On Effective Existence of Symmetric Differentials of Complex Hyperbolic Space Forms
• On the Betti Numbers of Finite Volume Hyperbolic Manifolds
• Orbifold K{ä}hler Groups related to arithmetic complex hyperbolic lattices