Effective results for complex hyperbolic manifolds

Abstract

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of cusps.

Some changes according the comments of the referee and references updated

Cited by in corpus (6)

• The Kodaira dimension of complex hyperbolic manifolds with cusps
• On Effective Existence of Symmetric Differentials of Complex Hyperbolic Space Forms
• On the Betti Numbers of Finite Volume Hyperbolic Manifolds
• Classification and arithmeticity of toroidal compactifications with $3{\bar{c}}_2={\bar{c}}_1^2=3$
• On the classification of toroidal compactifications with $3{\bar{c}}_2={\bar{c}}_1^2$ and ${\bar{c}}_2=1$
• The fundamental group of a toroidal compactification of a Hermitian locally symmetric space