Effective results for complex hyperbolic manifolds

TL;DR

This paper investigates the geometry of cusped complex hyperbolic manifolds, providing effective bounds and characterizations related to their compactifications, volume, and topological invariants.

Contribution

It offers new effective results on the ampleness, volume bounds, and Picard numbers of complex hyperbolic manifolds with cusps, advancing understanding of their geometric structure.

Findings

Derived effective very ampleness results for toroidal compactifications.

Estimated the number of ends of hyperbolic manifolds based on volume.

Provided bounds on the Picard numbers in terms of volume and cusps.

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.