The Kodaira dimension of complex hyperbolic manifolds with cusps

TL;DR

This paper establishes bounds on curves near cusps in complex hyperbolic manifolds, proving that higher-dimensional hyperbolic manifolds are of general type and exploring implications for algebraic geometry and hyperbolic geometry.

Contribution

It introduces a hybrid geometric and algebraic approach to bound curves near cusps and proves higher-dimensional hyperbolic manifolds are of general type, extending known results beyond dimension 2.

Findings

Bound relating volume of a curve near a cusp to its multiplicity

Proved $K_{ar X}+(1-rac{n+1}{2 extpi}) D$ is nef for toroidal compactifications

Established that hyperbolic manifolds of dimension ≥3 are of general type

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 $n\geq 6$. By an independent algebraic argument, we prove that every hyperbolic manifold of dimension $n\geq 3$ 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.