On the canonical divisor of smooth toroidal compactifications

TL;DR

This paper investigates the properties of the canonical divisor in smooth toroidal compactifications of complex hyperbolic manifolds, establishing nefness, bounds on numerical dimension, and conditions for ampleness, with implications for cusp counts.

Contribution

It proves the nefness of the canonical divisor in higher dimensions, establishes lower bounds on its numerical dimension, and refines cusp count estimates, extending classical results.

Findings

Canonical divisor is nef for dimensions ≥ 3

Numerical dimension ≥ n-1 for n-dimensional compactifications

Refined cusp count bounds for hyperbolic manifolds

Abstract

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if $n\geq 3$ we show that the numerical dimension of the canonical divisor of a smooth $n$-dimensional compactification is always bigger or equal to $n-1$. We also show that up to a finite \'etale cover all such compactifications have ample canonical class, therefore refining a classical theorem of Mumford and Tai. Finally, we improve in all dimensions $n\geq 3$ the cusp count for finite volume complex hyperbolic manifolds given in [DD15a].