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# Mathematical Research Letters

## Volume 24 (2017)

### Number 4

### On the canonical divisor of smooth toroidal compactifications

Pages: 1005 – 1022

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n4.a4

#### Authors

#### Abstract

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be $\mathrm{nef}$ 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 étale 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 “Effective results for complex hyperbolic manifolds”, *J. London Math. Soc.* 91 (2015), no. 1, 89–104.

Received 11 May 2015

Published 9 November 2017