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Mathematical Research Letters
Volume 24 (2017)
Number 4
On the canonical divisor of smooth toroidal compactifications
Pages: 1005 – 1022
DOI: https://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
Accepted 2 May 2016
Published 9 November 2017