On the canonical divisor of smooth toroidal compactifications

Gabriele Di Cerbo; Luca F. Di Cerbo

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arXiv · 1502.06258

On the canonical divisor of smooth toroidal compactifications

Gabriele Di Cerbo·Luca F. Di Cerbo

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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].

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BibTeX RIS

Gabriele Di Cerbo, Luca F. Di Cerbo. 2015-02-22. On the canonical divisor of smooth toroidal compactifications. https://arxiv.org/abs/1502.06258

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