Mathematical Research Letters

Volume 24 (2017)

Number 4

On the canonical divisor of smooth toroidal compactifications

Pages: 1005 – 1022



Gabriele Di Cerbo (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Luca F. Di Cerbo (Department of Mathematics, Notre Dame University, Notre Dame, Indiana, U.S.A.; and Max Planck Institute for Mathematics, Bonn, Germany)


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