On the volumes of complex hyperbolic manifolds with cusps

A sharp cusp count for complex hyperbolic surfaces and related results

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.

Hyperbolic four-manifolds with one cusp

We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of $k$-cusped hyperbolic four-manifolds with volume smaller than V grows like $C^{V log V}$ for any fixed $k$. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.

具有一个通道的复双曲2流形

One-cusped complex hyperbolic 2-manifolds

本文通过构造具有特定性质的复双曲2流形，展示了其在体积和几何边界上的贡献。

The Kodaira dimension of complex hyperbolic manifolds with cusps

We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for an $n$-dimensional toroidal compactification $\bar X$ with boundary $D$, $K_{\bar X}+(1-\frac{n+1}{2π}) D$ is nef, and in particular that $K_{\bar X}$ is ample for $n\geq 6$. By an independent algebraic argument, we prove that every hyperbolic manifold of dimension $n\geq 3$ is o…

On the cuspidalization problem for hyperbolic curves over finite fields

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necess…

Cusps and the family hyperbolic metric

The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian co…