One-cusped complex hyperbolic 2-manifolds

TL;DR

This paper constructs explicit examples of one-cusped complex hyperbolic 2-manifolds with arbitrarily large volume using geometric methods, revealing new insights into their structure and bounding properties.

Contribution

It introduces a novel geometric construction of one-cusped complex hyperbolic 2-manifolds with large volume and establishes their connection to bounding properties of certain nilmanifolds.

Findings

Constructed infinite families of one-cusped complex hyperbolic 2-manifolds.

Demonstrated these manifolds can have arbitrarily large volume.

Showed that specific nilmanifolds bound geometrically.

Abstract

This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d \ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible curve $E_d$ on $Z_d$ of genus one so that $Z_d \smallsetminus E_d$ admits a finite volume uniformization by the unit ball $\mathbb{B}^2$ in $\mathbb{C}^2$. This produces one-cusped complex hyperbolic $2$-manifolds of arbitrarily large volume. As a consequence, the $3$-dimensional nilmanifold of Euler number $12d$ bounds geometrically for all odd $d \ge 1$.