One-cusped complex hyperbolic $2$-manifolds

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 $1$ so that $Z_d\setminus E_d$ 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 $d\ge 1$.