One-cusped complex hyperbolic $2$-manifolds

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} \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 .