On the classification of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}$ and $\bar{c}_{2}=1$

Abstract

We classify the smallest finite volume complex hyperbolic surfaces with cusps which admit smooth toroidal compactifications and which are not birational to a bi-elliptic surface. Remarkably, there is only one such surface which appears to be the compactification of a Picard modular surface.