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

TL;DR

This paper classifies minimal finite volume complex hyperbolic surfaces with cusps that admit smooth toroidal compactifications, identifying a unique example related to a Picard modular surface.

Contribution

It provides a classification of specific complex hyperbolic surfaces with particular Chern class relations and identifies a unique such surface linked to Picard modular surfaces.

Findings

Only one such surface exists with the specified properties.

The identified surface is related to a Picard modular surface.

The classification excludes surfaces birational to bi-elliptic surfaces.

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.