Classification and arithmeticity of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}=3$

TL;DR

This paper classifies the smallest volume smooth complex hyperbolic surfaces with smooth toroidal compactifications, explicitly constructs them, and shows they are all arithmetic and related to bielliptic surfaces.

Contribution

It provides a complete classification of minimal volume such surfaces, constructs explicit examples, and establishes their arithmetic nature and relation to bielliptic surfaces.

Findings

Five such surfaces exist, all arithmetic.

All associated lattices are commensurable.

First example is a blowup of an Abelian surface.

Abstract

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an Abelian surface at one point. The others are bielliptic surfaces blown up at one point. These appear to be the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.