#### Volume 22, issue 4 (2018)

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Classification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{\mskip2mu 2} = 3$

### Luca F Di Cerbo and Matthew Stover

Geometry & Topology 22 (2018) 2465–2510
##### 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; ie 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. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.

##### Keywords
toroidal compactifications, complex hyperbolic manifolds, minimal volume manifolds, arithmetic lattices
##### Mathematical Subject Classification 2010
Primary: 22E40, 20H10, 57M50
Secondary: 14M27, 32Q45, 11F60