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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.

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Keywords
toroidal compactifications, complex hyperbolic manifolds, minimal volume manifolds, arithmetic lattices
Mathematical Subject Classification 2010
Primary: 22E40, 20H10, 57M50
Secondary: 14M27, 32Q45, 11F60
References
Publication
Received: 6 February 2017
Revised: 20 June 2017
Accepted: 20 July 2017
Published: 5 April 2018
Proposed: Richard Thomas
Seconded: Dan Abramovich, Ian Agol
Authors
Luca F Di Cerbo
Mathematics Section
International Centre for Theoretical Physics
Trieste
Italy
Matthew Stover
Department of Mathematics
Temple University
Philadelphia, PA
United States