Geometry & Topology

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

Luca F Di Cerbo and Matthew Stover

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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.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 2465-2510.

Received: 6 February 2017
Revised: 20 June 2017
Accepted: 20 July 2017
First available in Project Euclid: 13 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 57M50: Geometric structures on low-dimensional manifolds
Secondary: 14M27: Compactifications; symmetric and spherical varieties 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds 11F60: Hecke-Petersson operators, differential operators (several variables)

toroidal compactifications complex hyperbolic manifolds minimal volume manifolds arithmetic lattices


Di Cerbo, Luca F; Stover, Matthew. Classification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{2} = 3$. Geom. Topol. 22 (2018), no. 4, 2465--2510. doi:10.2140/gt.2018.22.2465.

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