Geometry & Topology

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

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.

Article information

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

Dates
Revised: 20 June 2017
Accepted: 20 July 2017
First available in Project Euclid: 13 April 2018

https://projecteuclid.org/euclid.gt/1523584828

Digital Object Identifier
doi:10.2140/gt.2018.22.2465

Mathematical Reviews number (MathSciNet)
MR3784527

Zentralblatt MATH identifier
06864343

Citation

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. https://projecteuclid.org/euclid.gt/1523584828

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