Classification and arithmeticity of toroidalcompactifications with 3c2=c12= 3

2015/05/31 by Luca F. Di Cerbo, Luca Di Cerbo, Matthew Stover
Mathematics · #Algebraic Geometry and Number Theory #Ball (mathematics) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Orbifold #Quotient #Smooth surface #Surface (topology) #Toroid #math.AG #math.DG #math.GT

paper · pdf · doi:10.2140/gt.2018.22.2465

published as Geom. Topol. 22 (2018) 2465-2510 · To appear in Geom. Topol., 41 pages. This paper supersedes arXiv:1309.5516

openalex created_date 2016/06/24 · arxiv created 2017/07/20 · openalex publication_date 2018/04/05 · arxiv updated 2018/04/18 · openalex updated_date 2026/08/05

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.

Citations

• On the classification of toroidal compactifications with 3c₂=c²₁ and c₂=1
• Complex Algebraic Surfaces
• On the Chern numbers of surfaces of general type
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I
• Algebraic Geometry
• Compact Complex Surfaces