Abstract
Let Ωbe a bounded symmetric domain and Γ⊂ Aut(Ω) be an irreducible nonuniform torsion-free discrete subgroup. When Γis of rank ≥ 2, Γis necessarily arithmetic, and X :=Ω/Γ?admits a Satake-Baily-Borel compactification. When Ω?is of rank 1, i.e., the complex unit ball Bn of dimension n≥ 1,Γmay be nonarithmetic.When n≥2, by a general result of Siu and Yau, X is pseudoconcave and it can be compactified to a Moishezon space by adding a finite number of normal isolated singularities. In this article we show that for X := Bn/Γ the latter compactification is in fact projective-algebraic.We do this by showing that, just as in the arithmetic case of rank-1, X admits a smooth toroidal compactification \(\bar{X}\)M obtained by adjoining an Abelian variety to each of its finitely many ends, and \(\bar{X}\)M can be blown down to a normal projective-algebraic variety \(\bar{X}\)Min by solving \(\bar{\partial}\)with L2-estimates with respect to the canonical Kähler-Einstein metric and by normalization. As an application, we give an alternative proof of results of Koziarz-Mok on the submersion problem in the case of complex-hyperbolic space forms of finite volume by adapting the cohomological arguments in the compact case to general hyperplane sections of the minimal projective-algebraic compactifications which avoid the isolated singularities.
Research partially supported by the CERG grant HKU7034/04P of the HKRGC, Hong Kong.
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Acknowledgments
The author would like to thank the organizers of the conference “Perspectives in Analysis, Geometry and Topology” in honor of Professor Oleg Viro on May 19–26, 2008, held in Stockholm, especially Professor Jöricke and Professor Passare, for inviting him to give a lecture at the conference. He also wishes to thank Vincent Koziarz for discussions that provided a motivation for finding an analytic proof of projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, making it applicable to the nonarithmetic case.
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Mok, N. (2012). Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume. In: Itenberg, I., Jöricke, B., Passare, M. (eds) Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol 296. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8277-4_14
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DOI: https://doi.org/10.1007/978-0-8176-8277-4_14
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