Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume

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 X¯ M obtained by adjoining an Abelian variety to each of its finitely many ends, and X¯ M can be blown down to a normal projective-algebraic variety X¯ Min by solving ∂¯ 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.