Quotients of the unit ball of ℂn for a free action of ℤ

Quotients of bounded homogeneous domains by cyclic groups

“…Therefore we assume that Γ is infinite cyclic. In the case that D is biholomorphically equivalent to the unit ball B n it is proven in [4] and [5] that X = D/ ϕ is Stein for hyperbolic and parabolic automorphisms ϕ. We will generalize this result to arbitrary bounded homogeneous domains.…”

“…In this section we discuss the automorphism group and the normal j-algebra of the unit ball B n := {z ∈ C n ; z < 1} in C n . It has been proven in [4] and [5] that the quotient manifold B n / ϕ is Stein for hyperbolic and parabolic automorphisms ϕ ∈ Aut O (B n ). We will give here a different proof of this fact.…”

A classification of taut, Stein surfaces with a proper $${\mathbb{R}}$$ -action

“…Thus such R-actions push down to proper R-actions on the quotient X /Z, whose fundamental group is Z. Observe that X /Z is taut by Proposition 2.5 and it is Stein by [5]. Finally note that the restrictions X → C and X → H to X of the projections of the associated holomorphic principal C-bundles are Z-equivariant.…”

“…t (u, v) := (u + t, v), t * (u, v) := (u − 2tv + it 2 , v − it).Such actions appear in[5] as normal forms of parabolic elements in the automorphism group of X . It is clear that the globalization with respect to the first action is C 2 and its C-quotient is C.…”

Properly discontinuous actions on bounded domains

“…Acknowledgement. After the manuscript was completed, the author became aware of the existence of [6] where the Steinness of quotients of the unit ball for cyclic, free and properly discontinuous actions is proved (actually they are even biholomorphic to bounded Stein domains!). He thanks Professor Takeo Ohsawa for pointing out this reference and valuable suggestions.…”