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Quotients of bounded homogeneous domains by cyclic groups. (English) Zbl 1197.32007
Let $$D\subset \mathbb C^n$$ be a bounded domain and let $$\varphi$$ be an automorphism of $$D$$ such that $$\Gamma:=\langle\varphi\rangle$$ is a discrete subgroup of $$\operatorname{Aut}_{\mathcal O}(D)$$. Then $$\Gamma$$ acts properly on $$D$$ and $$X:=D/\Gamma$$ is a complex space. The author investigates conditions on $$D$$ which guarantee that $$X$$ is Stein.
Write $$\varphi=\varphi_e \varphi_h \varphi_u$$ where $$\varphi_e$$ is elliptic, $$\varphi_h$$ is hyperbolic and $$\varphi_u$$ is unipotent. Then $$\Gamma':=\langle\varphi_h \varphi_u\rangle$$ is discrete in $$G:=\operatorname{Aut}_{\mathcal O}(D)^0$$ and $$X'=D/\Gamma'$$ is Stein if and only if $$X$$ is Stein. $$\Gamma'$$ is contained in a maximal split solvable subgroup $$S$$ of $$G$$ which acts simply transitively on $$D$$. By exploiting the structure theory of $$S$$ one obtains the existence of an equivariant holomorphic submersion $$\pi: D \to D'$$ onto a bounded homogeneous domain $$D'$$ whose fibers are biholomorphically equivalent to the unit ball $$\mathbb B_m$$. The final step consists in the study of the action of $$\Gamma'$$. So it is proved that $$X=D/\Gamma$$ is Stein.
The paper contains good explanations and adequated references to follow these constructions.

##### MSC:
 32M10 Homogeneous complex manifolds 32E10 Stein spaces, Stein manifolds
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