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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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