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Quotients of the unit ball of \(\mathbb C^n\) for a free action of \(\mathbb Z\). (English) Zbl 1008.32011

Let \(D\) be a bounded strictly holomorphically convex domain in \(\mathbb C^n\) and \(\Gamma\) an infinite cyclic subgroup of Aut\(D\) operating freely on \(D\). It is known that \(D\) is biholomorphically equivalent to the open unit ball \(\mathbb B_n\) in \(\mathbb C^n\) and the authors are interested in the global complex structure of the quotient manifold \( D/\Gamma\simeq\mathbb B_n/\Gamma\) and prove that \(\mathbb B_n/\Gamma\) can be realized as a bounded Stein domain in \(\mathbb C^n\). The group \(\Gamma\simeq\mathbb Z\) is generated by a hyperbolic or by a parabolic element \(\gamma\) of Aut\(\mathbb B_n\). In a former paper the first named author got the above result in the hyperbolic situation [see C. de Fabritiis, Complex Variables, Theory Appl. 36, No. 3, 233-252 (1998; Zbl 1022.32007)], while in this paper the authors consider the parabolic case. By using special normal forms for suitable conjugates of \(\gamma\) they construct a biholomorphic map from \(\mathbb B_n/\Gamma\) onto a bounded holomorphically convex domain in \(\mathbb C^n\).

MSC:

32M10 Homogeneous complex manifolds
32Q28 Stein manifolds
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds

Citations:

Zbl 1022.32007
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References:

[1] M. Abate,Iteration Theory of Holomorphic Mappings on Taut Manifolds, Mediterranean Press, Rende, 1989. · Zbl 0747.32002
[2] C. de Fabritiis,A family of complex manifolds covered by {\(\delta\)} n, Complex Variables36 (1998), 233–252. · Zbl 1022.32007
[3] C. de Fabritiis,Commuting holomorphic functions and hyperbolic automorphisms, Proc. Amer. Math. Soc.124 (1996), 3027–3037. · Zbl 0866.32004 · doi:10.1090/S0002-9939-96-03729-X
[4] C. de Fabritiis and A. Iannuzzi,Quotients of the unit ball of \(\mathbb{C}\) n for a free action of \(\mathbb{Z}\), Preprint. · Zbl 1008.32011
[5] F. Docquier and H. Grauert,Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann.140 (1960), 94–123. · Zbl 0095.28004 · doi:10.1007/BF01360084
[6] T. Franzoni and E. Vesentini,Holomorphic Maps and Invariant Distances, North-Holland, Amsterdam, 1980. · Zbl 0447.46040
[7] I. Kaplansky,Linear Algebra and Geometry, Chelsea, New York, 1974. · Zbl 0294.17003
[8] W. Rudin,Function Theory in the Unit Ball of \(\mathbb{C}\) n, Springer, Berlin, 1980. · Zbl 0495.32001
[9] B. Wong,Characterization of the unit ball in \(\mathbb{C}\) n by its automorphism group, Invent. Math.41 (1977), 253–257. · Zbl 0385.32016 · doi:10.1007/BF01403050
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