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A counterexample to the Serre problem with a bounded domain of \({\mathbb{C}}^ 2\) as fiber. (English) Zbl 0585.32030
The authors give an example of a holomorphic fibre bundle with base \({\mathbb{C}}^*\) and fibre a bounded Stein domain in \({\mathbb{C}}^ 2\), whose total space is not Stein, thus answering in the negative a conjecture of Y. T. Siu [Bull. Am. Math. Soc. 84, 481-512 (1978; Zbl 0423.32008)]. The bundle is described explicitly as the quotient of a domain in \({\mathbb{C}}^ 3\) by the action of a discrete group of automorphisms.
Reviewer: P.Newstead

MSC:
32L05 Holomorphic bundles and generalizations
32E10 Stein spaces, Stein manifolds
32M05 Complex Lie groups, group actions on complex spaces
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32U05 Plurisubharmonic functions and generalizations
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