A counterexample to the Serre problem with a bounded domain of \({\mathbb{C}}^ 2\) as fiber. (English) Zbl 0585.32030

Cœuré, Gérard; Loeb, Jean-Jacques

Ann. Math. (2) 122, 329-334 (1985).

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
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

Keywords:

Reinhardt domain; plurisubharmonic function; Serre problem; holomorphic fibre bundle; bounded Stein domain

Citations:

Zbl 0423.32008