## 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.
 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