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The Serre problem with Reinhardt fibers

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The Serre problem with Reinhardt fibers AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Peter PFLUG &Wlodzimierz ZWONEK The Serre problem with Reinhardt fibers Tome 54, no 1 (2004), p. 129-147. <http://aif.cedram.org/item?id=AIF_2004__54_1_129_0> © Association des Annales de l’institut Fourier, 2004, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ 129- THE SERRE PROBLEM WITH REINHARDT FIBERS by Peter PFLUG &#x26; Wlodzimierz ZWONEK 1. Introduction and the main result. Our aim is to discuss the Serre problem, i.e., the problem whether a holomorphic fiber bundle 7r : E - B with a Stein base B and a Stein fiber F is Stein. For a comprehensive list of positive partial results to this problem see e.g. [Siu]. In our paper we consider this problem under the additional assump- tion that the fiber F is a pseudoconvex hyperbolic Reinhardt domain in ~2. Note that the first example showing that the answer to the Serre prob- lem is in general negative were constructed for Reinhardt fibers (see [Sko], [Dem], and [Loeb]). Also first counterexamples with bounded domains as fibers were found in the class of pseudoconvex Reinhardt domains (see [Coe-Loeb]). We are interested in the problem, which bounded pseudoconvex Reinhardt domains as fibers guarantee that the holomorphic fiber bundle with Stein basis is Stein, in other words for which bounded pseudoconv

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