Extension of holomorphic bundles to the disc (and Serre’s Problem on Stein bundles)

Rosay, Jean-Pierre

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 517-523.

On montre que des fibrés holomorphes, à fibre ${ℂ}^n$, définis sur des ouverts de $ℂ$ par des automorphismes de transition localement constants se prolongent en fibrés holomorphes définis sur la sphère de Riemann. Ceci permet en particulier d’obtenir un exemple de fibré non de Stein sur le disque, avec automorphismes de transition polynomiaux.

Holomorphic bundles, with fiber ${ℂ}^n$, defined on open sets in $ℂ$ by locally constant transition automorphisms, are shown to extend to holomorphic bundles on the Riemann sphere. In particular, it allows us to give an example of a non-Stein holomorphic bundle on the unit disc, with polynomial transition automorphisms.

Classification : 32L05, 32Q28
Keywords: Holomorphic bundles, Stein manifolds, groups of automorphisms of $\mathbb{C}^n$
Mot clés : fibrés holomorphes, variétés de Stein, groupes d’automorphismes de $\mathbb{C}^n$

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