Extension of holomorphic bundles to the disc (and Serre’s Problem on Stein bundles)
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 517-523.

Holomorphic bundles, with fiber n , defined on open sets in 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.

On montre que des fibrés holomorphes, à fibre n , définis sur des ouverts de 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.

DOI: 10.5802/aif.2267
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 $\mathbb{C}^n$

Rosay, Jean-Pierre 1

1 University of Wisconsin Department of Mathematics Madison WI 53706 (USA)
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Rosay, Jean-Pierre. Extension of holomorphic bundles to the disc (and Serre’s Problem on Stein bundles). Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 517-523. doi : 10.5802/aif.2267. https://aif.centre-mersenne.org/articles/10.5802/aif.2267/

[1] Ahern, P.; Flores, M.; Rosay, J-P. On + and complete holomorphic vector fields, Proc. A.M.S., Volume 128 (2000), pp. 3107-3113 | DOI | MR | Zbl

[2] Ahern, P.; Forstnerič, F. One parameter automorphism groups on 2 , Complex Varables Theory Appl., Volume 27 (1995), pp. 245-268 | MR | Zbl

[3] Andersén, E.; Lempert, L. On the group of holomorphic automorphisms of n , Inven. Math., Volume 110 (1992), pp. 371-388 | DOI | MR | Zbl

[4] Buzzard, G.; Fornaess, J. E. Complete holomorphic vector fields and time-1 maps, Indiana Math. J., Volume 44 (1995), pp. 1175-1182 | MR | Zbl

[5] Coeuré, G.; Loeb., J-J. A counterexample to the Serre problem with a bounded domain in 2 ,, Ann. Math., Volume 122 (1985), pp. 329-334 | DOI | MR | Zbl

[6] Demailly, J-P.; Springer Différents exemples de fibrés holomorphes non de Stein, Séminaire P. Lelong, H. Skoda 1976/77, Volume 694 (1978), pp. 15-41 | MR | Zbl

[7] Demailly, J-P. Un exemple de fibré holomorphe non de Stein à fibre 2 ayant pour base le disque ou le plan, Inven. Math., Volume 48 (1978), pp. 293-302 | DOI | MR | Zbl

[8] Forstnerič, F.; Prezelj, J. Oka’s principle for holomorphic bundles with sprays, Math. Ann., Volume 317 (2000), pp. 117-154 | DOI | MR | Zbl

[9] Forstnerič, F.; Rosay, J-P. Approximation of biholomorphic mappings by automorphisms of n , Inven. Math., Volume 112 (1993), pp. 323-349 Erratum in Inven. Math. 118 (1994), 573-574 | DOI | MR | Zbl

[10] Gromov, M. Oka’s principle for holomorphic sections of elliptic bundles, Journal A.M.S., Volume 2 (1989), pp. 851-897 | MR | Zbl

[11] Jung, H. Über ganze birationale Transformationen der Ebene, Reine Angew. Math., Volume 184 (1942), pp. 161-174 | DOI | MR | Zbl

[12] Skoda, H. Fibrés holomorphes à base et à fibre de Stein, Inven. Math., Volume 43 (1977), pp. 97-107 | DOI | MR | Zbl

[13] Skoda, H. Fibrés holomorphes à base et fibre de Stein, C. R. Acad. Sci. Paris Série AB, Volume 284 (1977) no. 19, pp. 1199-1202 | MR | Zbl

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