Oka's principle for holomorphic submersions with sprays

By F. Forstnerič, Jasna Prezelj. 2001

65 citations recorded by the source corpus.

We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/38753d8588aebdda89d4f8b9af9c7b8a0bab66ee

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In Applied Mathematics

1. F. Kutzschebauch
2. F. Forstnerič
3. Jasna Prezelj
4. Björn Ivarsson
5. F. Lárusson
6. E. F. Wold
7. J. Fornaess
8. S. Kaliman

In Geometry and Topology

1. F. Forstnerič
2. F. Kutzschebauch
3. Jasna Prezelj
4. Björn Ivarsson
5. E. F. Wold
6. F. Lárusson
7. J. Fornaess
8. S. Kaliman

Citations in the corpus

• Extending holomorphic sections from complex subvarieties (2001, methodology): “Our methods are similar to those developed in the papers of Gromov [Gro], Henkin and Leiterer [HL2] and the authors [FP1], [FP2].”
• The Oka principle for sections of subelliptic submersions (2001, methodology): “Grassmanians. Similar arguments apply to part (b) and we omit the details (compare with the proof of Corollary 1.8 in [ FP2 ]).”
• Oka's principle for holomorphic fiber bundles with sprays (2000, methodology): “In [FP] we show that the h-principle fails in these cases.”
• Survey of Oka theory (2010, unknown): “Prezelj and the first-named author in [21, 22, 23].”
• Holomorphic factorization of mappings into SL_n(C) (2008, methodology): “We will introduce the concept of a spray associated with a holomorphic submersion following [15] and [5].”