nLab
Oka principle

Idea

The Oka-Grauert principle states that for any Stein manifold $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. The original reference is (Grauert 58).

The principle should maybe better be called the Oka-Grauert-Gromov principle/theory. Gromov viewed it in his book on partial differential relations as one of the examples of h-principle.

Related concepts

• Oka manifold.

References

• K. Oka, Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9, 7–19 (1939)

• H. Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, 263–-273 (1958) doi

• M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–-897.
• Finnur Lárusson, What is an Oka manifold, Notices AMS, pdf
• Franc Forstnerič, Finnur Lárusson, Survey of Oka theory, arxiv/1009.1934
• F. Forstnerič, The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Quarterly (Special Issue in honor of Joseph J. Kohn), 6 (2010), no. 3, 843–874, arxiv/0705.0591

There is now a model category structure on a category of presheaves of simplicial version of a Stein site where Oka maps are a fibration:

• Finnur Lárusson, Model structures and the Oka principle, math.CV/0303355

Theorem (Lárusson) The category of complex manifolds and holomorphic maps can be embedded into a Quillen model category such that:

• a holomorphic map is a weak equivalence in the ambient model category if and only if it is a homotopy equivalence in the usual topological sense.
• a holomorphic map is a fibration if and only if it is an Oka map. In particular, a complex manifold is fibrant if and only if it is Oka.
• a complex manifold is cofibrant if and only if it is Stein.
• a Stein inclusion is a cofibration.

This construction stems from some observations from Jardine, and uses his intermediate model structure from

• J. F. Jardine, Intermediate model structures for simplicial presheaves, Canad. Math. Bull. 49 (2006), no. 3, 407–413, MR2007d:18021

Some other articles on Oka principle:

• Tyson Ritter, A strong Oka principle for embeddings of some planar domains into $C\times C^{*}$, arxiv/1011.4116