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.

Statement in higher complex analytic geometry

In (Larusson 01, Larusson 03) this is formulated in terms of higher complex analytic geometry of complex analytic infinity-groupoids.

Say that a complex manifold $X$ is an Oka manifold if for every Stein manifold $\Sigma$ the canonical morphism

$$Maps_{hol}(\Sigma, X) \longrightarrow Maps_{top}(\Sigma, X)$$

from the mapping space of holomorphic functions to that of continuous functions (both equipped with the compact-open topology) is a weak homotopy equivalence.

Theorem

This is the case precisely if $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies descent with respect to finite covers.

(Larusson 01, theorem 2.1)

Theorem

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 an Oka manifold.

• a complex manifold is cofibrant if and only if it is Stein.

• a Stein inclusion is a cofibration.

(Larusson 03)

Related concepts

• Oka manifold.

References

Original articles include

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

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

• Mikhail Gromov, Oka’s principle for holomorphic sections

  of elliptic bundles_, J. Amer. Math. Soc. 2 (1989), 851–-897.

• Franc 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

Surveys and reviews include

• Finnur Lárusson, What is an Oka manifold, Notices AMS, pdf

• Franc Forstnerič, Finnur Lárusson, Survey of Oka theory, arxiv/1009.1934

• Franc Forstnerič, section 5.3 of Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011

Discussion in terms of higher complex analytic geometry and complex analytic infinity-groupoids is in

• Finnur Lárusson, Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle (arXiv:math/0101103)

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

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

• John Frederick 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

Related MO discussion: by Georges Elencwajg