Special and general types
The Oka-Grauert principle states that for any Stein manifold the holomorphic and the topological classification of complex vector bundles on 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 is an Oka manifold if for every Stein manifold the canonical morphism
from the mapping space of holomorphic functions to that of continuous functions (both equipped with the compact-open topology) is a weak homotopy equivalence.
This is the case precisely if satisfies descent with respect to finite covers.
(Larusson 01, theorem 2.1)
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.
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
Discussion in terms of higher complex analytic geometry and complex analytic infinity-groupoids is in
This construction stems from some observations from Jardine, and uses his intermediate model structure from
Some other articles on Oka principle:
- Tyson Ritter, A strong Oka principle for embeddings of some planar domains into , arxiv/1011.4116
Related MO discussion: by Georges Elencwajg
Revised on November 17, 2015 15:54:02
by Urs Schreiber