geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
In (Larussen 01, Larussen 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
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 $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ satisfies descent with respect to finite covers.
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
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
Some other articles on Oka principle:
Related MO discussion: by Georges Elencwajg