nLab
Oka principle

Context

Complex geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Oka-Grauert principle states that for any Stein manifold XX the holomorphic and the topological classification of complex vector bundles on XX 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 (Larussen 01, Larussen 03) this is formulated in terms of higher complex analytic geometry of complex analytic infinity-groupoids.

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

Maps hol(Σ,X)Maps top(Σ,X) 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)Psh (SteinSp)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.

(Larussen 03)

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

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 C×C *C\times C^*, arxiv/1011.4116

Related MO discussion: by Georges Elencwajg

Revised on June 7, 2014 02:54:03 by Urs Schreiber (89.204.130.218)