nLab
Oka principle

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.

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:

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×C *C\times C^*, arxiv/1011.4116

Revised on May 23, 2013 17:47:29 by Urs Schreiber (84.153.217.147)