nLab Oka manifold

Idea

Oka manifolds (the term is due to Forstnerič 2009a) comprise the class of complex analytic manifolds which, when used as classifying spaces, satisfy the weak homotopy equivalence form of the Oka principle.

The relative notion are Oka maps (Forstnerič 2009a), these are fibrations in the Jardine-Lárusson model structure on the category of simplicial presheaves on the simplicial Stein site. Those cofibrant objects which are representable by complex manifolds are in fact Stein manifolds.

Examples

Proposition

(complex projective spaces are Oka manifolds)
Every complex projective space $\mathbb{C}P^n$, $n \in \mathbb{N}$, is an Oka manifold. More generally every Grassmannian over the complex numbers is an Oka manifold.

(review in Forstnerič & Lárusson 11, p. 9, Forstnerič 2013, Ex. 2.7)

Yet more generally:

Proposition

(coset spaces of complex Lie groups are Oka manifolds)
Every complex Lie group and every coset space (homogeneous space) of complex Lie groups is an Oka manifold.

(review in Forstnerič & Lárusson 11, p. 9, Forstnerič 2013, Thm. 2.6)

Example

The complement of any compact polynomially convex subset of $\mathbf{C}^n$ ($n\gt1$) is an Oka manifold.

(Kusakabe 20, Theorem 1.2, Corollary 1.3)

References

Introduction and review:

• Finnur Lárusson, What is an Oka manifold?, Notices AMS Volume 57, Number 1, 2010 (pdf, pdf)

• Franc Forstnerič, Finnur Lárusson, Survey of Oka theory, New York J. Math., 17a (2011), 1-28 (arXiv:1009.1934, eudml:232963)

• Franc Forstnerič, Section 5 of: Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)

• Franc Forstnerič (appendix by Finnur Lárusson), Oka manifolds: From Oka to Stein and back, Annales de la Faculté des sciences de Toulouse, Mathématiques, Série 6, Tome 22 (2013) no. 4, pp. 747-809 (numdam:AFST_2013_6_22_4_747_0)

• Franc Forstnerič, Developments in Oka theory since 2017 (arXiv:2006.07888)

See also:

• Finnur Lárusson, Affine simplices in Oka manifolds, Documenta Mathematica (2009) Volume: 14, page 691-697 (eudml:228723)

• Yuta Kusakabe, Oka properties of complements of holomorphically convex sets (arXiv:2005.08247).

Proof of the homotopy-theoretic Oka principle:

• Franc Forstnerič, Oka manifolds, Comptes Rendus Mathematique, Acad. Sci. Paris 347 (2009), 1017–20 (arXiv:0906.2421, doi:10.1016/j.crma.2009.07.005)

• Franc Forstnerič, Oka maps, Comptes Rendus Mathematique, Acad. Sci. Paris, Ser. I 348 (2010) 145-148 (arxiv/0911.3439, doi:10.1016/j.crma.2009.12.004)