nLab
Oka manifold

Contents

Context

Complex geometry

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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 P n\mathbb{C}P^n, nn \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 C n\mathbf{C}^n (n>1n\gt1) is an Oka manifold.

(Kusakabe 20, Theorem 1.2, Corollary 1.3)

References

Introduction and review:

See also:

Proof of the homotopy-theoretic Oka principle:

Last revised on July 19, 2021 at 05:24:09. See the history of this page for a list of all contributions to it.