nLab Oka principle

Contents

Context

Complex geometry

Homotopy theory

Cohomology

Idea
Homotopical Oka principle
In higher complex analytic geometry
Related concepts
References

Idea

In complex geometry, the Oka-Grauert principle states that over complex manifolds $S$ which are Stein manifolds, the non-abelian cohomology-classification of holomorphic vector bundles coincides with that of topological vector bundles,

S \;\text{is Stein} \;\;\;\;\; \Rightarrow \;\;\;\;\; H^1 \Big( S; \, \underline{GL(n, \mathbb{C})}_{hol} \Big) \,\simeq\, H^1 \Big( S; \, \underline{GL(n, \mathbb{C})}_{top} \Big) \,.

This was originally proven for holomorphic line bundles by Oka 1939 (in which case it says that holomorphic line bundles over Stein manifolds are fully classified by their first Chern class) and generalized by Grauert 1958 to holomorphic vector bundles and further to holomorphic principal bundles with structure group any complex Lie group.

As a principle, this Oka-Grauert principle is sometimes stated as follows (Forstnerič 2012):

Analytic problems on Stein spaces which can be cohomologically formulated have only topological obstructions.

More generally, for suitable complex manifolds $A$ now called Oka manifolds (Forstnerič 2009a) – including (see here) the complex Grassmannians that serve as classifying spaces for complex vector bundles –, the inclusion into the space of continuous maps $S \to A$ , out of a Stein manifold $S$ , of the subspace of holomorphic functions is a weak homotopy equivalence:

Maps_{hol}(S, \, A) \xhookrightarrow{ \simeq_{whe} } Maps(S, \, A) \;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\; \underset{n \in \mathbb{N}}{\forall} \;\; \pi_n \big( Maps_{hol}(S, \, A) \big) \xrightarrow{\sim} \pi_n \big( Maps(S, \, A) \big) \,.

(review in Forstnerič & Lárusson 2011, Cor. 3.5, Forstnerič 2013, (1.1))

More generally, an analogous statement applies to suitable fiber bundles of Oka manifolds over Stein manifolds and their spaces of sections (Forstnerič 2009b).

This homotopy theoretic weak homotopy equivalence Oka principle goes back to results of Gromov 89, where (?) it is viewed an an example of the h-principle.

The duality between Stein manifolds and Oka manifolds in this homotopy-theoretic Oka principle is fully brought out by the existence of a model category for complex analytic ∞-groupoids in which a complex manifold is cofibrant/fibrant object if it is Stein/Oka, respectively (Lárusson 2001, 03).

Homotopical Oka principle

Proposition

(weak homotopy equivalence Oka principle) For

$S$ a Stein manifold,
$X$ an Oka manifold,

the inclusion

Maps_{hol} \big( S, \, X \big) \xhookrightarrow{\;\simeq_{whe}\;} Maps \big( S ,\, X \big)

of the subspace of holomorphic functions into the mapping space of their underlying topological spaces (with the compact-open topology) is a weak homotopy equivalence.

(review in Forstnerič & Lárusson 2011, Cor. 3.5)

More generally, for $Z \xrightarrow{\;} S$ a stratified holomorphic fiber bundle of Oka manifolds, the corresponding inclusion of spaces of sections

\Gamma_{hol}\big(S, \, Z \big) \xhookrightarrow{\;\simeq_{whe}\;} \Gamma\big(S, \, Z \big)

is a weak homotopy equivalence.

(Forstnerič 2011, Cor. 5.4.8)

In higher complex analytic geometry

In (Lárusson 2001, Lárusson 2003) 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

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 $Maps_{hol}(-,X) \in Psh_\infty(SteinSp)$ 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.

(Larusson 03 – apparently this follows an observation due to J. F. Jardine and uses his intermediate model structure on simplicial presheaves)

Related concepts

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 5.3 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)
Franc Forstnerič, Gromov’s contribution to the Oka principle, 2012 (pdf, pdf)