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Oka Manifolds

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Stein Manifolds and Holomorphic Mappings

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Abstract

This is the first of the two core chapters of the book. We begin with a historical introduction to the Oka-Grauert principle which says that the topological classification of principal fibre bundles over Stein spaces agrees with the holomorphic classification. Next we describe a reduction to the problem of deforming families of continuous sections to families of holomorphic sections in principal fibre bundles over Stein spaces. This naturally leads to the theory of Oka manifolds. A complex manifold \(Y\) is said to be an Oka manifold if every holomorphic map from any convex set in a Euclidean space \({\mathbb {C}}^{n}\) can be approximated by entire maps \({\mathbb {C}}^{n}\to Y\). The main result is that sections of any stratified holomorphic fibre bundle with Oka fibres over a reduced Stein space enjoy all forms of the Oka principle. Since every complex homogeneous manifold is Oka, this generalizes the classical Oka-Grauert theory. We give a complete proof, proceeding in steps from the simplest to the most general case. We also discuss properties and examples of Oka manifold and give several nontrivially equivalent characterizations of this class.

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    Franc Forstnerič

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Forstnerič, F. (2017). Oka Manifolds. In: Stein Manifolds and Holomorphic Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-319-61058-0_5

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