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Oka manifolds: From Oka to Stein and back

Franc Forstnerič (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert’s classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989.In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations...

On holomorphically separable complex solv-manifolds

Alan T. Huckleberry, E. Oeljeklaus (1986)

Annales de l'institut Fourier

Let G be a solvable complex Lie group and H a closed complex subgroup of G . If the global holomorphic functions of the complex manifold X : G / H locally separate points on X , then X is a Stein manifold. Moreover there is a subgroup H ^ of finite index in H with π 1 ( G / H ^ ) nilpotent. In special situations (e.g. if H is discrete) H normalizes H ^ and H / H ^ is abelian.

On nonimbeddability of Hartogs figures into complex manifolds

E. Chirka, S. Ivashkovich (2006)

Bulletin de la Société Mathématique de France

We prove the impossibility of imbeddings of Hartogs figures into general complex manifolds which are close to an imbedding of an analytic disc attached to a totally real collar. Analogously we provide examples of the so called thin Hartogs figures in complex manifolds having no neighborhood biholomorphic to an open set in a Stein manifold.

On the complex and convex geometry of Ol'shanskii semigroups

Karl-Hermann Neeb (1998)

Annales de l'institut Fourier

To a pair of a Lie group G and an open elliptic convex cone W in its Lie algebra one associates a complex semigroup S = G Exp ( i W ) which permits an action of G × G by biholomorphic mappings. In the case where W is a vector space S is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain D ⊆ S is Stein is and only if it is of the form G Exp ( D h ) , with D h ⊆ i W convex, that each holomorphic function on D extends to the smallest biinvariant Stein domain containing D ,...

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