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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 actions of ℂ * on algebraic spaces

Andrzej Bialynicki-Birula (1993)

Annales de l'institut Fourier

The main result of the paper says that all schematic points of the source of an action of C * on an algebraic space X are schematic on X .

On Compact Complex Manifolds with Finite Automorphism Group

Konrad Czaja (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

It is known that compact complex manifolds of general type and Kobayashi hyperbolic manifolds have finite automorphism groups. We give criteria for finiteness of the automorphism group of a compact complex manifold which allow us to produce large classes of compact complex manifolds with finite automorphism group but which are neither of general type nor Kobayashi hyperbolic.

On compact orbits in singular Kähler spaces

Jörg Winkelmann (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For a complex solvable Lie group acting holomorphically on a Kähler manifold every closed orbit is isomorphic to a torus and any two such tori are isogenous. We prove a similar result for singular Kähler spaces.

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 invariant domains in certain complex homogeneous spaces

Xiang-Yu Zhou (1997)

Annales de l'institut Fourier

Given a compact connected Lie group K . For a relatively compact K -invariant domain D in a Stein K ℂ -homogeneous space, we prove that the automorphism group of D is compact and if K is semisimple, a proper holomorphic self mapping of D is biholomorphic.

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Wilhelm Kaup, Dmitri Zaitsev (2006)

Journal of the European Mathematical Society

We present a large class of homogeneous 2-nondegenerate CR-manifolds M , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains U , V in M extends to a global real-analytic CR-automorphism of M . We show that this class contains G -orbits in Hermitian symmetric spaces Z of compact type, where G is a real form of the complex Lie group Aut ( Z ) 0 and G has an open orbit that is a bounded symmetric domain of tube type.

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