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On holomorphic maps into compact non-Kähler manifolds

Masahide Kato, Noboru Okada (2004)

Annales de l’institut Fourier

We study the extension problem of holomorphic maps σ : H → X of a Hartogs domain H with values in a complex manifold X . For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain Ω σ of extension for σ over Δ is contained in a subdomain of Δ . For such manifolds, we define, in this paper, an invariant Hex n ( X ) using the Hausdorff dimensions of the singular sets of σ ’s and study its properties to deduce informations on the complex structure of X .

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 locally biholomorphic mappings from multi-connected onto simply connected domains

Piotr Liczberski, Victor V. Starkov (2005)

Annales Polonici Mathematici

We continue E. Ligocka's investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex plane ℂ.

On locally biholomorphic surjective mappings

Ewa Ligocka (2003)

Annales Polonici Mathematici

We prove that each open Riemann surface can be locally biholomorphically (locally univalently) mapped onto the whole complex plane. We also study finite-to-one locally biholomorphic mappings onto the unit disc. Finally, we investigate surjective biholomorphic mappings from Cartesian products of domains.

On meromorphic functions with maximal defect sum

Pham Duc Thoan, Le Thanh Tung (2011)

Annales Polonici Mathematici

The purpose of this article is twofold. The first is to give necessary conditions for the maximality of the defect sum. The second is to show that the class of meromorphic functions with maximal defect sum is very thin in the sense that deformations of meromorphic functions with maximal defect sum by small meromorphic functions are not meromorphic functions with maximal defect sum.

On perturbations of pluriregular sets generated by sequences of polynomial maps

Maciej Klimek (2003)

Annales Polonici Mathematici

It is shown that an infinite sequence of polynomial mappings of several complex variables, with suitable growth restrictions, determines a filled-in Julia set which is pluriregular. Such sets depend continuously and analytically on the generating sequences, in the sense of pluripotential theory and the theory of set-valued analytic functions, respectively.

On proper discs in complex manifolds

Barbara Drinovec Drnovšek (2007)

Annales de l’institut Fourier

Let X be a complex manifold of dimension at least 2 which has an exhaustion function whose Levi form has at each point at least 2 strictly positive eigenvalues. We construct proper holomorphic discs in X through any given point and in any given direction.

On some rigidity properties of mappings between CR-submanifolds in complex space

Francine Meylan, Nordine Mir, Dimitri Zaitsev (2003)

Journées équations aux dérivées partielles

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

On the automorphism group of strongly pseudoconvex domains in almost complex manifolds

Jisoo Byun, Hervé Gaussier, Kang-Hyurk Lee (2009)

Annales de l’institut Fourier

In contrast with the integrable case there exist infinitely many non-integrable homogeneous almost complex manifolds which are strongly pseudoconvex at each boundary point. All such manifolds are equivalent to the Siegel half space endowed with some linear almost complex structure.We prove that there is no relatively compact strongly pseudoconvex representation of these manifolds. Finally we study the upper semi-continuity of the automorphism group of some hyperbolic strongly pseudoconvex almost...

On the automorphisms of the spectral unit ball

Jérémie Rostand (2003)

Studia Mathematica

Let Ω be the spectral unit ball of Mₙ(ℂ), that is, the set of n × n matrices with spectral radius less than 1. We are interested in classifying the automorphisms of Ω. We know that it is enough to consider the normalized automorphisms of Ω, that is, the automorphisms F satisfying F(0) = 0 and F'(0) = I, where I is the identity map on Mₙ(ℂ). The known normalized automorphisms are conjugations. Is every normalized automorphism a conjugation? We show that locally, in a neighborhood of a matrix with...

On the complexification of real-analytic polynomial mappings of ℝ²

Ewa Ligocka (2006)

Annales Polonici Mathematici

We give a simple algebraic condition on the leading homogeneous term of a polynomial mapping from ℝ² into ℝ² which is equivalent to the fact that the complexification of this mapping can be extended to a polynomial endomorphism of ℂℙ². We also prove that this extension acts on ℂℙ²∖ℂ² as a quotient of finite Blaschke products.

On the CR-structure of certain linear group orbits in infinite dimensions

Wilhelm Kaup (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits  M explicitly and show as main result that every continuous CR-function on  M has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite...

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