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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 commuting polynomial automorphisms of C2.

Cinzia Bisi (2004)

Publicacions Matemàtiques

We charocterize the commuting polynomial automorphisms of C2, using their meromorphic extension to P2 and looking at their dynamics on the line at infinity.

On complexification and iteration of quasiregular polynomials which have algebraic degree two

Ewa Ligocka (2005)

Fundamenta Mathematicae

We prove that each degree two quasiregular polynomial is conjugate to Q(z) = z² - (p+q)|z|² + pqz̅² + c, |p| < 1, |q| < 1. We also show that the complexification of Q can be extended to a polynomial endomorphism of ℂℙ² which acts as a Blaschke product (z-p)/(1-p̅z) · (z-q)/(1-q̅z) on ℂℙ²∖ℂ². Using this fact we study the dynamics of Q under iteration.

On D*-extension property of the Hartogs domains.

Do Duc Thai, Pascal J. Thomas (2001)

Publicacions Matemàtiques

A complex analytic space is said to have the D*-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X x C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex,...

On fixed points of holomorphic type

Ewa Ligocka (2002)

Colloquium Mathematicae

We study a linearization of a real-analytic plane map in the neighborhood of its fixed point of holomorphic type. We prove a generalization of the classical Koenig theorem. To do that, we use the well known results concerning the local dynamics of holomorphic mappings in ℂ².

On fundamental groups of algebraic varieties and value distribution theory

Katsutoshi Yamanoi (2010)

Annales de l’institut Fourier

If a smooth projective variety X admits a non-degenerate holomorphic map ℂ → X from the complex plane ℂ , then for any finite dimensional linear representation of the fundamental group of X the image of this representation is almost abelian. This supports a conjecture proposed by F. Campana, published in this journal in 2004.

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.

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