A Note on Strongly Starlike Mappings in Several Complex Variables

Letfbe a normalized biholomorphic mapping on the Euclidean unit ball𝔹ninℂnand letα∈0,1. In this paper, we will show that iffis strongly starlike of orderαin the sense of Liczberski and Starkov, then it is also strongly starlike of orderαin the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimensionn≥2.

Analytic cycles and generically finite holomorphic maps

We study the behaviour of analytic cycles under generically finite holomorphic mappings between compact analytic spaces and prove that if two compact and normal complex analytic spaces have the same analytic homology groups, then any generically one to one holomorphic map between them must be a biholomorphic mapping. This generalises an old theorem of Ax and Borel.

Spectral properties of holomorphic automorphism with fixed point

The Hilbert space methods in the theory of biholomorphic mappings were applied and developed by S. Bergman [1, 2]. In this approach the central role is played by the Hilbert space L2H(D) consisting of all functions which are square integrable and holomorphic in a domain D ⊂ ℂN. A biholomorphic mapping φ:D ⃗ G induces the unitary mapping Uφ:L2H(G) ⃗ L2H(D) defined by the formulaHere ∂φ/∂z denotes the complex Jacobian of φ. The mapping Uϕ is useful, since it permits to replace a problem for D by a problem for its biholomorphic image G (see for example [11], [13]). When ϕ is an automorphism of D we obtain a unitary operator Uϕ on L2H(D).

Holomorphic automorphisms and cancellation theorems

Let X be a complex analytic space of positive dimension and A a complex analytic subvariety of X. We call A a direct factor of X if there exist a complex analytic space B and a biholomorphic mapping f: A × B → X such that, for some b ∊ B, f(a, b) = a on A, and a complex analytic space X to be primary if X has no direct factor, not equal to X itself, of positive dimension.

A geometric characterization of Cn and open balls

The purpose of this paper is to give a result concerning the problem of geometric characterizations of the Euclidean n-space Cn and bounded domains. It is well known that a simply connected Riemann surface is biholomorphic to one of the Riemann sphere, the complex plane and the unit disc. And there are several results concerning the geometric characterization of these spaces. To show that some simply connected open Riemann surface is biholomorphic to the complex plane or the unit disc, it is sufficient to see that there exist non constant bounded sub-harmonic functions or not. But in the higher dimensional case, there is no uniformization theorem. By this reason to show that some complex manifold is biholomorphic to Cn or an open ball, we must construct a biholomorphic mapping directly.