A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in ℂ² with the logarithmic image equal to a strip or a half-plane is given.

We present a result on the existence of some kind of peak functions for ℂ-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for A(D) under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.

The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus $V_f$ is factored by automorphisms if and only if $f_{*}\left(\pi ₁(D\setminus f^{-1}\left(f\left(V_f\right)\right),x)\right)$ is a normal subgroup of $\pi ₁(\Omega \setminus f\left(V_f\right),b)$ for some $b\in \Omega \setminus f\left(V_f\right)$ and $x\in f^{-1}\left(b\right)$.

We characterize proper holomorphic self-mappings 𝔾₂ → 𝔾₂ for the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂): |λ₁|,|λ₂| < 1} ⊂ ℂ².

Nous proposons une généralisation d’un résultat de F. Berteloot et G. Patrizio [1], aux cas des applications holomorphes propres entre domaines quasi-disqués et non nécessairement bornés.

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$. The generalized Fock-Bargmann-Hartogs domain is defined by inequality ${\mathrm{e}}^{{\mu \parallel z\parallel }^2}{\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}<1$, where $(z,\omega )\in {ℂ}^n\times {ℂ}^m$. In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$ becomes a holomorphic automorphism if and only if it keeps the function ${\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}{\mathrm{e}}^{{\mu \parallel z\parallel }^2}$ invariant.

In this paper we prove some compactness theorems of families of proper holomorphic correspondences. In particular we extend the well known Wong-Rosay's theorem to proper holomorphic correspondences. This work generalizes some recent results proved in [17].

In this note we compute the Bergman kernel of the unit ball with respect to the smallest norm in ${ℂ}^n$ that extends the euclidean norm in ${ℝ}^n$ and give some applications.

We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n):{ℂ}^n\to {ℂ}^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod }_{i=1}^{n}deg{f}_i-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_i\right)$.

We prove that the symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains.