Proper holomorphic mappings between generalized pseudoellipsoids

Abstract

We characterize the existence of proper holomorphic mappings between pseudoellipsoids of the form\(\Sigma (\gamma ) = \left\{ {z \in C^n :\sum\limits_{j = 1}^n {\left| {z_j } \right|^{2\gamma j}< 1} } \right\},\gamma = (\gamma _1 ,\gamma _2 ,...,\gamma _n ) \in (R^ + )^n \). If f:Σ(α)→Σ(β) is any such mapping, the existence of a subgroup Γ of Aut (Σ(α)) such that\(f^{ - 1} f(z) = \bigcup\limits_{\psi \in \Gamma } {\left\{ {\psi (z)} \right\}} \) is shown equivalent to a condition on the branch locus of f.