A Rigidity Theorem for Proper Holomorphic Mappings between Generalized Pseudoellipsoids

Abstract

Let $E(\alpha) \subset \mathbb{C}^{m+1}$ and $E(\beta) \subset \mathbb{C}^{n+1}$ be generalized pseudoellipsoids. Assume that the inequality $m<n$ holds. They are parametrized by $N$-tuples of positive integers $\alpha=(\alpha_1, \dots, \alpha_N)$ and $\beta=(\beta_1, \dots, \beta_N)$. (See introduction for the definition of a generalized pseudoellipsoid) Assume that there exists a proper holomorphic mapping between them. In this article, two facts are proved. Firstly, under the assumptions of the existence of such a mapping, certain nondegeneracy conditions of a submatrix of the Jacobian matrix and additional inequalities on dimensions, the parameters $(\alpha_1, \dots, \alpha_N)$ and $(\beta_1, \dots, \beta_N)$ coincide; $\alpha_1=\beta_1, \dots, \alpha_N=\beta_N$ after re-ordering if necessary. Secondly, such a proper holomorphic mapping is a linear embedding up to automorphisms of a source and a target domains.