Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

Abstract

The Hartogs domain over homogeneous Siegel domain D_N,s (s > 0) is defined by the inequality ∥ζ∥² < K_D(z, z)^−s, where D is a homogeneous Siegel domain of type II, (z, ζ) ∈ D × ℂ^N and K_D(z, z) is the Bergman kernel of D. Recently, Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains D_N,s and D′_N′,s′ are biholomorphisms for N ≥ 2. In this article, we find a counter-example to show that the rigidity result is not true for D_1,s and obtain a classification of proper holomorphic mappings between D_1,s and D′_1,s′.