On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least $2$

Abstract

Proper holomorphic mappings among domains on Euclidean spaces is a classical topic in Several Complex Variables. The literature can date back to the earliest results like the theorem of H. Alexander [1] which says that any proper holomorphic self-map of the complex unit n-ball is a biholomorphism if n ≥ 2. Since then, the study of the proper holomorphic mappings between complex unit balls of different dimensions has become a very popular topic in the field. Many important inputs from various perspectives have been made, like Algebraic Geometry, Chern-Moser Theory, Segre variety and Bergman kernel, etc. It is apparent by now that the complexity of the problem grows with the codimension and one in general must impose certain regularity assumptions on the proper maps in order to give any satisfactory classification.