Proper holomorphic maps between bounded symmetric domains revisited

TL;DR

This paper proves that proper holomorphic maps between equal-dimensional bounded symmetric domains, with one irreducible, are biholomorphisms, unifying previous special cases and extending to non-transitive automorphism groups.

Contribution

It provides a unified proof that proper holomorphic maps between certain symmetric domains are biholomorphisms, covering previously known special cases and new classes of domains.

Findings

Proper holomorphic maps between equal-dimensional irreducible bounded symmetric domains are biholomorphisms.

A unified argument encompasses various known cases.

Application to domains with non-transitive automorphism groups.

Abstract

We prove that a proper holomorphic map between two bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively.