Stein Spaces Characterized by their Endomorphisms

TL;DR

This paper characterizes finite dimensional Stein spaces with a proper holomorphic embedding of the complex line through their holomorphic endomorphism semigroup, showing that semigroup isomorphisms imply biholomorphic or antibiholomorphic maps.

Contribution

It provides a unique characterization of certain Stein spaces based on their endomorphism semigroup, linking algebraic structure to geometric properties.

Findings

Semigroup isomorphisms induce biholomorphic or antibiholomorphic maps.

Finite dimensional Stein spaces with specific embeddings are uniquely characterized.

Endomorphism semigroup determines the complex structure of these Stein spaces.

Abstract

Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.