Holomorphic embeddings and immersions of Stein manifolds: a survey

TL;DR

This survey reviews key results on holomorphic embeddings and immersions of Stein manifolds into complex manifolds, highlighting recent advances including conditions for proper embeddings via homotopy and Stein structure deformation.

Contribution

It introduces a new result showing that any continuous map between Stein manifolds can be homotoped to a proper holomorphic embedding under certain dimension conditions.

Findings

Proper holomorphic embeddings exist under dimension constraints.

Homotopic deformation of Stein structures enables embeddings.

New theorem on homotopy to proper holomorphic embeddings.

Abstract

In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most results pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map $X\to Y$ between Stein manifolds is homotopic to a proper holomorphic embedding provided that $\mathrm{dim} Y > 2\mathrm{dim}\, X$ and we allow a homotopic deformation of the Stein structure on $X$.