nLab Narasimhan embedding theorem

Statements

Every Stein manifold of dimension $n$ admits an injective proper holomorphic immersion into $\mathbf{C}^{2n+1}$.

Every holomorphically complete complex space? of dimension $n$ admits an injective proper holomorphic map into $\mathbf{C}^{2n+1}$ that is an immersion at every uniformizable point.

If for some $N\gt n$ a holomorphically complete complex space? $X$ is locally isomorphic to an analytic subset of an open set in $\mathbf{C}^N$, then there is an injective proper holomorphic map $\phi\colon X\to\mathbf{C}^{N+n}$ that is an isomorphism onto its image.

The relevant spaces of embeddings are dense in the space of all holomorphic mappings into the corresponding cartesian spaces equipped with the compact convergence topology.

Related concepts

• Whitney embedding theorem

• Grauert embedding theorem

• Nash embedding theorem

References

The original reference is

• Raghavan Narasimhan?, Imbedding of Holomorphically Complete Complex Spaces, American Journal of Mathematics, Vol. 82, No. 4 (Oct., 1960), pp. 917-934, doi.