Holomorphic separability

In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.

Formal definition

A complex manifold or complex space $X$ is said to be holomorphically separable, if whenever x ≠ y are two points in $X$, there exists a holomorphic function $f\in {𝒪}(X)$, such that f(x) ≠ f(y).

Often one says the holomorphic functions separate points.

Usage and examples

• All complex manifolds that can be mapped injectively into some ${\mathbb{ℂ}}^n$ are holomorphically separable, in particular, all domains in ${\mathbb{ℂ}}^n$ and all Stein manifolds.
• A holomorphically separable complex manifold is not compact unless it is discrete and finite.
• The condition is part of the definition of a Stein manifold.

References

• Kaup, Ludger; Kaup, Burchard (9 May 2011). Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory. Walter de Gruyter. ISBN 9783110838350. https://books.google.com/books?id=4YVXCgewhTIC. 
• Narasimhan, Raghavan (1960). "Holomorphic mappings of complex spaces". Proceedings of the American Mathematical Society 11 (5): 800–804. doi:10.1090/S0002-9939-1960-0170034-8. 
• Noguchi, Junjiro (2011). "Another Direct Proof of Oka's Theorem (Oka IX)". J. Math. Sci. Univ. Tokyo 19 (4). https://www.ms.u-tokyo.ac.jp/journal/pdf/jms190407.pdf. 
• Remmert, Reinhold (1956). "Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes" (in fr). Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris 243: 118–121. https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.image.r.