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 $f \in \mathcal O(X)$ , such that f(x) ≠ f(y).{{Cite book |last=Grauert |first=Hans |title=Theory of Stein Spaces |last2=Remmert |first2=Reinhold |publisher=Springer-Verlag |year=2004 |isbn=3-540-00373-8 |edition=Reprint of the 1979 |publication-date=2004 |pages=117 |translator-last=Huckleberry |translator-first=Alan}}

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

{{cite book |isbn=9783110838350|title=Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|last1=Kaup|first1=Ludger|last2=Kaup|first2=Burchard|date=9 May 2011|publisher=Walter de Gruyter |url={{Google books|4YVXCgewhTIC|Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|plainurl=yes}}}}
{{cite journal|jstor=2034564 |doi=10.1090/S0002-9939-1960-0170034-8|title=Holomorphic mappings of complex spaces|year=1960|last1=Narasimhan|first1=Raghavan|journal=Proceedings of the American Mathematical Society|volume=11|issue=5|pages=800–804|doi-access=free}}
{{cite journal |arxiv=1108.2078|last1=Noguchi|first1=Junjiro|title=Another Direct Proof of Oka's Theorem (Oka IX)|year=2011|mr=3086750|journal=J. Math. Sci. Univ. Tokyo|url=https://www.ms.u-tokyo.ac.jp/journal/pdf/jms190407.pdf|volume=19|issue=4}}
{{cite journal |title =Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes |journal= Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris| pages=118–121| last1=Remmert|first1=Reinhold|year=1956|volume=243|url=https://gallica.bnf.fr/ark:/12148/bpt6k3195v/f118.image.r|language=fr|zbl=0070.30401}}

Category:Complex analysis

Category:Several complex variables