Holomorphic separability explained

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

X is said to be holomorphically separable, if whenever x ≠ y are two points in X , there exists a holomorphic function f \in l O(X) , such that f(x) ≠ f(y).^[1]

Often one says the holomorphic functions separate points.

Usage and examples

• All complex manifolds that can be mapped injectively into some

C ^{n} are holomorphically separable, in particular, all domains in 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

• Book: 9783110838350. [{{Google books|4YVXCgewhTIC|Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory|plainurl=yes}} Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory]. Kaup. Ludger. Kaup. Burchard. 9 May 2011. Walter de Gruyter .
• 2034564 . 10.1090/S0002-9939-1960-0170034-8. Holomorphic mappings of complex spaces. 1960. Narasimhan. Raghavan. Proceedings of the American Mathematical Society. 11. 5. 800–804. free.
• 1108.2078. Noguchi. Junjiro. Another Direct Proof of Oka's Theorem (Oka IX). 2011. 3086750. J. Math. Sci. Univ. Tokyo. 19. 4.
• Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris. 118–121. Remmert. Reinhold. 1956. 243. fr. 0070.30401.

Notes and References

1. Book: Grauert, Hans . Theory of Stein Spaces . Remmert . Reinhold . Springer-Verlag . 2004 . 3-540-00373-8 . Reprint of the 1979 . 2004 . 117 . Huckleberry . Alan.