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).

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. ISBN 9783110838350.
• 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. JSTOR 2034564.
• Noguchi, Junjiro (2011). "Another Direct Proof of Oka's Theorem (Oka IX)" (PDF). J. Math. Sci. Univ. Tokyo. 19 (4). arXiv:1108.2078. MR 3086750.
• Remmert, Reinhold (1956). "Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris (in French). 243: 118–121. Zbl 0070.30401.