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 {\displaystyle X} is said to be holomorphically separable, if whenever x ≠ y are two points in X {\displaystyle X}, there exists a holomorphic function f ∈ O ( X ) {\displaystyle 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 C n {\displaystyle \mathbb {C} ^{n}} are holomorphically separable, in particular, all domains in C n {\displaystyle \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.
  • Kaup, Ludger; Kaup, Burchard (9 May 2011). . Walter de Gruyter. ISBN 9783110838350.
  • Narasimhan, Raghavan (1960). . Proceedings of the American Mathematical Society. 11 (5): 800–804. doi:. JSTOR .
  • Noguchi, Junjiro (2011). (PDF). J. Math. Sci. Univ. Tokyo. 19 (4). arXiv:. MR .
  • Remmert, Reinhold (1956). . Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris (in French). 243: 118–121. Zbl .