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Holomorphically separable

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

Contents

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 ∈ 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 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

    Holomorphically separable Wikipedia
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