Holomorphic separable manifold

In the area of function theory , a branch of mathematics , one is interested in the multiplicity of holomorphic functions on complex manifolds . One concept is that of holomorphic separability or holomorphic separability . If a complex manifold is holomorphically separable, then it is ensured that other holomorphic functions exist on this manifold in addition to the constant functions. On the sphere, which is the standard example of a manifold , only the constant functions are holomorphic; the sphere is therefore not holomorphically separable.

Formal definition

Let it be a -dimensional complex manifold and denote the ring (or the sheaf ) of holomorphic functions . The manifold is called holomorphically separable if there is a completely holomorphic function for any two points with such that it holds. ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {O}}(X)}$ ${\displaystyle f\colon X\to \mathbb {C} }$ ${\displaystyle X}$ ${\displaystyle x,y\in X}$ ${\displaystyle x\neq y}$ ${\displaystyle X}$ ${\displaystyle f\in {\mathcal {O}}(X)}$ ${\displaystyle f(x)\neq f(y)}$

They say: "The holomorphic functions separate the points."

Examples

• If a complex manifold or a complex space can be mapped injectively (and holomorphically) , the space is holomorphically separable. ${\displaystyle \mathbb {C} ^{n}}$

• Hence every domain is separable in holomorphic. ${\displaystyle \mathbb {C} ^{n}}$
• Every Stein manifold is holomorphically separable. (There is a definition of Stein's manifolds that calls "holomorphically separable" as a condition.)

• Spaces that have nondiscrete, compact, complex subspaces or submanifolds are not holomorphically separable.
  • Consequently, the sphere and the torus , or more generally the Jacobi variety , are not holomorphically separable.

literature

• Klaus Fritzsche, Hans Grauert : From Holomorphic Functions to Complex Manifolds (= Graduate Texts in Mathematics 213). Springer, New York NY et al. 2002, ISBN 0-387-95395-7 .