Holomorphically separable

In the area of ​​function theory, a branch of mathematics, are more interested in the diversity of holomorphic functions on complex manifolds. One concept is that of holomorphic separability or holomorphic separability. If a complex manifold holomorphically separable, then it is ensured that on this manifold except the constant functions further holomorphic functions exist. 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

It should be a -dimensional complex manifold and denote the ring (or the sheaf ) of holomorphic functions. The variety is called holomorphically separable, if there are any two points on a very holomorphic function such that the following holds.

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

Examples

  • Is it possible to map a complex manifold or a complex space injective ( and holomorphic ) after, so the space is holomorphically separable.
  • Consequently, each field is separable in holomorphic.
  • Each manifold is Einstein's holomorphically separable. (There is a definition of rock shear manifolds, the " holomorphic separable calls" as a condition. )
  • The sphere and the torus, or more generally, the Jacobian variety, are not holomorphically separable.
  • It is even true that spaces non- discrete, compact, complex subspaces or submanifolds possess are not holomorphically separable.
Stein manifold Hans Grauert
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