In
mathematics in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, the concept of holomorphic separability is a measure of the richness of the set of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s on a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
or
complex-analytic space
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible
and (or) reduced or complex analytic space is a general ...
.
Formal definition
A
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
or
complex space is said to be holomorphically separable, if whenever ''x'' ≠ ''y'' are two points in
, there exists a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
, such that ''f''(''x'') ≠ ''f''(''y'').
Often one says the
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s ''separate points''.
Usage and examples
*All complex manifolds that can be mapped
injectively into some
are holomorphically separable, in particular, all
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
s in
and all
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a St ...
s.
*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 In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a St ...
.
References
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{{DEFAULTSORT:Holomorphically Separable
Complex analysis
Several complex variables