complex analytic manifold
A subset is called a complex analytic submanifold of if is closed in and if for every point there is a coordinate neighbourhood in with coordinates such that for some integer .
Obviously is now also a complex analytic manifold itself.
For a complex analytic manifold, dimension always means the complex dimension, not the real dimension. That is is of dimension when there are neighbourhoods of every point homeomorphic to . Such a manifold is of real dimension if we identify with . Of course the tangent bundle is now also a complex vector space.
Examples of complex analytic manifolds are for example the Stein manifolds or the Riemann surfaces. Of course also any open set in is also a complex analytic manifold. Another example may be the set of regular points of an analytic set.
Complex analytic manifolds can also be considered as a special case of CR manifolds where the CR dimension is maximal.
Complex manifolds are sometimes described as manifolds carrying an or . This refers to the atlas and transition functions defined on the manifold.
- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Title||complex analytic manifold|
|Date of creation||2013-03-22 15:04:40|
|Last modified on||2013-03-22 15:04:40|
|Last modified by||jirka (4157)|
|Defines||complex analytic submanifold|