complex analytic manifold
Definition.
A manifold is called a complex analytic manifold (or sometimes just a complex manifold) if the transition functions are holomorphic.
Definition.
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.
A function is said to be holomorphic on if it is a holomorphic function when considered as a function of the local coordinates.
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.
References
- 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 |
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Canonical name | ComplexAnalyticManifold |
Date of creation | 2013-03-22 15:04:40 |
Last modified on | 2013-03-22 15:04:40 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 6 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32Q99 |
Synonym | complex manifold |
Defines | complex analytic submanifold |
Defines | complex submanifold |
Defines | analytic structure |
Defines | holomorphic structure |