Definition 1   A manifold is called a complex analytic manifold (or sometimes just a complex manifold) if the transition functions are holomorphic.

Definition 2   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 analytic structure or holomorphic structure. This refers to the atlas and transition functions defined on the manifold.

Bibliography

1

Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.

2

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.