complex analytic manifold (original) (raw)
Definition.
Definition.
A subset N⊂M is called a complex analytic submanifold of Mif N is closed in M and if for every point z∈N there is a coordinate neighbourhood U in M with coordinates z1,…,zn such thatU∩N={p∈U∣zd+1(p)=…=zn(p)} for some integer d≤n.
Obviously N 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 M is of dimension n when there are neighbourhoods of every point homeomorphic to ℂn. Such a manifold is of real dimension 2n if we identify ℂn withℝ2n. Of course the tangent bundle
is now also a complex vector space.
A function f is said to be holomorphic on M 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 ℂn 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.