# complex analytic manifold

###### Definition.

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

###### Definition.

A subset $N\subset M$ is called a complex analytic submanifold of $M$ if $N$ is closed in $M$ and if for every point $z\in N$ there is a coordinate neighbourhood $U$ in $M$ with coordinates $z_{1},\ldots,z_{n}$ such that $U\cap N=\{p\in U\mid z_{d+1}(p)=\ldots=z_{n}(p)\}$ for some integer $d\leq n$.

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 ${\mathbb{C}}^{n}$. Such a manifold is of real dimension $2n$ if we identify ${\mathbb{C}}^{n}$ with ${\mathbb{R}}^{2n}$. 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 ${\mathbb{C}}^{n}$ 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 ComplexAnalyticManifold 2013-03-22 15:04:40 2013-03-22 15:04:40 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 32Q99 complex manifold complex analytic submanifold complex submanifold analytic structure holomorphic structure