complex analytic manifold

Definition.

A manifold $M$ is called a complex analytic manifold (or sometimes justa 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 $z_1,\mathrm{\dots },z_n$ such that$U\cap N=\{p\in U\mid z_{d+1}(p)=\mathrm{\dots }=z_n(p)\}$ for some integer $d\le n$.

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 ${ℂ}^n$. Such a manifold is of real dimension $2n$ if we identify ${ℂ}^n$ with${ℝ}^{2n}$.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 ${ℂ}^n$ 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.