ComplexAnalyticManifold 2005-02-22 13:37:20 2006-07-14 02:21:18 Definition complex analytic submanifold complex submanifold analytic structure holomorphic structure % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \theoremstyle{definition} \newtheorem*{defn}{Definition} \begin{defn} A manifold $M$ is called a {\em complex analytic manifold} (or sometimes just a {\em complex manifold}) if the transition functions are holomorphic. \end{defn} \begin{defn} A subset $N \subset M$ is called a {\em 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 $d \leq n$. \end{defn} 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 {\em \PMlinkescapetext{analytic structure}} or {\em \PMlinkescapetext{holomorphic structure}}. This refers to the atlas and transition functions defined on the manifold. \begin{thebibliography}{9} \bibitem{Hormander:several} Lars H\"ormander. {\em \PMlinkescapetext{An Introduction to Complex Analysis in Several Variables}}, North-Holland Publishing Company, New York, New York, 1973. \bibitem{Krantz:several} Steven~G.\@ Krantz. {\em \PMlinkescapetext{Function Theory of Several Complex Variables}}, AMS Chelsea Publishing, Providence, Rhode Island, 1992. \end{thebibliography}