DAngeloFiniteType 2007-12-05 14:04:36 2009-05-01 22:17:32 Definition % 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} \newtheorem*{example}{Example} \newtheorem*{prop}{Proposition} \theoremstyle{definition} \newtheorem*{defn}{Definition} \theoremstyle{remark} \newtheorem*{rmk}{Remark} Let $M \subset {\mathbb{C}}^n$ be a real analytic submanifold of real codimension 1. We say $M$ is of \emph{\PMlinkescapetext{finite type}} in the sense of D'Angelo if there does not exist any germ of a complex analytic subvariety at $p \in M$, that is contained in $M$. The Diederich-Fornaess theorem can be then restated to say that every compact real analytic subvariety of ${\mathbb{C}}^n$ is of D'Angelo finite type at every point. \begin{thebibliography}{9} \bibitem{ber:submanifold} M.\@ Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, Princeton University Press, Princeton, New Jersey, 1999. \bibitem{DAngelo} D'Angelo, John~P. {\em \PMlinkescapetext{Several complex variables and the geometry of real hypersurfaces}}, CRC Press, 1993. \end{thebibliography}