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tangential Cauchy-Riemann complex of smooth forms

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\begin{document}
\subsection{Tangential Cauchy-Riemann complexes}

\subsubsection{Introduction:} \emph{Cauchy-Riemann ($CR$) manifolds and generic submanifolds}

 Let $X$ be a complex manifold of complex dimension $n$. If $M$ is a $\mathcal{C}^{\infty}$-smooth
real submanifold of real codimension $k$ in $X$, let us denote by $T_{\tau}^{\mathbb{C}} (M)$ the
\emph{tangential complex space at $\tau \in M$}. Such a manifold $M$ can be locally represented in the form:
$ M = { z \in \Omega | \rho_1(z)=...= \rho_k(z)=0}$, where all $\rho_i , 1 \leq i \leq k$ are real
$\mathcal{C}^{\infty}$--functions in an open subset $\Omega$ of X. The submanifold $M$ is called \emph{$CR$} if the
number $dim_{\mathbb{C}} T_{\tau}^{\mathbb{C}} (M)$ is independent of the point $\tau \in M$. A submanifold $M_g$
is called \emph{CR generic} if $dim_{\mathbb{C}} T_{\tau}^{\mathbb{C}} (M_g)= (n-k)$ for every $\tau \in M$.

\begin{definition}
\emph{Tangential Cauchy-Riemann complexes}

Let us consider $M_g$ to be an oriented $\mathcal{C}^{\infty}$-smooth $CR$ \emph{generic submanifold} of real
codimension $k$ in an $n$-dimensional complex manifold $X$, and let us denote by $\mathsf{S_M}$
the ideal sheaf in the \PMlinkname{Grassmann algebra}{GrassmanHopfAlgebrasAndTheirDualCoAlgebras} ${\E}$ of germs of complex valued $\mathcal{C}^{\infty}$--forms on
$X$, that are \emph{locally generated by functions} (which vanish on $M_g$), and by their
anti-holomorphic differentials. One also has on $X$ the \emph{Dolbeault complexes} for the
sheaves of germs of smooth forms: 

\[\xymatrix{
{\E}^{p,*} : 0 \to {\E}^{p,0}\ar[r]^{~~~~~~~\overline{\partial}} & {\E}^{p,1} \ar[r]^{\overline {\partial}} & \cdots \ar[r]^{\overline {\partial}} & {\E}^{p,n}\ar[r] & 0
}\]

, where ${\E}^{p,j}$ is the sheaf of germs of complex valued $\mathcal{C}^{\infty}$--forms of bidegree $(p,j)$, for $p,j \leq n$. Let us also set $\mathsf{S_M}^{p,j} = \mathsf{S_M} \bigcup {\E}^{p,j} $.
As $\overline{\partial}\mathsf{S_M}^{p,j} \subset \mathsf{S_M}^{p,j+1}$, for each $0 \leq p \leq n$
we now have the categorical sequence of subcomplexes of the complex ${\E}^{p,*}$ written as :

\[\xymatrix{
{\mathsf{S_M}^{p,*}}: 0 \to {\mathsf{S_M}^{p,0}} \ar[r]^{~~~~~~~\overline{\partial}} & {\mathsf{S_M}^{p,1}} \ar[r]^{\overline{\partial}} & \cdots \ar[r]^{\overline{\partial}} & {\mathsf{S_M}^{p,n}}\ar[r] & 0.}
\]

Therefore, we also have the \emph{quotient complexes} ${\E}^{p,*}$ defined by the exact sequences of
fine sheaves complexes:

\[\xymatrix{
{0} \to {\mathsf{S_M}^{p,*}} \ar[r]& {\E}^{p,*} \ar[r]& \cdots \ar[r] & [{\E}^{p,*}]\ar[r] & 0.
}\]

With the \emph{induced differentials} denoted by $\overline{\partial_M}$ we can now write
the quotient complex--which is called the \emph{tangential Cauchy-Riemann complex of $\mathcal{C}^{\infty}$--smooth forms}-- as follows:

\[\xymatrix{
[{\E}^{p,*}]: 0 \to [{E}^{p,0}]~\ar[r]^{~~~~~~~\overline{\partial_M}} & [{\E}^{p,1}] \ar[r]^{\overline{\partial_M}} & \cdots \ar[r]^{\overline{\partial_M}} & [{\E}^{p,n}]\ar[r] & 0.
}\]
\end{definition}

\textbf{Remarks:}
There are two distinct ways of defining the tangential Cauchy-Riemann complex:
\begin{itemize}
\item an extrinsic approach that uses the $\overline{\partial_M}$ of the ambient $C^n$;

\item an intrinsic approach that does not utilize the ambient $C^n$, and thus generalizes to abstract
$CR$ manifolds (\emph{viz.} A. Bogess, 2000).
\end{itemize}

For further, full details the reader is referred to the recent textbook by Burgess (2000) on this subject.

The \emph{cohomology groups} of $[{\E}^{p,*}]$ on $M \bigcap U$, for $U$ being an open subset
of $X$, are then appropriately denoted here as $H_{\infty}^{p,j}(M\bigcap U)$.


\begin{thebibliography}{9}

\bibitem{CLJL2k}
Christine Laurent-Thi\'ebaut and J\''urgen Leiterer: Dolbeault Isomorphism for CR Manifolds ({\em preprint}).
Pr\'epublication de l'Institut Fourier no. 521 (2000).

\bibitem{NMVG87}
M. Nacinovich and G. Valli, Tangential Cauchy-Riemann complexes on distributions, \emph{Ann. Math. Pure Appl.},
146 (1987): 123--169.

\bibitem{AB2k}
A. Boggess, 2000. \emph{$CR$ Manifolds and the Tangential Cauchy-Riemann Complex}, Boca Raton: CRC Press
(\PMlinkexternal{Book Abstract and Contents on line}{http://books.google.com/books?printsec=frontcover&id=UPQKdeVkLCwC};
\PMlinkexternal{see also the PM book reference}{http://planetmath.org/?op=getobj&from=books&id=154}).

\bibitem{SDGT2k6}
Sorin Dragomir and Giuseppe Tomassini, 2006. Differential geometry and analysis on CR manifolds,
{\em Progress in Mathematics}, vol. 246, Birkh\''auser, Basel.
(\PMlinkexternal{avail. review in PDF}{http://www.ams.org/bull/2008-45-01/S0273-0979-07-01160-3/S0273-0979-07-01160-3.pdf})

\end{thebibliography}
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