# PlanetPhysics/Tangential Cauchy Riemann Complex
### Tangential Cauchy-Riemann complexes
#### Introduction: 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 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 $CR$ 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 CR generic if $dim_{\mathbb {C} }T_{\tau }^{\mathbb {C} }(M_{g})=(n-k)$ for every $\tau \in M$.

### Definition of Tangential Cauchy-Riemann complexes

Let us consider $M_{g}$ to be an oriented ${\mathcal {C}}^{\infty }$-smooth $CR$ 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 Grassmann algebra Failed to parse (unknown function "\E"): {\displaystyle {\E}} of germs of complex valued ${\mathcal {C}}^{\infty }$--forms on $X$, that are locally generated by functions (which vanish on $M_{g}$), and by their anti-holomorphic differentials. One also has on $X$ the Dolbeault complexes for the sheaves of germs of smooth forms:
   Failed to parse (unknown function "\begin{xy}"): {\displaystyle \begin{xy} *!C\xybox{ \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 } }\end{xy}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \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 Failed to parse (unknown function "\E"): {\displaystyle {\E}^{p,*}} written as :
   Failed to parse (unknown function "\begin{xy}"): {\displaystyle \begin{xy} *!C\xybox{ \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.} }\end{xy}}
Therefore, we also have the quotient complexes Failed to parse (unknown function "\E"): {\displaystyle {\E}^{p,*}} defined by the exact sequences of fine sheaves complexes:
   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {0} \to {\mathsf{S_M}^{p,*}} \ar[r]& {\E}^{p,*} \ar[r]& \cdots \ar[r] & [{\E}^{p,*}]\ar[r] & 0. } }\end{xy}}
With the induced differentials denoted by ${\overline {\partial _{M}}}$ we can now write the quotient complex--which is called the tangential Cauchy-Riemann complex of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \mathcal{C ^{\infty}} --smooth forms}-- as follows:
   Failed to parse (unknown function "\begin{xy}"): {\displaystyle \begin{xy} *!C\xybox{ \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{xy}}
Remarks: There are two distinct ways of defining the tangential Cauchy-Riemann complex:
* an extrinsic approach that uses the ${\overline {\partial _{M}}}$ of the ambient $C^{n}$;
* an intrinsic approach that does not utilize the ambient $C^{n}$, and thus generalizes to abstract $CR$ manifolds (viz. A. Bogess, 2000).
For further, full details the reader is referred to the recent textbook by Burgess (2000) on this subject.
The cohomology groups of Failed to parse (unknown function "\E"): {\displaystyle [{\E}^{p,*}]} on $M\bigcap U$, for $U$ being an open subset of $X$, are then appropriately denoted here as $H_{\infty }^{p,j}(M\bigcap U)$.