PlanetPhysics/Tangential Cauchy Riemann Complex

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 $${\E}$$ 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:

$$\begin{xy} \xymatrix{ {\E}^{p,*} : 0 \to {\E}^{p,0}\ar[r]^{~\overline{\partial}} & {\E}^{p,1} \ar[r]^{\overline {\partial}} & \cdots \ar[r]^{\overline {\partial}} & {\E}^{p,n}\ar[r] & 0 } }\end{xy}$$
 * !C\xybox{

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 :

$$\begin{xy} \xymatrix{ {\mathsf{S_M}^{p,*}}: 0 \to {\mathsf{S_M}^{p,0}} \ar[r]^{~\overline{\partial}} & {\mathsf{S_M}^{p,1}} \ar[r]^{\overline{\partial}} & \cdots \ar[r]^{\overline{\partial}} & {\mathsf{S_M}^{p,n}}\ar[r] & 0.} }\end{xy}$$
 * !C\xybox{

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

$$\begin{xy} \xymatrix{ {0} \to {\mathsf{S_M}^{p,*}} \ar[r]& {\E}^{p,*} \ar[r]& \cdots \ar[r] & [{\E}^{p,*}]\ar[r] & 0. } }\end{xy}$$
 * !C\xybox{

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 $$\mathcal{C ^{\infty}$$--smooth forms}-- as follows:

$$\begin{xy} \xymatrix{ [{\E}^{p,*}]: 0 \to [{E}^{p,0}]~\ar[r]^{~\overline{\partial_M}} & [{\E}^{p,1}] \ar[r]^{\overline{\partial_M}} & \cdots \ar[r]^{\overline{\partial_M}} & [{\E}^{p,n}]\ar[r] & 0. } }\end{xy}$$
 * !C\xybox{

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