CauchyRiemannManifold

Let

be a complex manifold of complex dimension $n$

. If

is a $\mathcal{C}^{\infty}$ -smooth real submanifold of real codimension $k$

, 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: $M = { z \in \Omega \vert \rho_1(z)=...= \rho_k(z)=0}$ , 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

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

Definition 0.1

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 ${\mathcal E}$ 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:

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

where ${\mathcal E}^{p,j}$ 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 $\mathsf{S_M}^{p,j} = \mathsf{S_M} \bigcup {\mathcal E}^{p,j}$ . 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 ${\mathcal E}^{p,*}$ written as :

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

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

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

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

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