tangential Cauchy-Riemann complexes

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 \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 $ 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

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:

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

Remarks: There are two distinct ways of defining the tangential Cauchy-Riemann complex:

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

The cohomology groups of $ [{\mathcal 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)$ .

Bibliography

1
Christine Laurent-Thiébaut and J'urgen Leiterer: Dolbeault Isomorphism for CR Manifolds (preprint). Prépublication de l'Institut Fourier no. 521 (2000).

2
M. Nacinovich and G. Valli, Tangential Cauchy-Riemann complexes on distributions, Ann. Math. Pure Appl., 146 (1987): 123-169.

3
A. Boggess, 2000. $ CR$ Manifolds and the Tangential Cauchy-Riemann Complex, Boca Raton: CRC Press (Book Abstract and Contents on line; see also the PM book reference).

4
Sorin Dragomir and Giuseppe Tomassini, 2006. Differential geometry and analysis on CR manifolds, Progress in Mathematics, vol. 246, Birkh'auser, Basel. (avail. review in PDF)



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