PlanetPhysics/Tangential Cauchy Riemann Complex

Tangential Cauchy-Riemann complexes

Introduction: Cauchy-Riemann (CR) manifolds and generic submanifolds

Let $X$ be a complex [[../NoncommutativeGeometry4/|manifold]] of complex dimension $n$. If $M$ is a ${𝒞}^{\mathrm{\infty }}$-smooth real submanifold of real codimension $k$ in $X$, let us denote by $T_{\tau }^{ℂ}(M)$ the tangential complex space at $\tau \in M$ . Such a manifold $M$ can be locally represented in the form: $M=z\in \mathrm{\Omega }|{\rho }_1(z)=...={\rho }_k(z)=0$, where all ${\rho }_i,1\le i\le k$ are real ${𝒞}^{\mathrm{\infty }}$--functions in an open subset $\mathrm{\Omega }$ of X. The submanifold $M$ is called $CR$ if the number $di{m}_{ℂ}{T}_{\tau }^{ℂ}(M)$ is independent of the point $\tau \in M$. A submanifold $M_g$ is called CR generic if $di{m}_{ℂ}{T}_{\tau }^{ℂ}(M_g)=(n-k)$ for every $\tau \in M$.

Definition of Tangential Cauchy-Riemann complexes

Let us consider $M_g$ to be an oriented ${𝒞}^{\mathrm{\infty }}$-smooth $CR$ generic submanifold of real codimension $k$ in an $n$-dimensional complex manifold $X$, and let us denote by ${𝖲}_{𝖬}$ the ideal sheaf in the Grassmann algebra Failed to parse (unknown function "\E"): {\displaystyle {\E}} of germs of complex valued ${𝒞}^{\mathrm{\infty }}$--forms on $X$, that are locally generated by [[../Bijective/|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 (unknown function "\E"): {\displaystyle {\E}^{p,j}} is the sheaf of germs of complex valued ${𝒞}^{\mathrm{\infty }}$--forms of bidegree $(p,j)$, for $p,j\le n$. Let us also set Failed to parse (unknown function "\E"): {\displaystyle \mathsf{S_M}^{p,j} = \mathsf{S_M} \bigcup {\E}^{p,j} } . As $\bar{\partial }{{𝖲}_{𝖬}}^{p,j}\subset {{𝖲}_{𝖬}}^{p,j+1}$, for each $0\le p\le n$ we now have the [[../HomologicalSequence2/|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 (unknown function "\begin{xy}"): {\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 (syntax error): {\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 [[../CohomologyTheoryOnCWComplexes/|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 $H_{\mathrm{\infty }}^{p,j}(M\bigcap U)$.