tangential Cauchy-Riemann complexes

0.1 Tangential Cauchy-Riemann complexes

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

Let X be a complex manifold of complex dimension n. If M is a 𝒞^∞-smooth real submanifold of real codimension k in X, let us denote by T_τ^ℂ(M) the tangential complex space at τ ∈ M. Such a manifold M can be locally represented in the form: M = z ∈ Ω|ρ₁(z) = ... = ρ_k(z) = 0, where all ρ_i, 1 ≤ i ≤ k are real 𝒞^∞–functions in an open subset Ω of X. The submanifold M is called CR if the number dim_ℂT_τ^ℂ(M) is independent of the point τ ∈ M. A submanifold M_g is called CR generic if dim_ℂT_τ^ℂ(M_g) = (n − k) for every τ ∈ M.

0.2 Definition of Tangential Cauchy-Riemann complexes

Definition 0.1.

Let us consider M_g to be an oriented 𝒞^∞-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 ℰ of germs of complex valued 𝒞^∞–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:

p,∗ p,0--∂-- p,1--∂-- --∂-- p,n----- ℰ : 0 → ℰ ℰ ⋅ ⋅⋅ ℰ 0

where ℰ^p,j is the sheaf of germs of complex valued 𝒞^∞–forms of bidegree (p,j), for p,j ≤ n. Let us also set S_M^p,j = S_M ⋃ ℰ^p,j. As ∂S_M^p,j ⊂ S_M^p,j+1, for each 0 ≤ p ≤ n we now have the categorical sequence of subcomplexes of the complex ℰ^p,∗ written as :

p,∗ p,0 -∂--- p,1--∂-- --∂-- p,n ----- SM : 0 → SM SM ⋅⋅⋅ SM 0.

Therefore, we also have the quotient complexes ℰ^p,∗ defined by the exact sequences of fine sheaves complexes:

0 → SMp,∗-----ℰ p,∗----- ⋅⋅⋅-----[ℰp,∗]----- 0.

With the induced differentials denoted by ∂_M we can now write the quotient complex–which is called the tangential Cauchy-Riemann complex of 𝒞^∞–smooth forms– as follows:

p,∗ p,0 ∂M- p,1 ∂M ∂M- p,n [ℰ ] : 0 → [E ] -----[ℰ ]-----⋅⋅⋅-----[ℰ ]-----0.

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

an extrinsic approach that uses the ∂_M of the ambient Cⁿ;
an intrinsic approach that does not utilize the ambient Cⁿ, 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 ⋂U, for U being an open subset of X, are then appropriately denoted here as H_∞^p,j(M ⋂U).

References

[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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