Geometric significance of closed primitive horizontal (k,k)-forms on pseudoconvex CR manifolds

Determine the geometric role and place within established CR-geometry frameworks of the finite-dimensional spaces X^{k+1,k}(M) defined on a pseudoconvex CR manifold M of real dimension 4k+1 by X^{k+1,k}(M) = { γ ∈ Ω_H^{k,k} : ω ∧ γ = 0 and d_H γ = 0 }, where H is the contact distribution, θ is a contact form with Levi form ω = dθ, and d_H is the horizontal differential. In particular, ascertain how these spaces relate to known CR cohomology theories (such as tangential ∂̄_b cohomology) and characterize their geometric significance.

Background

The paper introduces, for a general CR manifold M, subspaces L_T^{p,q} ⊂ Λ_T^{p,q} that can be regarded as (p+q)-forms on M, and defines spaces X^{p,q}(M) = {χ ∈ L_T^{p,q} ⊂ Ω^{p+q}(M) : dχ = 0}. When dim M = 5 and (p,q) = (2,1), this is the complexification of the boundary deformation space considered earlier in the paper.

For a pseudoconvex CR structure on a manifold of real dimension 4k+1, the authors single out the case (p,q) = (k+1,k). With a choice of contact form θ (Levi form ω = dθ), the space can be identified as X^{k+1,k}(M) ≅ {γ ∈ Ω_H^{k,k} : ω ∧ γ = 0, d_H γ = 0}, i.e., primitive horizontal (k,k)-forms that are d_H-closed. The algebraic condition ω ∧ γ = 0 defines the primitive subspace P_H^{k,k}, which consists of anti-self-dual forms, implying d_H γ = 0 ⇒ d_H^* γ = 0. Arguments analogous to those used earlier show that these spaces are finite dimensional.

While maps from these spaces to de Rham and tangential cohomology are discussed, the authors explicitly note that the position of these spaces within the broader CR-geometry framework and their geometric meaning are not yet understood.