Pseudoconformal structures and the example of Falbel's Cross--Ratio variety

We introduce pseudoconformal structures on 4--dimensional manifolds and study their properties. Such structures are arising from two different complex operators which agree in a 2--dimensional subbundle of the tangent bundle; this subbundle thus forms a codimension 2 ${\rm CR}$ structure. A special case is that of a strictly pseudoconformal structure: in this case, the two complex operators are also opposite in a 2-dimensional subbundle which is complementary to the ${\rm CR}$ structure. A non trivial example of a manifold endowed with a (strictly) pseudoconformal structure is Falbel's cross--ratio variety $\mathfrak{X}$; this variety is isomorphic to the ${\rm PU}(2,1)$ configuration space of quadruples of pairwise distinct points in $S^3$. We first prove that there are two complex structures that appear naturally in $\mathfrak{X}$; these give $\mathfrak{X}$ a pseudoconformal structure which coincides with its well known ${\rm CR}$ structure. Using a non trivial involution of $\mathfrak{X}$ we then prove that $\mathfrak{X}$ is a strictly pseudoconformal manifold. The geometric meaning of this involution as well as its interconnections with the $\rm{CR}$ and complex structures of $\mathfrak{X}$ are also studied here in detail.