Paired $\mathrm{CR}$ structures and the example of Falbel’s cross-ratio variety

Abstract

We introduce paired $\mathrm{CR}$ 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 $\mathrm{CR}$ structure. A special case is that of a strictly paired $\mathrm{CR}$ structure: in this case, the two complex operators are also opposite in a 2-dimensional subbundle which is complementary to the $\mathrm{CR}$ structure. A non-trivial example of a manifold endowed with a (strictly) paired $\mathrm{CR}$ structure is Falbel’s cross-ratio variety ${\mathfrak {X}}$; this variety is isomorphic to the $\mathrm{PU}(2,1)$ 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 paired $\mathrm{CR}$ structure which agrees with its well known $\mathrm{CR}$ structure. Using a non-trivial involution of ${\mathfrak {X}}$ we then prove that ${\mathfrak {X}}$ is a strictly paired $\mathrm{CR}$ manifold. The geometric meaning of this involution as well as its interconnections with the $\mathrm {CR}$ and complex structures of ${\mathfrak {X}}$ are also studied here in detail.