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

• DOI: 10.1007/s10711-015-0123-3
• Publication Date: 2015-10-20T08:52:18Z
• Authors: Ioannis D. Platis
• Keywords: Codimension

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.