Paired CR structures and the example of Falbel’s cross-ratio variety

• Published in: Geometriae dedicata Vol. 181; no. 1; pp. 257 - 292
• Main Author: Platis, Ioannis D.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2016
• Subjects: Algebraic Geometry; Convex and Discrete Geometry; Differential Geometry; Hyperbolic Geometry; Mathematics; Mathematics and Statistics; Original Paper; Projective Geometry; Topology; CR structures; Paired CR structures; Cross-ratios; 32Q55; 53B99
• ISSN: 0046-5755; 1572-9168
• DOI: 10.1007/s10711-015-0123-3

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