Paired \(\mathrm{CR}\) structures and the example of Falbel's cross-ratio var

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• 作者：Ioannis D. Platis
• 关键词：Cross ; ratios ; CR structures ; Paired CR structures
• 刊名：Geometriae Dedicata
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：181
• 期：1
• 页码：257-292
• 全文大小：702 KB
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  7.Falbel, E., Platis, I.D.: The \(\text{ PU }(2,1)\) configuration space of four points in \(S^3\) and the cross-ratio variety. Math. Ann. 340(4), 935–962 (2008)MathSciNet CrossRef MATH
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  13.Parker, J.R., Platis, I.D.: Complex hyperbolic Fenchel–Nielsen coordinates. Topology 47, 101–135 (2008)MathSciNet CrossRef MATH
  14.Parker, J.R., Platis, I.D.: Global geometrical coordinates on Falbel’s cross-ratio variety. Canad. Math. Bull. 52(2), 285–294 (2009)MathSciNet CrossRef MATH
  15.Tu, L.W.: An introduction to manifolds. Springer Science. Springer, New York (2008)
  
• 作者单位：Ioannis D. Platis (1)
  
  1. Department of Mathematics and Applied Mathematics, University Campus, University of Crete, Voutes, 70013, Heraklion Crete, Greece
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

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.