Complex and CR-structures on compact Lie groups associated to Abelian actions

2006/10/30 by Loeb, J. -J., Manjarin, M., Nicolau, M.
#32C10 #32C16 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0610915

Abstract

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description of these structures. In this article we present an alternative and more geometric construction of this type of invariant structures on a compact Lie group K when it is semisimple. We prove that each left-invariant complex structure, or each CR-structure of maximal dimension with a transverse CR-action by R, is induced by a holomorphic C^l action on a quasi-projective manifold X naturally associated to K. We then show that X admits more general Abelian actions, also inducing complex or CR structures on K which are generically non-invariant.

Related

• Levi-flat CR structures on compact Lie groups
• Non-invariant deformations of left-invariant complex structures on compact Lie groups
• Left-invariant CR structures on 3-dimensional Lie groups
• Invariant and non-invariant almost complex structures on compact quotients of Lie groups
• The chains of left-invariant CR structures on the 3-dimensional Lie groups