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

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

2006-10-30

arxiv.org/abs/math/0610915v1

Mathematics

Differential Geometry

Scientific paper

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.

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