Complex and CR-structures on compact Lie groups associated to Abelian actions
Complex and CR-structures on compact Lie groups associated to Abelian actions
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与阿贝尔作用相关的紧李群上的复数结构和 CR 结构
DOI：
10.1007/s10455-007-9067-7
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发表时间：
2006
影响因子：
0.7
通讯作者：
M. Nicolau
中科院分区：
文献类型：
--
作者：
J. Loeb;Mònica Manjarín;M. Nicolau
Samelson[一类复解析流形]证明了这一点。葡萄牙数学12,129-132(1953)]和Wang[具有均匀复杂结构的封闭流形]。阿默。J.Math.76，1-32(1954)]证明了偶数维的每个紧李群K都有左不变复结构。当K具有奇数维时，它允许最大维左不变的CR-结构。这一点最近已被Charbonnel和Khalgui[Class DES Structures CR Condiantes Pour les Groupe de Lie Comacters]所证明。J.Lie Theory14,165-198(2004)]也给出了这些结构的完整的代数描述。在这篇文章中，我们给出了紧李群K上当它是半单时这类不变结构的另一种更几何的构造。我们证明了每个左不变复结构，或每个具有横向CR作用的最大维CR-结构，都是由与K自然相关的拟射影流形上的全纯作用诱导的。然后，我们证明了X允许更一般的Abel作用，也诱导了K上一般不变的复结构或CR-结构。
It was shown by Samelson [A class of complex-analytic manifolds. Portugaliae Math.12, 129–132 (1953)] and Wang [Closed manifolds with homogeneous complex structure. Amer. J. Math.76, 1–32 (1954)] 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 [Classification des structures CR invariantes pour les groupes de Lie compactes. J. Lie theory14, 165–198 (2004)] 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, is induced by a holomorphic-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.