We introduce a process, which we call Levi–Kähler reduction, for constructing Kähler manifolds and orbifolds from CR manifolds (of arbitrary codimension) with a transverse torus action. Most of the paper is devoted to the study of Levi–Kähler reductions of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of the orbifolds obtained by such reductions, and find that the Levi–Kähler reductions of products of $3$-spheres are extremal in a weighted sense introduced by G. Maschler and the first author [11], and further studied by A. Futaki and H. Ono [34].

中文翻译：

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Levi–Kähler减少CR结构，球的乘积和复曲面几何

我们引入了一个称为Levi–Kähler约简的过程，该过程从具有横向圆环作用的CR歧管（任意余维）构造Kähler流形和球面。大部分论文专门研究复曲面CR歧管的Levi-Kähler约简，尤其是奇维球体的乘积。我们获得了通过此类折减获得的球面的明确描述和特征，并发现G. Maschler和第一作者[11]在加权意义上将Levi-Kähler折减$ 3 $球的乘积是极值。由A. Futaki和H. Ono研究[34]。