Complex manifolds with maximal torus actions
Complex manifolds with maximal torus actions
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DOI：
10.1515/crelle-2016-0023
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发表时间：
2013-02
期刊：
影响因子：
--
通讯作者：
Hiroaki Ishida
中科院分区：
文献类型：
--
作者：
Hiroaki Ishida
本文引入紧环面在光滑流形上的极大作用的概念，研究了具有紧环面极大作用的紧连通复流形。我们给出了这种流形在组合对象上的完全分类，组合对象是{\mathfrak{G}}中非奇异完全扇形Δ的三元组{(\Delta,\mathfrak{h}，G)}， {\mathfrak{G}^{\mathbb{C}}}的复向量子空间{\mathfrak{h}}和满足一定条件的紧环面G。我们也给出了类的等价，并给出了这些族中态射的适当定义，如环面几何。作为范畴等价的应用，我们得到了几个结果；矩角流形上的复结构，复维数n的紧连通复流形上全纯非退化{\mathbb{C}^{n}}-作用的分类，non-Kähler流形的具体实例构造。
In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.