Complex geometry of moment-angle manifolds
Complex geometry of moment-angle manifolds
复制标题
矩角流形的复杂几何
DOI：
10.1007/s00209-016-1658-1
复制
发表时间：
2013
影响因子：
0.8
通讯作者：
M. Verbitsky
中科院分区：
文献类型：
--
作者：
T. Panov;Yury Ustinovskiy;M. Verbitsky
力矩角流形提供了大量非凯勒紧凑复流形的示例。复矩角流形 $$\mathcal {Z}$$Z 是通过某些组合数据构造的，称为完全单纯扇。在有理扇的情况下，流形 $$\mathcal {Z}$$Z 是具有纤维紧复环的环面簇上的全纯丛的总空间。一般来说，复杂的矩角流形 $$\mathcal {Z}$$Z 配备有规范的全纯叶 $${\mathcal {F}}$$F，它与 $$({\mathbb {C}}^\times )^m$$(C×)m 作用等变。矩角流形的例子包括 Vaisman 型 Hopf 流形、Calabi-Eckmann 流形及其变形。在对组合数据的一些限制下，我们在矩角流形上构造横向凯勒度量。我们证明，这样的矩角流形中的任何凯勒子流形（或者更一般地说，Fujiki 类 $$\mathcal {C}$$C 子流形）都包含在叶子 $${\mathcal {F}}$$F 的叶子中。对于其组合类中的通用矩角流形$$\mathcal {Z}$$Z，我们证明所有子类型都是较小维数的矩角流形，并且它们的数量有限。这特别意味着 $$\mathcal {Z}$$Z 的代数维数为零。
Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold $$\mathcal {Z}$$Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold $$\mathcal {Z}$$Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold $$\mathcal {Z}$$Z is equipped with a canonical holomorphic foliation $${\mathcal {F}}$$F which is equivariant with respect to the $$({\mathbb {C}}^\times )^m$$(C×)m-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi–Eckmann manifolds, and their deformations. We construct transversely Kähler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kähler submanifold (or, more generally, a Fujiki class $$\mathcal {C}$$C subvariety) in such a moment-angle manifold is contained in a leaf of the foliation $${\mathcal {F}}$$F. For a generic moment-angle manifold $$\mathcal {Z}$$Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of $$\mathcal {Z}$$Z is zero.
环面流形拓扑
DOI：
--
发表时间：
2010
期刊：
影响因子：
--
作者：
Rappleye;J.;S.Choi
通讯作者：
S.Choi