摘要翻译：
尼尔流形是幂零群$G$与一个共紧离散子群的商。复零流形是具有$G$-不变复结构的零流形。我们证明了复Nilgionold具有平凡的规范丛。它用于研究超复Nilfiolds（具有满足四元数关系的三个$G$-不变复结构的Nilfiolds）。证明了一个超复Nilfiold允许HKT（hyperkahler with torsion）度量当且仅当其下的超复结构是阿贝尔的。此外，Nilfiold上的任何$G$-不变HKT-度量对于所有相关的复杂结构都是平衡的。
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英文标题：
《Canonical bundles of complex nilmanifolds, with applications to
  hypercomplex geometry》
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作者：
Maria Laura Barberis, Isabel G. Dotti and Misha Verbitsky
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Differential Geometry        微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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英文摘要：
  A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.
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PDF链接：
https://arxiv.org/pdf/0712.3863