标题： 六维几乎幂零可解流形上的埃尔米特结构 显示英文标题

标题： Hermitian structures on six-dimensional almost nilpotent solvmanifolds

摘要： 我们完成了对六维强单模几乎幂零李代数在允许复结构下的分类。 对于几种情况，我们描述了复结构在同构意义下的空间。 作为结果，我们确定了允许不变复结构的六维几乎幂零可解流形，并研究了特殊类型的赫米特度量的存在性，包括SKT、平衡、局部共形凯勒和强高斯度量。 特别是，我们确定了新的平衡可解流形，并验证了第一作者与Vezzoni关于六维强单模几乎幂零情况下SKT和平衡结构的猜想。 此外，我们证明了一些关于被辛形式驯服的复结构的负面结果，特别地，表明在每个维度上这样的结构不能存在于非凯勒几乎交换李代数上。 显示英文摘要

摘要： We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally K\"ahler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-K\"ahler almost abelian Lie algebras.