Stability of Abelian Complex Structures

具有大阿贝尔子代数的复辛李代数

Complex Symplectic Lie Algebras with Large Abelian Subalgebras

该研究提出两种构造大阿贝尔理想李代数上复辛结构的方法，完成几乎阿贝尔李代数复辛结构的分类，得到多类非（超）凯勒的紧复辛流形及带拉格朗日环面纤维的实例。

几乎阿贝尔李群与可解流形上的调和殆复结构

Harmonic almost complex structures on almost abelian Lie groups and solvmanifolds

该研究针对几乎阿贝尔李群，刻画了其上殆复结构关于埃尔米特度量调和的殆埃尔米特结构，适配了Gray-Hervella分类，并在相关紧可解流形上给出不同类的调和殆复结构实例。

Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbi…

非负Ricci曲率、无穷远处的几乎稳定性与基本群结构

Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

本文研究非负Ricci曲率开流形的基本群，在万有覆盖无穷远切锥截面几乎稳定的条件下，证明基本群有限生成且含有限指数正规阿贝尔子群，并给出指数上界。

关于幂零李代数上的阿贝尔复结构

On abelian complex structures on Nilpotent Lie Algebras

本文提出幂零李代数上阿贝尔复结构等价分类的新方法，应用该方法得到4、6维已知分类，给出8维1-阿贝尔复结构分类及一般情况部分结果。

Odd dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triple…

Abelian complex structures on solvable Lie algebras

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie groups are complex affine manifolds. It turns out that all 4-dimensional Lie algebras carrying abelian complex structures are central extensions of such affine Lie algebras.

Pseudo-Kähler Lie algebras with abelian complex structures

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-Kähler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen tha…

Pluriclosed and Strominger Kähler-like metrics compatible with abelian complex structures

We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be $2$-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a $2$-step nilpotent Lie group preserves the Strominger Kähler-like condition.

6维李代数上阿贝尔复结构的分类

Classification of abelian complex structures on 6-dimensional Lie algebras

本文分类了可赋予阿贝尔复结构的6维李代数，并在相差全纯同构的意义下对每个代数上这类结构的空间进行了参数化。

Ordered abelian groups over a CW complex

If X is a CW complex, one can assign to each point of X an ordered abelian group of finite rank whose subset of positive elements depends continuously on the points of X. A locally trivial bundle which arises in this way we denote by E(X). In the present work we establish a topological classification of such bundles in terms of the first cohomology group of X with coefficients in the ring Z_2. This result has an amazing application in the theory of characteristic classes of foliations on compact manifolds.