Compatibility between non-Kähler structures on complex (nil)manifolds

Stability of pure Nilpotent Structures on collapsed Manifolds

The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a $n$-manifold of bounded sectional curvature are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeomorphism, the underlying nilpotent Killing structures coincide with each other or one is embedded into another as a subsheaf. It improves Cheeger-Fukaya-Gromov's locally compatibility of pure nilpotent Killing structures for one collapsed metric of bounded sectional curvature to …

Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure

Let $E$ be a holomorphic vector bundle. Let $θ$ be a Higgs field, that is a holomorphic section of $End(E)\otimesΩ^{1,0}_X$ satisfying $θ^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle $(E,θ)$. The tuple $(E,θ,h)$ is called a harmonic bundle. Let $X$ be a complex manifold, and $D$ be a normal crossing divisor of $X$. In this paper, we study the harmonic bundle $(E,θ,h)$ over $X-D$. We regard $D$ as the singularity of $(E,θ,h)$, and we are particularly interested in the asymptotic behaviour of the harmonic bundle around $D$. We will see that it is similar to the asymptotic behaviou…

Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M=G/K is diffeomorphic to a product U/K x N, where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such tha…

关于拟统计流形上的 Kähler 和 anti-Kähler 结构的注记

Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds

本文研究了拟统计流形上的近 Hermite 和近反 Hermite 结构，给出了近复结构可积性条件，并证明了其在特定条件下成为 Kähler、anti-Kähler 及拟 Kähler-Norden 流形的充要或必要条件。

Locally conformal SKT structures

A Hermitian metric on a complex manifold is called SKT (strong Kähler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $α$ such that $d H = α\wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices…