Horn problem for quasi-hermitian Lie groups

平坦的Hermite李代数是Kähler的

Flat Hermitian Lie algebras are Kähler

本文证明平坦的Hermite李代数必为Kähler，通过分析2步可解李群的Hermite几何性质，得出平坦Hermite结构的李群必为Kähler的结论。

Hypercomplex structures with Hermitian-Norden metrics on four-dimensional Lie algebras

Integrable hypercomplex structures with Hermitian and Norden metrics on Lie groups of dimension 4 are considered. The corresponding five types of invariant hypercomplex structures with hyper-Hermitian metric, studied by M.L. Barberis, are constructed here. The different cases regarding the signature of the basic pseudo-Riemannian metric are considered.

Special Hermitian metrics and Lie groups

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form $ω$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfrak g$ {and using} a Hermitian flat connection on $\mathfrak g$ we construct a $4n$-dimensional SKT Lie algebra. We apply this method to some 4-dimensional SKT Lie algebras. Moreover, we classify symplectic forms taming complex structures on 4-dimensional Lie algebras.

Lie groups of dimension 4 and almost hypercomplex manifolds with Hermitian-Norden metrics

Object of investigation are almost hypercomplex manifolds with Hermitian-Norden metrics of the lowest dimension. The considered manifolds are constructed on 4-dimensional Lie groups. It is established a relation between the classes of a classification of 4-dimensional indecomposable real Lie algebras and the classification of the manifolds under study. The basic geometrical characteristics of the constructed manifolds are studied in the frame of the mentioned classification of the Lie algebras.

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger- Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

余维数为 $2$ 的交换理想的李代数的 Hermitian 几何

Hermitian geometry of Lie algebras with abelian ideals of codimension $2$

本文研究含余维数二交换理想的幺模李代数上的 Hermitian 度量，并分类了 Bismut Kähler-like 与挠率平行度量。

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-Kähler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-Kähler is given.

Curvature flows for almost-hermitian Lie groups

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2n-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to …

Natural almost Hermitian structures on conformally foliated 4-dimensional Lie groups with minimal leaves

Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\F$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\F$. We classify such structures $J$ which are almost Kähler $(\A\K)$, integrable $(\I)$ or Kähler $(\K)$. Hereby we construct several new multi-dimensional examples in each class.

Maximal representations of uniform complex hyperbolic lattices in exceptional Hermitian Lie groups

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G$, then $n=2$ and $G={\rm E}_6$, and we describe completely the representation $ρ$. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author (arxiv:1506.07274). However we do not focus immediately on the ex…

Dolbeault Harmonic $(1,1)$-forms on $4$-dimensional compact quotients of Lie Groups with a left invariant almost Hermitian structure

We study Dolbeault harmonic $(1,1)$-forms on compact quotients $M=Γ\backslash G$ of $4$-dimensional Lie groups $G$ admitting a left invariant almost Hermitian structure $(J,ω)$. In this case, we prove that the space of Dolbeault harmonic $(1,1)$-forms on $(M,J,ω)$ has dimension $b^-+1$ if and only if there exists a left invariant anti self dual $(1,1)$-form $γ$ on $(G,J)$ satisfying $id^cγ=dω$. Otherwise, its dimension is $b^-$. In this way, we answer to a question by Zhang.

Abelian Balanced Hermitian structures on unimodular Lie algebras

Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dimension of the center of $\mathfrak{g}$. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent…

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on comp…

Abelian Hermitian geometry

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a…

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…

Natural Almost Hermitian Structures on Conformally Foliated 4-Dimensional Lie Groups with Minimal Leaves

Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\mathcal{F}$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\mathcal{F}$. The corresponding Lie algebra $\mathfrak{g}$ must then belong to one of 20 families $\mathfrak{g}_1,\dots,\mathfrak{g}_{20}$ according to S. Gudmundsson and M. Svensson. We classify such structures $J$ which are almost Kähler $(\mathcal{A}\mathcal{K})$, integrable $(\mathcal{I})$ or Kähler $(\mathcal{K})$. Hereby, we construct 16 multi-dimensional almost Kähler fami…

埃尔米特约化李代数中的涡旋丝

Vortex Filaments in Hermitian Reductive Lie Algebras

本文以纯几何方式将涡旋丝理论推广至埃尔米特约化李代数，所得涡旋模型在等坍缩为埃尔米特对称李代数时可对应退化，丰富了相关几何与物理研究内容。

Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisim…

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedull…

Graded Lie algebra of Hermitian tangent valued forms

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the…

Eight-Dimensional Hermitian Lie Groups Conformally Foliated by Minimal $\textbf{SU}(2) \times \textbf{SU}(2)$ Leaves

We investigate the $8$-dimensional Riemannian Lie groups $G^8$, carrying a left-invariant, conformal and minimal foliation $\mathcal{F}$, with leaves diffeomorphic to the subgroup $\textbf{SU}(2) \times \textbf{SU}(2)$ of $G^8$. Such groups have been classified by E. Ghandour, S. Gudmundsson and T. Turner in their recent work. They show that these $8$-dimensional Lie groups form a real $13$-dimensional family. For each left-invariant Hermitian structure $J_\mathcal{V}$ on $\textbf{SU}(2) \times \textbf{SU}(2)$, we extend this to an almost Hermitian structures $J$ on $G$ adapted to the foliatio…

具有殆 Hermitian 结构的李群上的左不变 Hermitian 联络

Left-invariant Hermitian connections on Lie groups with almost Hermitian structures

本文研究任意李群上左不变殆 Hermitian 结构下的 Hermitian 与 Gauduchon 联络，给出其与 (1,1) 型形式空间的一一对应及挠率显式公式，并在特殊直积情形下构造满足强 Kähler 带挠率条件且含平坦联络的实例。

正 Hermitian 曲率流在 2 步幂零李群上的行为

Positive Hermitian curvature flow on 2-step nilpotent Lie groups

本文研究2步幂零李群上左不变度量的正Hermitian曲率流，证明其长期存在并收敛到非平坦半代数孤子，且此类孤子至多唯一。

具有二维交换子群的李群上的正交结构

Hermitian and Kähler Structures of Two Types on Lie Groups with Two-Dimensional Derived Algebras

本文研究了具有二维交换子群的李群上的左不变正交结构，分类了两种类型，并计算了凯勒结构的比姆特连接，同时给出了相关例子。

Submaximally Symmetric Quaternion Hermitian Structures

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

关于具有余维二交换理想的Hermitian李代数的曲率性质

Curvature property on Hermitian Lie algebras with abelian ideals of codimension two

本文研究含余维二交换理想的单模Hermitian李代数，证明常Chern全纯截面曲率蕴含Chern平坦，且其他典范联络的常全纯截面曲率导致Kähler平坦及商代数的交换性。

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。

以李代数 $\mathfrak{gl}^{+}(2,\mathbb{C})$ 中 Hermitian 矩阵为生成空间的 ${\rm GL}^{+}(2,\mathbb{C})$ 上的次洛伦兹测地线

Sub-Lorentzian geodesics on ${\rm GL}^{+}(2,\mathbb{C})$ with the generating space of Hermitian matrices in the Lie algebra $\mathfrak{gl}^{+}(2,\mathbb{C})$

本文在正实行列式矩阵李群 GL+(2,C) 上，以 Hermitian 矩阵子空间的 Minkowski 结构定义左不变次洛伦兹度量，借助时间反最优控制与 Pontryagin 最小值原理，并利用 SL(2,C) 上的次黎曼测地线，求出了非类空测地线和最长弧。

Hermitian vector fields and special phase functions

We start by analysing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase …

Lie Groups with flat Gauduchon connections

We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in \mathbb{R}$ and $\nabla^c$ and $\nabla^b$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a conne…

Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar …

Locally conformal SKT almost abelian Lie algebras

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure $(J, g)$ whose Bismut torsion 3-form $H$ satisfies the condition $dH = α\wedge H$, for some closed non-zero 1-form $α$. This condition was introduced in [6] as a generalization of the SKT (or pluriclosed) condition $dH= 0$. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure $(J, g)$ such that $dH = α\wedge H$, for some closed 1-form $α$. As an application we classifiy LCSKT almost abelian Lie algebras in dimension $6$. Finally, we also study on almost abelian Lie algebras the …

Hermitian curvature flow on complex homogeneous manifolds

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nil…

Tight homomorphisms and Hermitian symmetric spaces

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore we classify all tight embeddings of the Poincare' disk.

Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

We obtain the full classification of coisotropic and polar actions of compact Lie group on irreducible Hermitian symmetric spaces.

On almost complex Lie algebroids

The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the $E$-Chern form of $E^{1,0}$ associated to an almost complex connection $\nabla$ on $E$ can be expressed in terms of the matrix $J_ER$, where $J_E$ is the almost complex structure of $E$ and $R$ is the curvature of $\nabla$. Also, we cons…

Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures. Some natural algebraic problems are approached and…

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of uniform lattices of SU(m,1), m>1, into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G=SU(p,2) with p>=2m and the representation is redu…

On the Chern-Ricci flow and its solitons for Lie groups

This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgr…

Positive Hermitian Curvature Flow on special linear groups and perfect solitons

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

Almost Hermitian Geometry on Six Dimensional Nilmanifolds

The fundamental 2-form of an invariant almost Hermitian structure on a 6-dimensional Lie group is described in terms of an action by SO(4)xU(1) on complex projective 3-space. This leads to a combinatorial description of the classes of almost Hermitian structures on the Iwasawa and other nilmanifolds.

Lie bialgebras of complex type and associated Poisson Lie groups

In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G* are complex Lie groups. We also prove that a Hermitian structure on the Lie algebra $\mathfrak{g}$ with ad-invariant metric induces a structure of the same type on the double Lie algebra ${\mathcal D}\mathfrak{g}= \mathfrak{g}\oplus\mathfrak{g}^*$, with respect to the canonical ad-invariant metric of neutral signature on ${\m…

两步幂零李群上的复结构

Complex structures on two-step nilpotent Lie groups

本文刻画了容许左不变复结构的 $2$ 步幂零李代数，研究了其 Hermitian 几何，给出了度量为 pluriclosed 的充要条件，并将结果推广至 hypercomplex 情形。

Complex and Kahler structures on hom-Lie algebras

Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a hom-left symmetric algebra, a phase space is provided and then a complex structure on it, is presented. Finally, the notion of Kahler hom-Lie algebra is introduced and then using a Kahler hom-Lie algebra, a phase space is constructed.

Hermitian structures on a class of almost nilpotent solvmanifolds

In this paper we investigate the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients $Γ\backslash G$, where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and $Γ$ is a lattice of $G$. We first obtain a characterization of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical $\mathfrak{n}$ has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In parti…

具有常数混合曲率的厄米流形

On Hermitian manifolds with constant mixed curvature

本文验证了Kai Tang关于非零常数混合曲率紧致厄米流形必为凯勒流形的猜想，针对复幂零流形、可解流形等特殊类型及平衡三维流形，提供了部分证据。

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…

Geometrical phases on hermitian symmetric spaces

For simple Lie groups, the only homogeneous manifolds $G/K$, where $K$ is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the complex Grassmann manifold and its noncompact dual is presented.It is shown that the multiplicative factor is identical with the two-cocycle considered by A. Guichardet and D. Wigner for simple Lie groups.

The Anomaly flow on unimodular Lie groups

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.

几类特殊埃尔米特流形上的Streets-Tian猜想

Streets-Tian Conjecture on several special types of Hermitian manifolds

本文针对几类特殊紧埃尔米特流形验证了Streets-Tian猜想，通过显式构造度量及形变路径的初等方法拓展了已有结果，展现了该猜想的代数复杂性。

Homogeneous Hermitian manifolds and special metrics

We consider non-Kaehler compact complex manifolds which are homogeneous under the action of a compact Lie group of biholomorphisms and we investigate the existence of special (invariant) Hermitian metrics on these spaces. We focus on a particular class of such manifolds comprising the case of Calabi-Eckmann manifolds and we prove the existence of an invariant Hermitian metric which is Chern-Einstein, namely whose second Ricci tensor of the associated Chern connection is a positive multiple of the metric itself. The uniqueness is also discussed.

Local rigidity of complex hyperbolic lattices in semisimple Lie groups

We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.

Static SKT metrics on Lie groups

An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form $ω$ satisfies $\de\debarω=0$. Streets and Tian introduced in \cite{sttiPlur} a Ricci-type flow that preserves the SKT condition. This flow uses the Ricci form associated to the Bismut connection, the unique Hermitian connection with totally skew-symmetric torsion, instead of the Levi-Civita connection. A SKT metric is static if the (1,1)-part of the Ricci form of the Bismut connection satisfies $\riccib=λω$ for some real constant $λ$. We study invariant static metrics on simply connected Lie groups, providing in…