Kahler, Poisson Geometry of CR Lie groups

On Lie groups as quasi-Kähler manifolds with Killing Norden metric

A 6-parametric family of 6--dimensional quasi-Kähler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically.

Left-invariant almost para-Kähler structures on six-dimensional nilpotent Lie groups

In this paper, we consider left-invariant para-complex structures on six-dimensional nilpotent Lie groups. A complete list of six-dimensional nilpotent Lie groups that admit para-Kähler structures is obtained, explicit expressions for para-complex structures are found, and curvature properties of associated para-Kähler metrics are investigated. It is shown that paracomplex structures are nilpotent and the corresponding para-Kähler metrics are Ricci-flat.

On generalized Kähler geometry on compact Lie groups

We present some fundamental facts about a class of generalized Kähler structures defined by invariant complex structures on compact Lie groups. The main computational tool is the BH-to-GK spectral sequences that relate the bi-Hermitian data to generalized geometry data. The relationship between generalized Hodge decomposition and generalized canonical bundles for generalized Kähler manifolds is also clarified.

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine…

Canonical pseudo-Kähler structures on six-dimensional nilpotent Lie groups

In this paper we consider left-invariant pseudo-Kähler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-Kähler metrics are investigated. It is proved that the associated pseudo-Kähler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-Kähler structures obtained give basic models of pseudo-Kähler six-dimensio…

Generalized Kähler almost abelian Lie groups

We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $\partial \bar \partial$-closed. We obtain a classification of six-dimensional generalized Kähler almost abelian Lie groups and determine the 6-dimensional compact almost abelian solvmanifolds admitting an invariant generali…

Anti-Kählerian geometry on Lie groups

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-Kähler structure on $G$. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-Kähler structure $(g,J)$ where $J$ is abelian then the Lie algebra of $G$ is unimodular and $(G,g)$ is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric $g$ for which $J$ is an anti-isometry we obtain that the triple $(G, g, J)$ is an anti-Kähler manif…

左不变近伪Kähler结构与切李群

Left invariant nearly pseudo-Kähler structures and the tangent lie group

该微分几何研究证明，若李群上带有左不变近伪Kähler结构，则其带有诱导李群结构的切丛也存在左不变近伪Kähler结构，拓展了近伪Kähler结构的构造路径。

Locally conformally Kähler structures on unimodular Lie groups

We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is unimodular, while in the latter, we prove that if the Lie algebra is unimodular, then it is isomorphic to the product of $\mathbb R$ and a Heisenberg Lie algebra.

横向凯勒几乎接触度量李代数

Transversely Kähler almost contact metric Lie algebras

研究横向凯勒几乎接触度量李代数的结构，分类最大秩的η-爱因斯坦横向凯勒结构，探讨其与准萨克西安和反准萨克西安李代数的关系。

On the classification of left-invariant para-Kähler structures on four-dimensional Lie groups

A first classification of para-Kähler structures on four-dimensional Lie algebras was obtained by D. Calvaruzo in 2015. In this paper, we propose another classification based on the classification of symplectic Lie algebras. For each four-dimensional symplectic Lie algebra, compatible para-complex structures and the corresponding pseudo-Riemannian metrics are found in explicit form. This leads to the classification of Sasaki para-contact structures on five-dimensional contact Lie algebras with a nontrivial center.

Left invariant generalized complex and Kähler structures on simply connected four dimensional Lie groups: classification and invariant cohomologies

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized Kähler structures on four dimensional simply connected Lie groups.

Deformation classes in generalized Kähler geometry

We introduce natural deformation classes of generalized Kähler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. Lastly we show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

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。

On para-Kähler Lie algebroids and generalized pseudo-Hessian structures

In this paper, we generalize all the results obtained on para-Kähler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-Kähler Lie algebroids. In particular, we study exact para-Kähler Lie algebroids as a generalization of exact para-Kähler Lie algebras. This study leads to a natural generalization of pseudo-Hessian manifolds. Generalized pseudo-Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo-Hessian structures. Namely, we will show that give…

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…

Riemann-Poisson 与 Kähler-Poisson 复流形及结构

Riemann-Poisson and Kähler-Poisson complex manifolds and structures

研究由闭、非退化 $(1,1)$-形式生成的复 Poisson 括号结构，分析其与 Riemann 结构的相容性，并在 $\C^\ast$ 中给出示例。

关于六维幂零非辛李群上的左不变半凯勒结构

On left-invariant semi-Kähler structures on six-dimensional nilpotent nonsymplectic Lie groups

本文针对三类六维幂零非辛李群，定义并研究了其上的左不变半凯勒结构与近仿半凯勒结构及其几何性质。

Locally conformally Kähler structures on four-dimensional solvable Lie algebras

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the $4$-dimensional structures in our classification.

Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, while in the latter case we provide examples of such Lie groups admitting lattices in any even dimension.

Canonical almost pseudo-Kähler structures on six-dimensional nilpotent Lie groups

It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compatible complex structures and, therefore, define pseudo-Kähler metrics. In this paper we show that on the remaining 12 classes of six-dimensional nilpotent symplectic Lie groups there are left-invariant almost pseudo-Kähler metrics, and we study their geometrical properties.

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…

Kähler Lie algebroids 的 Kodaira 消失定理

Kodaira vanishing theorems for Kahler Lie algebroids

本文将复几何中的 Kodaira 消失定理推广到 Kähler Lie algebroids，证明当 p+q 足够大时正线丛值 (p,q)-形式上的 Laplace 算子核消失，并利用局部坐标计算推广 Kähler 恒等式。

关于李群紧商空间上 $p$-Kähler 结构的一则注记

A note on $p$-Kähler structures on compact quotients of Lie groups

本文研究带不变复结构的李群紧商空间上的 $p$-Kähler 结构，分析四维幂零流形的 $p=2$ 情形，并证明高维近阿贝尔可解流形在 $(n-2)$-Kähler 条件下必为 Kähler。

Classification of generalized Kähler-Ricci solitons on complex surfaces

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on non-standard Hopf surfaces, completing the existence question for such structures. Moreover this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex su…

几乎阿贝尔伪Kähler李代数

Almost abelian pseudo-Kähler Lie algebras

研究可解流形上几乎阿贝尔李代数的不变伪Kähler结构，分类所有维数下的酉同构类，并刻画存在复结构和辛结构但无相容伪Kähler度量的李代数。

On the geometry of Kähler-Poisson structures

We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illustration of the new concepts we give an algebraic proof of the statement that a bound on the (algebraic) Ricci curvature induces a bound on the eigenvalues of the (algebraic) Laplace operator, in analogy with the well-known theorem in Riemannian geometry. As the correspondence be…

On $k$-para-Kähler Lie algebras a subclass of $k$-symplectic Lie algebras

$k$-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-Kähler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized $S$-matrices. We determine then all the $k$-symplectic Lie algebras of dimension $(k+1)$ and all the six dimensional 2-pa…

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。

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…

复阿贝尔李超群上的超凯勒结构

Super Kähler structures on the complex Abelian Lie supergroups

本文对复阿贝尔李超群的复化空间上的G-不变超凯勒形式分类，将几何量子化推广到超情形构造酉G-表示，证明不可约表示由动量映射像决定，还构造了G的Gelfand模型。

Deformations of generalized complex and generalized Kahler structures

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structur…

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…

Kähler-Norden structures on Hom-Lie group and Hom-Lie algebras

In the present paper, we describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each other. We study Kähler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex struc…

Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras

We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. Our primary examples include four series of holomorphic Poisson structures on products of flag varieties and related spaces of complex semi-simple Lie groups.

Harmonic morphisms and moment maps on hyper-Kähler manifolds

We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kähler moment map $ϕ$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $ϕ$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the comple…

Generalized Kahler geometry, gerbes, and all that

This work is based on the talk delivered at Poisson 2008. We review the recent advances in Generalized Kahler geometry while stressing the use of Poisson and symplectic geometry. The derivation of the generalized Kahler potential is sketched and the relevant global issues are discussed.

Generalized Kähler metrics from Hamiltonian deformations

We give a new characterization of generalized Kähler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for arbitrary generalized Kähler structures. That is, we show that any generalized Kähler structure may be deformed in such a way that one of its underlying holomorphic Poisson structures remains fixed, while the other deforms via Hitchin's deformation. Finally, we indicate a close r…

Kählerian氧化与约化：具有Kähler结构的李代数

Kählerian oxidation and reduction of Lie algebras with Kähler structures

本文研究李代数上的Kähler结构，提出Kählerian约化与氧化框架，证明维数大于四的不可分解拟幂零Kählerian幂零李代数可由基代数经有限次中心平面氧化重构。

The classification of left-invariant para-Kähler structures on simply connected four-dimensional Lie groups

We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.

Complex nilmanifolds and Kähler-like connections

In this note, we analyze the question of when will a complex nilmanifold have Kähler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a Kähler metric. We give a classification in the first two cases and a partial description in the third case. It would be interesting to understand these questions for all Lie-Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric.

Formality in generalized Kahler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.

Singular Kähler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups

In this paper, we prove an existence result for Kähler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups. As an application, we classify $\mathbb Q$-Fano compactifications of $SO_4(\mathbb C)$ which admit a Kähler-Einstein metric with the same volume as that of a smooth Fano compactification of $SO_4(\mathbb C)$.

Invariant pseudo Kaehler metrics in dimension four

Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,ω)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $ω$ is a symplectic structure). Such structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is parallel. It is proved that most of these complex homogeneous spaces admit a pseudo Kähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular Kähler Lie algebras are flat in dimension four. Ot…

Realization of compact Lie algebras in Kähler manifolds

The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the Kähler structure. The…

Holomorphic vector fields and perturbed extremal Kähler metrics

We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.

Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifold…

Einstein 可解流形上的伪 Kähler 与伪 Sasaki 结构

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

研究可解李群上的左不变 Einstein 伪 Riemann Sasaki 度量，通过与带相容导子的伪 Kähler 幂零李代数对应来刻画并分类相关结构。

Hyper-Kähler quotients of solvable Lie groups

In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on $\H^p \times \H^q$ ($\H$: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.

Geometric structures on Lie groups with flat bi-invariant metric

Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η\in Λ^3L\subset Λ^3V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphism J with J^2=\e Id, then the Lie group {\cal L}(η) is endowed with a left-invariant nearly Kähler structure if \e =-1 and with a left-invariant nearly para-Kähler structure if \e =+1. This construction exhausts all complete simply connected flat nearly (para-)Kähler manifolds. If…

Linearization of Poisson Lie group structures

We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group multiplication in G^* and for Poisson Lie group morphisms, with similar assumptions.

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.

泊松K稳定性与常数量曲率的广义凯勒结构

Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence

研究紧致凯勒全纯泊松流形的泊松K稳定性，利用无限维动量映射技术，证明了凯勒 - 爱因斯坦法诺流形上泊松结构的“半经典YTD猜想”存在性部分，得到新的常数量曲率辛广义凯勒结构例子并在射影平面证猜想。

A note about invariant SKT structures and generalized Kahler structures on flag manifolds

We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.