Non Kaehler solvmanifolds with generalized Kaehler structure

Generalized Kaehler manifolds with split tangent bundle

We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.

On the local structure of generalized Kaehler manifolds

Let (g,b,J_+,J_-) be the bihermitian structure corresponding to a generalized Kaehler structure. We find natural integrability conditions under which db=0.

非卡拉维叶溶菌体上的阿普利-波茨-陈三重Massey积

Aeppli-Bott-Chern Massey products on non-Kähler solvmanifolds

本文研究了非卡拉维叶溶菌体上非平凡三重ABC-Massey积的显式计算，并证明了特定流形具有非消失的三重积且不具有近卡拉维叶度量。

From generalized Kaehler to generalized Sasakian structures

This is an expository paper, which provides a first introduction to geometric structures on $TM\oplus T^*M$. The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures. A few of these properties are new.

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…

Six dimensional solvmanifolds with holomorphically trivial canonical bundle

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family $(M,J_a)$ of compact complex manifolds such that $(M,J_a)$ satisfies the $\partial\bar\partial$-Lemma and admits a balanced metric for any $a\not=0$, but the central limit neither satisfies the $\partial\bar\partial$-Lemma nor …

Euler-Lagrange formulas for pseudo-Kaehler manifolds

Let $c$ be a characteristic form of degree $k$ which is defined on a Kaehler manifold of real dimension $m>2k$. Taking the inner product with the Kaehler form $Ω^k$ gives a scalar invariant which can be considered as a generalized Lovelock functional. The associated Euler-Lagrange equations are a generalized Einstein-Gauss-Bonnet gravity theory; this theory restricts to the canonical formalism if $c=c_2$ is the second Chern form. We extend previous work studying these equations from the Kaehler to the pseudo-Kaehler setting.

关于标准径向凯勒度量

On canonical radial Kaehler metrics

该研究证明了径向凯勒度量成为凯勒-爱因斯坦度量的充要条件，涉及极值性、凯勒-里奇溶菌及广义标量曲率的常数性。

A note on submanifolds and mappings in generalized complex geometry

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.

Almost Hermitian Manifolds with Vanishing Generalized Bochner Curvature Tensor

We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).

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.

殆阿贝尔可解流形的广义上同调

On the generalized cohomologies of almost abelian solvmanifolds

本文研究殆阿贝尔李代数与殆阿贝尔可解流形上的三类广义上同调，将其转化为Chevalley–Eilenberg上同调并推导组合公式，证明不变形式上同调可嵌入紧商空间的上同调。

Goto的广义Kaehler稳定性定理

Goto's generalized Kaehler stability theorem

本文给出了Goto的广义Kaehler稳定性定理的简短且更直接的证明，表明在J_2由无处为零的闭形式确定时，J_1与J_2的小变形可耦合以保持广义Kaehler结构。

几乎阿贝尔可解流形的几何与上同调

On the generalized geometry of almost abelian solvmanifolds

本文研究几乎阿贝尔李群及其可解流形上的左不变广义复结构，基于结构矩阵的谱完成分类，推导广义卡拉比-丘条件，计算广义多贝尔上同调，为相关几何结构的研究提供系统分类与上同调结果。

Instantons, Poisson structures and generalized Kaehler geometry

Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.

Generalized pseudo Kaehler structures

In the paper we consider pseudo bihermitian structures - a pair of complex structures compatible with a pseudo Riemannian metric. As in the positive definite case we establish its relations with generalized (pseudo) Kaehler geometry and holomorphic Poisson structures. We provide a list of compact complex surfaces which could admit such structure and also examples of bihermitian structures on some of them. We also consider a naturally defined null plane distribution on generalized pseudo Kaehler 4-manifold and show that under a mild restriction it determines an Engel structure.

通过映射环面构造广义Kähler流形

Generalized Kähler manifolds via mapping tori

本文利用3-环面与紧致Kähler或hyperKähler流形的乘积构造映射环面，得到分裂型与非分裂型广义Kähler流形，并研究其形式性与Dolbeault上同调。

瞬子模空间的广义Kaehler几何

Generalized Kaehler geometry of instanton moduli spaces

本文证明了紧致偶广义Kaehler四流形上瞬子模空间的Hitchin广义Kaehler结构可通过广义Kaehler约化获得，其底层Courant algebroid约化实现了Donaldson三次$μ$-映射。

Toric generalized Kaehler structures

Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler structures of symplectic type, without requiring them to be anti-diagonal. Such a structure is characterized by a triple $(τ, C, F)$ where $τ$ is a strictly convex function defined in the interior of the moment polytope $Δ$ and $C, F$ are two constant anti-symmetric matrices. We prove that underlying each such a structure is a \emph{canonical} toric K$\ddot{a}$hler s…

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…

具有余维二幂零根的可解李代数上的SKT结构

SKT solvable Lie algebras with codimension two nilradical

本文分类了具有余维二幂零根的六维可解李代数上的SKT结构，并给出将SKT幂零李代数扩张为高维可解李代数的一般构造，同时构造了新的紧可解流形例子。

The fundamental group, rational connectedness and the positivity of Kaehler manifolds

First we confirm a conjecture asserting that any compact Kähler manifold $N$ with $\Ric^\perp>0$ must be simply-connected by applying a new viscosity consideration to Whitney's comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension $n$ under the condition $\Ric_k>0$ (for some $k\in \{1, \cdots, n\}$, with $\Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a secon…

Invariant f-structures in the generalized Hermitian geometry

We collect the recent results on invariant f-structures in the generalized Hermitian geometry. Here the canonical f-structures on homogeneous k-symmetric spaces play a remarkable role. Specifically, these structures provide a wealth of invariant examples for the classes of nearly Kaehler f-structures, Hermitian f-structures and some others. Finally, we consider all invariant f-structures on the complex flag manifold SU(3)/T_{max} and describe them in the sense of generalized Hermitian geometry. In particular, it presents first invariant examples of Killing f-structures.

齐性广义 Ricci 流

The homogeneous generalized Ricci flow

本文基于括号流框架研究Lie群离散商上的广义Ricci流，证明了任意维solvmanifolds上的整体存在性，定义了exact Courant algebroids上的广义Ricci孤立子并分析了其在nilmanifolds上的极限行为，同时给出了不变广义度量的曲率新公式。

Branes on Poisson varieties

We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kaehler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is non-holomorphic in nature. Finally we show an equivalence between certain configurations of branes on Poisson varieties and generalized Kaehler structures, and use this to construct explicitly new families of generalized Kaehler structures on compact holomorphic Poisson manifolds equipped with positive Poisson line bundles (e.g. Fano manifolds). We end with so…

The index of symmetry of a flag manifold

We study the index of symmetry of a compact generalized flag manifold M=G/H endowed with an invariant Kaehler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces and we estimate their dimension.

Pseudo-Kaehler Quantization on Flag Manifolds

A unified approach to geometric, symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo-Kaehler structure is proposed. The Hilbert space of states is realized via the Bott-Borel-Weil theorem in the sheaf cohomology of the geometric quantization line bundle. The corresponding deformation quantization is a quantization with separation of variables. In particular cases we arrive at Berezin's quantization via covariant and contravariant symbols.

Differential Geometry of generalized almost quaternionic structures, 2

Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.

Dirac structures on generalized Riemannian manifolds

We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.

可解流形SYZ镜像的上同调

The Cohomology of Solvmanifold SYZ Mirrors

本文研究非凯勒SYZ镜像在可解流形上双扭纤结构中的几何和上同调性质，探讨了超对称膜在辛和复侧映射的关系，构建了由李代数数据决定的非凯勒SYZ镜像对，并分析了Tseng-Yau上同调在傅里叶-穆卡伊变换中的作用。

Equivariant APS index for Dirac operators of non-product type near the boundary

We consider a generalized APS boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly c…

Generalized Complex Submanifolds

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of an involution preserving a twisted generalized complex structure is a twisted generalized complex submanifold. We also prove that a twisted generalized complex manifold has a natural Poisson structure. We also discuss generalized Kaehler submanifolds.

Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost Kähler - Ricci geometric evolution equations. The se…

Reduction of metric structures on Courant algebroids

We use the procedure of reduction of Courant algebroids to reduce strong KT, hyper KT and generalized Kaehler structures on Courant algebroids. This allows us to recover results from the literature as well as explain from a different angle some of the features observed there in. As an example, we prove that the moduli space of instantons of a bundle over a SKT/HKT/generalized Kähler manifold is endowed with the same type of structure as the original manifold.

Generalized toric varieties for simple non-rational convex polytopes

We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a complex quasitorus. We associate to each simple polytope, rational or not, a family of complex quasifolds having same dimension as the polytope, each containing a dense open orbit for the action of a suitable complex quasitorus. We show that each of these spaces M is diffeomorphic to one of the symplectic quasifolds defined in http://arXiv.org/abs/math:SG/9904179,…