M. Paredes
We obtain new families of (1,2)-symplectic invariant metrics on the full complex flag manifolds F(n). For n > 4, we characterize n-3 different n-dimensional families of (1,2)-symplectic invariant metrics on F(n). Any of these families corresponds to a different class of non-integrable invariant almost complex structure on F(n).
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A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.
A description of the canonical symplectic form of the cotangent bundle of a generalized flag manifold
本文研究半单李群相关齐性空间的辛几何，重点是广义旗流形的余切丛，通过主丛的联络与曲率形式给出刘维尔辛形式的显式描述，建立了李理论性质与辛结构的几何联系。
For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures o…
The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determine the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action given by the Iwasawa decomposition of the general linear group over the quaternions.
For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a cho…
Weighted nonlinear flag manifolds as coadjoint orbits
本文研究加权非线性旗的Frechet流形几何，推广了加权非线性Grassmannians，并在辛流形环境下利用加权迷向非线性旗描述了Hamilton微分同胚群的一类余伴随轨道。
In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Our construction is based on a toric degeneration of flag varieties and a deformation of Kähler structure on toric varieties by symplectic potentials.
The $H$-flux on flag manifolds generated by infinitesimal $T$-duality
该研究定义了旗流形与$H$-通量配对的新对应关系，指出无穷小$T$-对偶可生成非平凡$H$-通量，并阐明该对应在$B$-变换下交换复结构与辛结构。
Quaternionic stochastic areas on quaternionic full flag manifolds and applications
本文研究了四元数全旗流形上布朗运动的随机面积及其极限分布，通过矩阵值扩散过程和辛布朗运动的关系，推导出正态分布的协方差矩阵，并应用于四元数缠绕问题。
We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study \textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With these we define a moduli space of $ω$-compatible complex structures. We recall the theory of flag manifolds in order to study the Siegel domain and other domains alike, which is the fibre of the referred twistor space. Finally the structure equations for the twistor of a Riemann surface with the canonical symplectic-metric connection are deduced, based on a given conf…
We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds.
Some invariant connections on symplectic reductive homogeneous spaces
本文研究辛约化齐性空间上的自然辛联络，引入一族不变联络并证明存在唯一自然辛联络，计算其曲率张量，并验证Wallach旗流形的首选联络与之重合且Ricci平行。
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples.
In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic…
The nonholonomic Riemann and Weyl tensors for flag manifolds
该研究定义了非完整结构下黎曼与外尔张量的类似物，借助Premet定理和SuperLie包计算了单李代数抛物子代数对应旗簇的相关张量，并得到G(2)-结构及其超对称对应物的平坦性障碍。
We indicate a geometric relation between Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating to each compact Riemannian symmetric space, via Marsden-Weinstein reduction, a generalized flag manifold which covers the space parametrizing all of its maximal totally geodesic tori. In the process we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in d…
This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with…
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