Normal forms of vector fields on Poisson manifolds
Philippe Monnier, Nguyen Tien Zung
本文研究了Poisson流形上奇点附近的径向和Hamilton向量场的形式和解析正规形。
We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.
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Action-angle variables and a KAM theorem for b-Poisson manifolds
本文证明了b-Poisson流形上b-可积系统的作用角定理，改进了文献[LMV 11]中关于一般Poisson流形的作用角定理。作为应用，我们证明了b-Poisson流形的一个KAM型定理。
Action-angle coordinates and KAM theory for singular symplectic manifolds
本专著研究$b^m$-Poisson流形上可积系统的分类与摄动问题，证明了其作用-角坐标定理与KAM定理，结合去奇异化技术推广得到折叠辛流形的KAM定理，还提出了带特异超曲面摄动的辛流形新KAM定理，为一般Poisson流形摄动理论研究奠定基础。
We prove that for regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group and we give some applications. In particular, we show that for regular unimodular Poisson manifolds top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what is a perfect Poisson manifold. We use these Poisson homology computations to provide families of perfect Poisson manifolds.
On the Lie-formality of Poisson manifolds
本文从描述Poisson流形上的上同调不变量的问题出发，证明了一个在某种意义上“行不通”的结果：光滑Poisson流形上的de Rham型的微分分次李代数是形式的.
Killing-Poisson tensors on Riemannian manifolds
我们在黎曼流形上引入了一类新的Poisson结构。这类中的泊松结构称为Killing-Poisson结构。Killing-Poisson结构类包含辛结构类、在{ \it（Differential Geometry and its Applications，{ \bf Vol.20，Issue 3}（2004），279- 291）}中研究的Poisson结构类以及由黎曼流形上的某些无穷小李代数作用诱导的Poisson结构类。我们证明了辛流形上的一些经典结果（李代数体结构与辛结构的可积性，紧流形上辛结构的非精确性）对正则Killing-Poisson结构仍然有效.
Notes on Geometric Morita equivalence of twisted Poisson manifolds
本文主要研究扭Poisson流形的Morita等价。我们回顾了一些Morita不变量，并证明了可积扭Poisson流形是规范等价的是Morita等价的。此外，我们引进了弱Morita等价的概念，并证明了如果两个扭Poisson流形是弱Morita等价的，则它们的扭辛叶之间存在一一对应。
Bott Connection and Generalized Functions on Poisson Manifold
本文将寻找泊松流形上哈密顿不变体积形式的问题推广到构造哈密顿不变广义函数的问题。为此，我们引入了广义中心的泊松代数，这是广义Casimir函数的空间的概念。我们研究了当广义函数的测试对象集是紧支撑的光滑函数空间时的情形，以及当测试对象是n-形式时的情形，其中n是Poisson流形的维数。我们描述了这个问题与泊松结构的同调性质的关系，与相应的辛叶理和模类的Bott连接。
On the modular classes of Poisson-Nijenhuis manifolds
本文证明了Poisson-Nijenhuis流形的一个性质，从而给出了Damianou和Fernandes定义的模向量场族的双哈密顿性质的新证明.
The Canonical Differential Complex of Poisson Manifold and Distributions
本文研究了Poisson代数的各种同调结构、奇异Poisson结构的典型微分复形和这类流形上的星算子的类似。给出了经典的Koszul微分的外形式作为具有某些二阶元的超交换子的解释。描述具有奇异泊松结构的流形上的不变分布空间。
Locally Lagrangian Symplectic and Poisson Manifolds
我们讨论辛流形，局部的结构是拉格朗日动力学中遇到的。给出了实例和特征性质。然后，我们讨论了拉格朗日子流形的Maslov类的计算。最后，我们指出这种类型的结构的推广泊松流形。
Algebraic definition of Holonomy on Poisson Manifold
给出了\cite{Ginzburg}中引入的Poisson流形辛叶上联络的代数构造。这种构造适用于正则辛叶理上的线性化完整性的定义。
We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection $D$ of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a k-cosymplectic structure, on the intersection of hypersurfaces in $D$.
We inquire into the relation between the curl operators, the Poisson coboundary operators and contravariant derivatives on Poisson manifolds to study the theory of differential operators in Poisson geometry. Given an oriented Poisson manifold, we describe locally those two differential operators in terms of Poisson connection whose torsion is vanishing. Moreover, we introduce the notion of the modular operator for an oriented Poisson manifold. For a symplectic manifold, we describe explicitly the modular operator in terms of the curvature 2-section of Poisson connection, analogously to the Wei…
Coupling Poisson and Jacobi structures on foliated manifolds
设M是一个赋有叶理F的微分流形。M上的Poisson结构P是F-耦合的，如果TF的零化子由P定义的锐态射的象是叶理F的法丛。这个概念扩展了杨-米尔斯场中粒子的斯滕伯格耦合辛形式。本文将Vorobiev的耦合Poisson结构理论从纤维丛推广到叶理上，给出了辛流形上传递李代数胚核簇上耦合Poisson结构的Vorobiev存在性和等价性定理的简单证明.然后讨论了耦合条件在叶流形上Jacobi结构的推广。
Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds
本文通过范畴化辛群胚约化方法，统一了拟泊松流形商的可积性理论，并构造了积分规范群胚上泊松结构的双重辛群胚。
We associate to any integrable Poisson manifold a stack, i.e. a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic.
Reduction and construction of Poisson quasi-Nijenhuis manifolds with background
我们扩展Falceto-Zambon版本的Marsden-Ratiu泊松约化的泊松拟Nijenhuis结构与背景上的流形。定义了具有背景的Poisson拟Nijenhuis结构的规范变换，研究了它们的一些性质，并证明了它们与约化过程是相容的。我们利用规范变换构造了具有背景的Poisson拟Nijenhuis结构。
Reduction of Poisson manifolds with Hamiltonian Lie algebroids
本文提出了具有哈密顿李代数胚的Poisson流形的约化定理，通过引入相容动量截面概念并证明其为李代数同态，得出其零水平集商空间为Poisson流形。
The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation
证明了对于任意正则Poisson流形，存在从第一个叶上同调空间到第一个Poisson上同调空间的内射自然线性映射，该映射将辛叶理的Reeb类映射到Poisson流形的模类.这些类的黎曼解释将允许我们表明，一个正则泊松流形的辛叶是余维1是么模的当且仅当它的辛叶是黎曼叶。它允许我们也建设的例子，单模泊松流形和其他不unimodular。最后，我们证明了第一叶上同调是Morita等价的不变量。
We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introd…
Deformations of log-Lagrangian submanifolds of Poisson manifolds
我们考虑某些偶数维类辛泊松流形中的类拉格朗日子流形。我们表明，在适当的横截性假设下，对由周围的泊松流形和子流形有通畅的变形和变形自动保持拉格朗日性质。
In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson cohomology groups of holomorphic toric Poisson manifolds, i.e., toric varieties endowed with $T$-invariant holomorphic Poisson structures.
Poisson Quasi-Nijenhuis Manifolds
我们引入Poisson拟Nijenhuis流形的概念，推广了Magri-Morosi的Poisson-Nijenhuis流形。我们还研究了Poisson拟Nijenhuis流形的积分问题。特别地，我们证明了在一定的拓扑假设下，Poisson（拟）-Nijenhuis流形与辛（拟）-Nijenhuis群胚一一对应。作为应用，我们研究了Poisson拟Nijenhuis流形上的广义复结构。证明了广义复流形对应于一类特殊的Poisson拟Nijenhuis结构。因此，我们证明了一个广义的复结构可积分到一个辛拟Nijenhuis广群，从而恢复了Crainic定理。
A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we computed the Poisson cohomology groups for all holomorphic toric Poisson structures on $CP^n$, with the stand Poisson structure on $CP^n$ as a special case. We also computed the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on $C^n$.
Coisotropic embeddings in Poisson manifolds
考虑了两类余迷向嵌入的存在性和唯一性，并证明了一类基本函数的Poisson代数的形变量子化的存在性。首先，我们证明了满足一定的常秩条件的泊松流形的任何子流形位于coisotropically一些更大的共辛子流形，这是自然赋予的泊松结构。然后给出了Dirac流形可余各向同性嵌入Poisson流形的条件，推广了Gotay的一个经典定理。
A contravariant pseudo-Hessian manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a contravariant Codazzi equation. When $h$ is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifolds and, in fact, to any contravariant pseudo-Hessian manifold $(M,\nabla,h)$ we associate naturally a Poisson tensor on $TM$. We investigate these properties and we study in details many classes of such structures in order to high…
We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foli…
Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
在简要总结了Poisson流形和李代数胚的主要性质之后，我们综述了Poisson流形和扭Poisson流形的模类、具有Poisson结构或扭Poisson结构的李代数胚的模类以及Poisson-Nijenhuis流形的模类的最新工作.回顾旋量方法的模块类的结论文件。
Homogeneous symplectic manifolds of Poisson-Lie groups
本文研究Poisson-李群的齐性空间辛流形。我们表明，这些空间是，在一定的假设下，覆盖空间的穿着轨道的泊松-李群的作用。泊松归纳过程的影响，这样的空间也检查，从而导致一个有趣的推广的概念的齐性空间。根据已有的结果讨论了Poisson-李群的齐性空间的一些例子。
Normal forms for Poisson maps and symplectic groupoids around Poisson transversals
Poisson断面是Poisson流形中与所有辛叶横截和辛相交的子流形。在前面的一个注记中，我们证明了围绕这类子流形的一个规范形定理。在这篇文章中，我们将这个结果推广到泊松断面周围的泊松映射的范式定理。泊松映射将泊松断面拉回到泊松断面，我们的第一个主要结果表明，在这样的断面周围存在同时的规范形，对于这种断面，泊松映射成为横向线性的，并且交织断面的规范形数据。我们的第二个主要结果涉及辛积分。我们证明了当Poisson断面本身可积时，Poisson断面的邻域是可积的，并且在这种情况下，我们证明了辛广群围绕其对Poisson断面的限制的一个规范形定理，该定理将其所有的结构映射置于规范形中.最后，我们用李代数的例子来说明我们的结果。
We prove that the information geometry's Frobenius manifold is a symplectic manifold having Poisson structures. By proving this statement, a bridge is created between the theories developed by Vinberg, Souriau and Koszul and the Frobenius manifold approach. Until now these approaches seemed disconnected.
We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.
We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $π$ is defined on the tubular neighbourhood of the singular locus $Z_ω$ of the 2-form $ω$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_ω$. We compute its smooth Poisson cohomology, which depends on the modular vector field and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure $π$ and the overtwisted contact structure associated to a near-symplectic 4-manifold…
E-structures and almost regular Poisson manifolds
本文研究E-结构与殆正则泊松结构的关联，分析二者泊松性质，给出E-结构对应泊松群胚上泊松结构的显式公式，揭示交换标架与Darboux-Carathéodory型表达式的联系。
On the Integrability of Orthogonal Distributions in Poisson Manifolds
研究了定义在正则Poisson流形上的分布作为辛叶理叶的切空间的正交补（关于某些（伪）黎曼度量）的可积性条件.这个分布的可积性和不可积性的例子。
Symplectic and Poisson geometry on b-manifolds
设$M^{2n}$是具有Poisson双向量场$Π$的Poisson流形.如果映射$Π^n:M\toΛ^{2n}(TM)$与零截面在余维1子流形$Z\subset M$上横交，则称$M$是b-Poisson.本文将系统地研究这类泊松流形。特别是，我们将详细研究$(M,Π)$的结构在附近的$Z$和使用辛技术定义拓扑不变量，确定结构同构。我们还研究了这些流形的德拉姆理论的一个变体及其与泊松上同调的联系。
On homotopy Poisson actions and reduction of symplectic Q-manifolds
本文给出了辛Q流形通过分次群作用约化的一般框架。在这个框架中，作用群上的同调结构是一个乘法多重向量场。
We study various properties of polarized vectorial Poisson structures subordinate to polarized k-symplectic manifolds, and also, we study the notion of polarized vectorial Poisson manifold. Some properties and examples are given.
Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras
作者在文献[1]中引入了具有相容伪度量的Poisson流形的概念。本文引入了一类新的李代数，我们称之为伪黎曼李代数。这两个概念是密切相关的：我们证明了李代数对偶上的线性Poisson结构具有相容的伪度量当且仅当该李代数是伪黎曼李代数，并且通过在一点线性化具有相容的伪度量的Poisson流形得到的李代数是伪黎曼李代数。进一步，我们给出了这类流形的辛叶的一些性质，并证明了每个具有相容度量的Poisson流形（每个Riemann-李代数）都是幺模的.最后，我们对二维和三维的伪黎曼李代数进行了分类。
Poisson manifolds of strong compact type over 2-tori
本文通过修改基于 K3 曲面理论的现有构造方法，为每个强整仿射 2-环面构建了以其为叶空间的强紧型 Poisson 流形，从而获得了一类新的非平凡示例。
Riemann Poisson manifolds and Kähler-Riemann foliations
作者在[1]中引入了Riemann Poisson流形，并在[2]中对其进行了更详细的研究。Kähler-Riemann叶理构成了Riemann叶理的一个有趣的子集，具有显著的性质（见[3]）。本文证明了在流形$M$上给出正则Riemann Poisson结构等价于在$M$上给出Kähler-Riemann叶状构造，使得叶辛形式对于所有局部叶状垂直向量场是不变的.我们还证明了叶上同调向量空间与基本形式向量空间之和是Poisson上同调空间的一个子空间。
We prove that if a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact Kähler manifolds with split tangent bundle.
Stability of higher order singular points of Poisson manifolds and Lie algebroids
研究了光滑Poisson结构和一般李代数胚的奇点的稳定性。给出了Poisson结构或李代数体在奇点上的第一（不一定是线性）逼近稳定的充分条件。这里使用的主要工具是经典的Lichnerowicz-Poisson上同调和最近由Crainic和Moerdijk引入的李代数胚的变形上同调。我们还提供了几个例子稳定奇点顺序$k \geq 1$泊松结构和李代数胚。最后，我们将我们的结果应用到狄拉克流形的前辛叶。
We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singular…
We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a $5$-dimensional orientable fibered manifolds with $2$-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.
Nonlinearizability of certain Poisson structures near a symplectic leaf
我们给出了一个内在的证明，Vorobjev的第一近似的泊松流形附近的辛叶是泊松流形。我们还表明，康恩的线性化结果不能扩展在Vorobjev的设置。
Morita equivalence of Poisson manifold via stack groupoids
这是一个未来工作的结果的浓缩阐述，基于第二作者在Oberwolfach研讨会上的讲话“泊松几何”，2007年4月30日至5月4日。
Poisson structures on complex flag manifolds associated with real forms
对于复半单群G及其实形式G 0，我们在G的旗簇上定义一个Poisson结构，使得所有的Bruhat胞腔（对于G的博雷尔子群的适当选择）以及所有的G 0-轨道都是Poisson子流形.我们表明，每个交叉点的G 0-轨道与Bruhat细胞是一个经常泊松子流形，我们计算其辛叶的尺寸。
Connections in Poisson Geometry I: Holonomy and Invariants
讨论了Poisson流形上的反变联络。对于向量丛，Izu Vaisman引入了相应的逆变导数的运算概念。我们表明，这些连接起着重要的作用，在研究的整体性质的泊松流形，我们使用它们来定义泊松完整和新的不变量的泊松流形。
Holonomy on Poisson Manifolds and the Modular Class
我们在Poisson流形上引入线性完整性。泊松结构的线性完整性推广了正则辛叶理上的线性完整性。然而，对于奇异Poisson结构，线性完整性被定义为余切丛的切向路径（余切路径）的提升。线性完整性与A.温斯坦也就是说，线性完整性的行列式的对数等于模向量场沿着这样的提升的积分。这个断言依赖于积分的概念，向量场沿余切路径上的泊松流形，这也是介绍的文件。本文的第二部分证明了局部么模Poisson流形的模类是Morita等价的不变量。
Dirac structures, moment maps and quasi-Poisson manifolds
我们将辛群胚的Poisson映射和作用之间的对应关系推广到Dirac几何领域，从而推广了动量映射和Hamilton作用之间的对应关系。作为一个例子，我们展示了如何汉密尔顿拟泊松流形适合这个框架，通过构建一个“反演”过程有关拟泊松双向量扭曲狄拉克结构。
Almost Homogeneous Poisson Spaces
证明了全纯Poisson流形的开辛叶是伪Kähler子流形，并定义了一个障碍来研究切向Poisson作用的动量映射的等变性.研究了几乎齐次Poisson流形的一些性质，证明了任何紧辛Poisson齐性空间都是Dressing轨道上的环面丛。
An extension of the Marsden-Ratiu reduction for Poisson manifolds
我们提出了一个推广的减少泊松流形的分布Marsden和Ratiu介绍。我们的建议克服了一些限制的原始程序，并使减少泊松结构有效地依赖于分布。不同的应用进行了讨论，以及代数解释的程序和它的制定狄拉克结构。
We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold…
Group orbits and regular partitions of Poisson manifolds
我们研究了一个大类的泊松流形，来自Manin三元组，为此我们构造明确的分区成定期泊松子流形相交某些组轨道。例子包括还原二次李代数$\d$的拉格朗日子代数的所有变种${\mathcal L}$，其泊松结构由$\d$的拉格朗日分裂定义。在$\g \oplus \g$的特殊情况下，其中$\g$是一个复杂的半单李代数，我们显式计算的${\mathcal L}$上的泊松结构的秩定义的任意拉格朗日分裂的${\mathfrak g} \oplus {\mathfrak g}$。这种拉格朗日分裂已被P. Delorme分类，它们包含Belavin-Drinfeld分裂作为特殊情况。
We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant partially complex structure, or $f$-structure, introducing the notion of (almost) Kähler--Poisson manifolds. In addition, we study some of their properties under structure preserving maps and symmetries.
We prove a normal form theorem for principal Hamiltonian actions on Poisson manifolds around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map.
Integrability of Poisson brackets
我们证明了各种关于泊松括号的可积性概念都是等价的，并给出了积分泊松流形的精确障碍。我们描述的集成作为一个辛商，在泊松西格玛模型的Cattaneo和费尔德的精神。对于正则泊松流形，我们用辛面积的变化来表示障碍物。作为这些结果的应用，我们证明了泊松流形允许一个完全的辛实现当且仅当它是可积的。我们还讨论了子流形的积分和Poisson流形的Morita等价。
This paper is about the role of Planck's constant, $\hbar$, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single "Heisenberg-Poisson" manifold. The new coordinate on this manifold is identified with $\hbar$. I present an explicit construction for a symplectic groupoid integrating a Heisenberg-Poisson manifold and discuss its geometric quantization. I show that in cases where $\hbar$ canno…
WZW-Poisson manifolds
我们观察到，WZW型的一个术语可以被添加到泊松-西格玛模型的拉格朗日中，使得第一类约束的代数保持封闭。这导致了泊松几何概念的自然推广。由此得到的“WZW-Poisson”流形M的特征在于一个双向量Pi和一个闭的三形式H使得[Pi，Pi]_Schouten = < H，Pi^3 >。
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