The geometry of symplectic pairs
G. Bande, D. Kotschick
本文研究了由两个常秩闭2-形式组成的辛对偶流形的几何性质，其核叶理是互补的。使用Boothby和王建设的变化，我们建立接触辛和接触对辛对。
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
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We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $β$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.
Simple Non-Rational Convex Polytopes via Symplectic Geometry
在这篇文章中，我们考虑流形和orbifolds的一个推广，我们称之为quasifolds;维数k的quasifolds通过离散群的作用与R^k的商局部同构--特别是它们不是Hausdorff拓扑空间。在这种几何中，环面的类似物是准环面。本文定义了拟托在辛拟褶皱上的Hamilton作用，并证明了任何简单的凸多胞体，不论有理与否，都是辛拟褶皱上一类有效Hamilton作用的矩映射的象，这些作用的维数是相应拟托的两倍.
Lagrangian submanifolds in affine symplectic geometry
我们揭示了仿射辛映射下的拉格朗日子流形的最低阶微分不变式，并发现当它们是常数时会发生什么。
Quantitative symplectic geometry
虽然辛流形没有局部不变量，但它们有许多整体数值不变量。其中最突出的是所谓的辛能力。不同的容量以不同的方式定义，因此容量之间的关系常常导致辛几何和哈密尔顿动力学的不同方面之间令人惊讶的关系。在本文中，我们提出了一个尝试，以更好地理解空间的所有辛容量，并讨论了一些进一步的一般性质的辛容量。我们还描述了一些新的关系，某些辛容量的椭球和多圆盘。在整个讨论过程中，我们提到了许多悬而未决的问题。
Complex of twistor operators in symplectic spin geometry
对于具有亚辛结构（黎曼自旋结构的辛模拟）的辛流形，利用辛无挠仿射联络构造了一个由微分算子组成的序列.除了一个外，所有这些算子都是一阶的。一阶的是从黎曼自旋几何已知的扭量算子的辛类似物。证明了在辛联络的辛Weyl曲率张量场为零的条件下，上述序列构成复形。这就产生了一个新的复形，所谓的里奇型辛流形，其中承认一个亚辛结构。
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
This paper studies the geometry of Cartan-Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan-Hartogs and their dual. Further, we compute their symplectic capacity and show that a Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.
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.
Symplectic geometry of projective structures on surfaces with boundary
研究带有边界的曲面的项目结构变形空间的辛几何结构及其与辛群oid的关系。
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…
We explain the geometric origin of the $L_{\infty}$-algebra controlling deformations of pre-symplectic structures.
Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and the Viterbo conjecture. This will give us a perfect excuse to introduce the reader to some important ideas in Riemannian and symplectic geometry.
In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-W…
Symplectic Geometry
这是一个概述文章选定的主题在辛几何书面的手册微分几何（第2卷，编辑F.J.E.迪伦和L.C.A. Verstraelen）。
In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.
$C^0$-rigidity of characteristics in symplectic geometry
本文给出了辛几何中特征叶理的$C^0$ -刚性结果.保光滑超曲面的辛同胚（在Eliashberg-Gromov意义下）也保持其特征叶理。
Symplectic maps and hyperKähler moment map geometry
本文建立了4维实环面辛微分同胚群与超Kähler动量映射零点轨迹的对应关系，提出修正动量映射流，推广至多面体设定并证明Duistermaat型定理，为多面体辛几何与Morse-Bott理论奠定基础。
Remarks on Symplectic Geometry
本文综述近五十年辛几何研究进展，涵盖其历史、核心理论问题、关键成果及相关分支，为该领域发展提供系统梳理。
An extension theorem in symplectic geometry
本文将有界星形域的辛嵌入的“限制后扩张原理”推广到无界星形域的一大类辛嵌入。
Estimated transversality in symplectic geometry and projective maps
本文的主要定理是关于殆复流形上近似全纯范畴中喷流空间的分层的横截性估计的一个结果。本文还讨论了殆复流形上复向量丛的渐近扩张性的概念，以及它在构造从紧辛流形到射影空间的具有通有奇点的映射中的应用.
Open problems in billiards and quantitative symplectic geometry
本文汇总了刚体与定量辛几何领域的开放问题，基于研讨会讨论整理而成。
Symplectic Geometry for Engineers, Fundamentals
本文面向工程师介绍辛几何基础，重点讨论平面辛几何在物理与工程教育中解决标准几何与力学问题的适用性，以填补二维情形相关文献的不足。
I describe some of McDuff's contributions to symplectic geometry, with a focus on symplectic embedding problems.
We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the $b_1=1$ case (in the coKähler/cosymplectic situation).
The symplectic geometry of polygons in hyperbolic 3-space
本文研究了三维双曲空间中定边长闭n维的模空间的辛几何。我们证明了这些模空间具有来自Poisson李理论的辛结构。我们通过沿对角线弯曲n维空间，在这些模空间上构造完全可积系统。本文的结果是前两位作者在JDG 44中关于欧氏空间中n边形连杆的结果的类似。通过变形证明了具有相同边长集的双曲三维空间和欧氏三维空间中的n边形连杆的模空间是辛同构的。
In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic but not symplectomorphic. These spaces inherit from the tiling its very interesting symmetries.
We study coisotropic submanifolds of $b$-symplectic manifolds. We prove that $b$-coisotropic submanifolds (those transverse to the degeneracy locus) determine the $b$-symplectic structure in a neighborhood, and provide a normal form theorem. This extends Gotay's theorem in symplectic geometry. Further, we introduce strong $b$-coisotropic submanifolds and show that their coisotropic quotient, which locally is always smooth, inherits a reduced $b$-symplectic structure.
This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them to study the symplectic geometry of cotangent bundles. We explain how we can recover the Gromov nonsqueezing theorem, the Gromov-Eliashberg rigidity theorem, the existence of graph selectors, we prove a three cusps conjecture about curves on the sphere and we recover more recent results on the topology of exact Lagrangian submanifolds of cotan…
We extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.
Let (M,ω) be a symplectic manifold, and Sigma a compact Riemann surface. We define a 2-form on the space of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M,ω) that ensure that the restriction of the form to the moduli space of simple immersed J-holomorphic Sigma-curves in a homology class A in H_2(M,\Z) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Sigma-curves in …
The geometry of recursion operators
本文研究了辛结构的自同态交织对的域。利用这些自同态，我们证明了一个类似的Moser定理的同时同位素的两个家庭的辛形式。我们还考虑了由辛形式的对和三重定义的几何结构，其中交织自同态的平方是正或负的单位。对于形式对，我们恢复辛对和全纯辛结构的概念。对于三元组，我们恢复了超辛结构的概念，我们还发现了三个新的结构，以前没有考虑过。其中之一是辛公式的超凯勒几何，这原来是一个严格的推广通常定义的微分或凯勒几何。
We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace …
The Lefschetz property, formality and blowing up in symplectic geometry
本文研究了Lefschetz性质在爆破构造下的性质。我们证明了，如果我们沿着一个合适的子流形满足Lefschetz性质，它是可能的，以减少的Lefschetz映射的核的维数。我们使用它，连同有关Massey产品的结果，构造满足Lefschetz性质的非形式（单连通）辛流形。
Stability theorems for symplectic and contact pairs
我们证明了Gray-Moser稳定性定理的互补对的形式的常数类定义辛对，接触辛对和接触对。我们还考虑了接触-辛结构和接触-切触结构的情形，其中一种形式的常类条件被其核超平面分布在E意义下具有常类的条件所取代。嘉当
Relative Flux Homomorphism in Symplectic Geometry
在这项工作中，我们定义了一个相对版本的通量同态，介绍了卡拉比在1969年，辛流形。我们用它来研究辛流形的辛同胚群（的泛覆盖）留下一个拉格朗日子流形不变。我们证明了这个群的某些子群在辛约化下是稳定的。
In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a holomorphic immersion, generalizing the known results for the moduli space of marked complex projective structures. Finally, we prove that the symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space …
BRST model applied to symplectic geometry
包括两个本地宏（gothic.sty和fleqn.sty）
Noncommutative Symplectic Geometry of the Endomorphism Algebra of a Vector Bundle
本文研究了结合Poisson代数的退化Poisson结构、Poisson子流形和商流形、辛叶理和辛叶等经典辛几何概念的非交换推广。我们考虑了向量丛的自同态代数的这种结构，并给出了这种代数的Poisson结构族的完整描述。
Positive divisors in symplectic geometry
本文给出了绝对和相对Gromov-Witten不变量之间的一些明确关系。证明了辛流形是辛有理连通的，如果它包含一个辛同构于$P^n$的正因子。
Fundamentals for Symplectic $\mathcal{A}$-modules
抽象微分几何以理论为基础，综合和推广了所有经典微分几何。在这里，我们承诺部分探讨抽象微分几何的经典辛几何的影响。完整的调查将在其他地方进行。
We prove a quantitative $h$-principle statement for subcritical isotropic embeddings. As an application, we construct a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least $6$.
Let Z be the real part of a K{ä}hler Hamiltonian manifold M. The O'Shea-Sjamaar's Theorem tells us that the moment polytope Delta(Z) corresponds to the anti-invariant part of the Kirwan polytope Delta(M). The purpose of the present paper is to explain how to parameterize the equations of the facets of Delta(Z) in terms of real Ressayre's pairs of Z.
This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension $4$ symplectic submanifolds ($C^0$-flexibility), while this is impossible for codimension $2$ symplectic submanifolds ($C^0$-rigidity). We also discuss $C^0$-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-…
In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric analysis, classical convex geometry, and the theory of normed spaces.
Nonlinear Hodge flows in symplectic geometry
本文提出非线性Hodge热流方法，用于研究辛类中辛形式的各向同性问题，并证明了流在接近几乎Kähler结构时的稳定性及收敛性。
This note collects a number of standard statements in Riemannian geometry and in Sobolev-space theory that play a prominent role in analytic approaches to symplectic topology. These include relations between connections and complex structures, estimates on exponential-like maps, and dependence of constants in Sobolev and elliptic estimates.
Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
本文从接触与辛几何角度刻画三维 Anosov 流，建立研究其动力学的几何拓扑框架，并给出双接触结构唯一性及基于 Reeb 流的 Anosov 性判据。
Geometry of symplectic intersections
本文综述了辛拓扑中的几种交与不交现象。我们讨论了它们的影响，最后概述了一些新的关系，代数几何的主题。
On certain canonical diffeomorphisms in symplectic and Poisson geometry
二重（=迭代）切丛的标准对合可以用不同的方法对偶化，以产生Tulczyjew同形、与余切空间上的标准辛结构相关的泊松锚和反转同形之间的关系。我们表明，这些地图的建设，产生非常普遍的双李代数胚的双李群胚，在那里他们发挥了至关重要的作用，双李代数胚和李双代数胚之间的关系。
Recently in symplectic geometry there arose an interest in bounding various functionals on spaces of matrices. It appears that Grothendieck's theorems about factorization are a useful tool for proving such bounds. In this note we present two such applications.
Symplectic Geometry, Poisson Geometry, and Beyond
本文以经典物理为动机，系统介绍辛几何与泊松几何的基本对象和技巧，并探讨其在李理论及数学物理其他领域中的应用。
Dirac geometry, deformation theory and shifted symplectic geometry
该研究发展狄拉克变形理论，在扭曲狄拉克几何与泊松几何间插值，证明其与狄拉克几何结构运算兼容，以紧致李群相关结构变形为例，还得到统一变形理论及约化空间的光滑变形。
Some incarnations of Hamiltonian reduction in symplectic geometry and geometric representation theory
本文介绍了辛几何与几何表示论中哈密顿约简的现代应用，包括交换化、通过辛群状的约简及摩尔-塔奇克瓦猜想。
Symplectic, Hofer and sub-Riemannian geometry
本文的目的是把Carnot-Caratheodory群上的次黎曼几何与辛几何联系起来。我们在这里将集中讨论海森堡群，尽管很明显，几乎所有的事情都可以在一般的卡诺-卡拉西奥多利群上进行。这种概括将是即将发表的一份说明的主题。
In this paper, we introduce a new kind of Siegel upper half space and consider the symplectic geometry on it explicitly under the action of the group of all holomorphic transformations of it. The results and methods will form a basis for our number theoretic applications later.
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.
A translation of "The Geometry of the Group of Symplectic Diffeomorphisms"
本书介绍霍费尔度量，研究哈密顿微分同胚群的直径、测地线及单参数子群增长，并应用于动力学和遍历理论。
In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.
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