Marsden-Weinstein reduction for symplectic connections

李代数胚的辛约化与 Darboux-Moser-Weinstein 定理

Symplectic reduction and a Darboux-Moser-Weinstein theorem for Lie algebroids

本文将 Marsden-Weinstein 约化定理与 Darboux-Moser-Weinstein 定理推广至辛李代数胚，并提出了相应的余迷向嵌入定理。

Darboux-Weinstein theorem for locally conformally symplectic manifolds

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω= θ\wedge ω$. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In this case it will be show…

The Weinstein Conjecture in Product of Symplectic Manifolds

In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in $\mathbb{C}\mathbb{P}^2\times T^*N$ holds.

Connections under Symplectic Reduction

In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.

空间曲线形变空间上的辛结构

Symplectic structures on the space of space curves

本文提出一种扩展经典Marsden-Weinstein结构的辛结构，结合Liouville 1-形式与数学形变分析中的Riemannian结构，推导了若干经典哈密顿函数的哈密顿向量场。

Generalized Weinstein tubular neighbourhood & Space of Lagrangian submersions

We present a generalized Weinstein Tubular Neighbourhood theorem for Lagrangian subbundles of Symplectic fibrations. This is then used to study the space of Lagrangian fibrations of symplectic manifolds.

The Weinstein Conjecture for Hamiltonian Fibrations

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type, under certain technical conditions. The proof is based on the product formula for Gromov-Witten invariants ($GW$-invariant) of Hamiltonian fibrations derived in \cite{H}.

Finiteness of the Hofer-Zehnder capacity of neighborhoods of symplectic submanifolds

We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite ($π_1$-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds near closed symplectic submanifolds in any symplectic manifold.

The Weinstein conjecture in the uniruled manifolds

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

Weinstein邻域定理对于分层子空间

Weinstein neighbourhood theorems for stratified subspaces

本文通过类比Weinstein定理，证明了分层子空间的辛邻域唯一性，并推广了现有构造，同时证明了Moser技巧的强版本及管状邻域定理。

温斯坦辛流形中的层量化

Sheaf quantization in Weinstein symplectic manifolds

研究在温斯坦辛流形中利用层的微局部理论关联范畴，通过构造微局部特殊化函子，证明精确拉格朗日量与范畴的关系及范畴在温斯坦同伦下的不变性。

Stabilized convex symplectic manifolds are Weinstein

We show that a stabilized convex symplectic (also called Liouville) manifold with the homotopy type of a half dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.

Subflexible symplectic manifolds

We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.

对称R-空间的极大Weinstein邻域及其辛容量

Maximal Weinstein neighborhoods of symmetric R-spaces and their symplectic capacities

本文证明对称R-空间的极大Weinstein管状邻域具有稠密性，并据此计算其圆盘（余）切丛的Gromov宽度与Hofer--Zehnder容量。

叠哈密顿作用与辛约化

Stacky Hamiltonian actions and symplectic reduction

本文引入了étale李群叠在étale辛叠上的哈密顿作用概念，并在该背景下证明了Kirwan凸性定理、Meyer-Marsden-Weinstein辛约化定理及Duistermaat-Heckman定理的版本。

Maximal contact and symplectic structures

We introduce a procedure for gluing Weinstein domains along Weinstein subdomains. By gluing along flexible subdomains, we show that any finite collection of high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a `maximal' Weinstein domain also with the same topology. As an application, we produce exotic cotangent bundles containing many closed regular Lagrangians that are formally Lagrangian isotopic but not Hamiltonian isotopic and also give a new construction of exotic Weinstein structures on Euclidean space. We describe a similar construction in the contact …

Reduction of a symplectic-like Lie algebroid with momentum map and its application to fiberwise linear Poisson structures

This article addresses the problem of developing an extension of the Marsden- Weinstein reduction process to symplectic Lie algebroids, and in particular to the case of the symplectic cover of a fiberwise linear Poisson structure, whose reduction process is the analogue to cotangent bundle reduction in the context of Lie algebroids.

Symplectic 4-manifolds admit Weinstein trisections

We prove that every symplectic 4-manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in $\mathbb{CP}^2$ can be symplectically isotoped into bridge position.

关于辛流形中的Weinstein领域

On Weinstein domains in symplectic manifolds

研究证明了Weinstein领域在辛流形中的存在性及边界障碍，揭示了Brieskorn球体在不同辛结构下的限制。

Weinstein拉格朗日管状邻域半径的一个下界

A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood

本文针对凯勒流形中浸入拉格朗日子流形的Weinstein管状邻域，给出了其半径上确界的一个显式下界，并补充了紧嵌入情形下的类似结果。

Symplectic cobordisms and the strong Weinstein conjecture

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.

共形辛Weinstein猜想

A conformal symplectic Weinstein conjecture

本文提出并部分验证了局部共形辛流形上的Weinstein猜想推广，引入Reeb 2-曲线，并利用Gromov-Witten理论构造Fuller指标，获得经典Reeb动力学新结果。

Reduction of multisymplectic manifolds

We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.

The Normal Form Theorem around Poisson Transversals

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.

A Note On Weinstein's Conjecture

We prove that there exists at least one close orbit in a given contact hypersurface in some symplectic manifolds.

多辛流形上可观测量 $L_\infty$-代数的约化

Reduction of $L_\infty$-Algebras of Observables on Multisymplectic Manifolds

本文提出预多辛流形上可观测量 $L_\infty$-代数的约化方案，可在辛情形重现 Marsden-Weinstein-Meyer 约化的 Poisson 代数，并通过多余切丛等例子展示其与既有约化理论的区别及场论应用。

使用填充构造的辛 Milnor 纤维中的精确拉格朗日环面

Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings

本文利用精确拉格朗日填充和 Weinstein 把体图，在正模数孤立超曲面奇点的 4 维 Milnor 纤维中构造了无限多个不同的精确拉格朗日环面，并推广了判断 Weinstein 4-流形辛同调非零的准则。

魏instein辛“范畴”的对偶双DG范畴及其在几何量子化中的应用

The Dual DG Category to Weinstein's symplectic "category" and applications to Geometric Quantization

本文构造了魏instein辛“范畴”的对偶双DG范畴，将其应用于几何量子化，还证明了超代数${\mathfrak {osp}}(1|2)$的作用及相关消失定理。

The symplectic reduced spaces of a Poisson action

During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a moment…

Open books and the Weinstein conjecture

We show the existence of a contractible periodic Reeb orbit for any contact structure supported by an open book whose binding can be realised as a hypersurface of restricted contact type in a subcritical Stein manifold. A key ingredient in the proof is a higher-dimensional version of Eliashberg's theorem about symplectic cobordisms from a contact manifold to a symplectic fibration.

Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem

In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The pr…

Symplectic embeddings of 4-manifolds via Lefschetz fibrations

In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, ω_{pr}),$ where $ω_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.

辛几何中的定位与柔化

Localization and flexibilization in symplectic geometry

本文提出了一种基于辛几何的P-柔化方法，通过局部化Weinstein无穷范畴，实现了对缠绕Fukaya范畴中P的可逆性控制，并展示了其在构造E-infinity交换代数中的应用。

Weinstein morphism的推广

Generalization of Weinstein's Morphism

本文提出Weinstein morphism的推广，并证明了在特定条件下辛同调群的非平凡性。

具有ADE型Milnor纤维的辛流形的奇异族

Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type

本文构造了无限多个微分同胚的不同Weinstein流形族，包含ADE型Milnor纤维及粘合余切丛的A型Milnor纤维构造的流形，且后者具有可分解的缠绕Fukaya范畴。

微分几何中的导出辛约减

Derived Symplectic Reduction in Differential Geometry

本文证明了Marsden-Weinstein-Meyer辛约减定理的导出版本，通过dg群oid建模辛商，构建群oid内的约减辛形式，并展示其满足导出非退化条件。

Triangular Poisson structures on Lie groups and symplectic reduction

We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).

Weinstein 流形上范畴熵与拓扑熵的比较

A comparison of categorical and topological entropies on Weinstein manifolds

针对紧支集辛自同构，本文证明 Weinstein 流形上 Fukaya category 自等价的范畴熵为其拓扑熵提供下界，并提出推广动力系统结果的猜想。

通过余辛矩映射对余辛群胚的约化

Reduction of Cosymplectic groupoids by cosymplectic moment maps

本文将余辛流形的 Marsden-Weinstein-Meyer 约化推广到群胚情形，证明了余辛群胚代数胚的积分与约化可交换，并研究了约化对典范辛子群胚及乘法陈类的影响。

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in T…

精确辛流形的约简与能量超曲面

Reduction of exact symplectic manifolds and energy hypersurfaces

本文提出两种精确辛流形上哈密顿系统的约简方法，并通过修改的马尔登-梅尔-韦instein定理证明其等价性。

Jeffrey-Kirwan-Witten Localization Formula for Reductions at Regular Co-adjoint Orbits

For Marsden-Weinstein reductions at the point 0 in the vector space dual to the Lie algebra, the well-known Jeffrey-Kirwan-Witten localization formula was proven and lastly modified by M. Vergne. We prove in this paper the same kind formula for the reductions at regular coadjoint orbits by using the universal orbital formula of characters.

Geometric Reductions, Dynamics and Controls for Hamiltonian System with Symmetry

This is a survey article, from the viewpoint of the completeness of the Marsden- Weinstein reduction, to introduce briefly some recent developments of the symmetric reductions and Hamilton-Jacobi theory of the regular controlled Hamiltonian systems, the nonholonomic controlled Hamiltonian systems and the controlled magnetic Hamiltonian systems. These research reveal the deeply internal relationships of the geometrical structures of phase spaces, the nonholonomic constraint, the dynamical vector fields and the controls of these systems.

A supergeometric approach to Poisson reduction

This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to describe the corresponding reductions.

Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_ϕ$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $ϕ$. $T_ϕ$ is another Weinstein domain and its contact boundary is independent of $ϕ$. In this paper, we distinguish $ϕ$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic co…

Fukaya Categories as Categorical Morse Homology

The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplecti…

The Stable Symplectic Category and Quantization

We study a stabilization of the symplectic category introduced by A. Weinstein as a domain for the geometric quantization functor. The symplectic category is a topological category with objects given by symplectic manifolds, and morphisms being suitable lagrangian correspondences. The main drawback of Weinstein's symplectic category is that composition of morphisms cannot always be defined. Our stabilization procedure rectifies this problem while remaining faithful to the original notion of composition. The stable symplectic category is enriched over the category of spectra (in particular, its…

Equivariant slices for symplectic cones

The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqu…

Symplectic homology product via Legendrian surgery

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken in [BEE1] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here we express the product and $\BV$-operator of symplectic homology in that context.

Normal modes in symplectic stratified spaces

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. More specifically we significantly improve a result proved earlier jointly with Tokieda (math.SG/9901102): we remove a technical hypothesis which amounted to an assumption that the symmetry group is a torus.

导出高阶群胚的平移辛几何

The shifted symplectic geometry of derived higher groupoids

该研究引入导出李n-群胚及其平移辛结构，定义平移拉格朗日结构并证明其合成条件，框架包含多种临界值约化过程，如经典哈密顿约化等，为相关研究提供新视角与方法。

The Symmetric Representation of the Generalized Rigid Body Equations and Symplectic Reduction

We show that a symplectic reduction of the symmetric representation of the generalized $n$-dimensional rigid body equations yields the $n$-dimensional Euler equation. This result provides an alternative to the more elaborate relationship between these equations established by Bloch, Crouch, Marsden, and Ratiu. Specifically, we exploit the inherent $\mathsf{Sp}(2n,\mathbb{R})$-symmetry in the symmetric representation to present its relationship with the Euler equation via symplectic reduction facilitated by the dual pair recently developed by Skerritt and Vizman.

Isotropic submanifolds and coadjoint orbits of the Hamiltonian group

We describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,ω)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegener…