Yakov Eliashberg, Noboru Ogawa, Toru Yoshiyasu
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.
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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.
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。
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。
We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive.
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.
This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.
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。
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.
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.
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。
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.
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformation…
Generalized Marsden-Weinstein symplectic structures
本文在奇异1-形式密度上定义辛结构，在增广三元组上定义预辛结构，作为可压缩流体中Marsden-Weinstein辛结构的模拟，并阐明其介于Weinstein与Marsden-Weinstein两种设置之间的中间地位。
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 …
Let $(X,ω_X)$ be a derived scheme with a 0-symplectic form and suppose there is a Hamiltonian $G$-action with a moment map for $G$ a reductive group. We prove, under no further assumptions, that symplectic reduction along any coadjoint orbit in the category of derived Artin stacks has a 0-symplectic form.
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.
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.
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}.
We show how to encode a Weinstein 4-manifold using a multisection diagram with divides, which is a sequence of cut systems on a surface, together with a separating collection of curves. We give two algorithms to construct a multisection diagram with divides of a given Weinstein manifold, one starting with a Kirby-Weinstein handle decomposition and one starting with a positive, allowable Lefschetz fibration (PALF). Through the connections with PALFs, we define a monodromy of a multisection and show how to symplectically carry out monodromy substitution on multisections with divides.
We show that the Siegel upper half space $Σ_{d}$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $T^{*}\mathbb{R}^{2d^{2}}$ with $\mathsf{O}(2d)$-symmetry. The reduced symplectic form on $Σ_{d}$ corresponding to the standard symplectic form on $T^{*}\mathbb{R}^{2d^{2}}$ turns out to be a constant multiple of the symplectic form on $Σ_{d}$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show …
Mikami-Weinstein Type Theorem for Cosymplectic Groupoid Actions
本文将 Mikami-Weinstein 辛约化定理推广到余辛情形，通过引入余辛群胚作用概念并证明相应的约化定理。
Tight contact structures without symplectic fillings are everywhere
该研究证明高维流形中同一殆切触类内存在紧但不可辛填充的切触结构，并构造了特定条件下Liouville可填充但Weinstein不可填充的切触结构，推进了切触拓扑与辛几何的关联研究。
In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map but in the Cartan model of equivariant cohomology. As a byproduct we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.
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.
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。
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。
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 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。
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.
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…
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。
We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.
We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed $n$-manifolds of dimension $n>2$ can be realized as exact Lagrangian submanifolds of $T^*S^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^* S^n$, infinitely many of which are distinct, are formed by a single handle attachment to the standard $2n…
Localization and flexibilization in symplectic geometry
本文提出了一种基于辛几何的P-柔化方法，通过局部化Weinstein无穷范畴，实现了对缠绕Fukaya范畴中P的可逆性控制，并展示了其在构造E-infinity交换代数中的应用。
The Dual DG Category to Weinstein's symplectic "category" and applications to Geometric Quantization
本文构造了魏instein辛“范畴”的对偶双DG范畴，将其应用于几何量子化，还证明了超代数${\mathfrak {osp}}(1|2)$的作用及相关消失定理。
We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler's program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse-Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler's arboreal singularities which are combinato…
A stabilized polydisc is a product of a symplectic polydisc and several copies of the complex plane. This paper gives a complete characterization of symplectic embeddings of stabilized polydiscs into other stabilized polydiscs.
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。
In this paper we prove the unobstructedness of the deformation of integral coisotropic submanifolds in symplectic manifolds, which can be viewed as a natural generalization of results of Weinstein for Lagrangian submanifolds.
We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained, including the uniqueness of diffeomorphism types of fillings for many co…
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。
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。
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n.
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。
On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high codimension, exact embedding into a cosphere bundle. Thicken to a hypersurface, and consider the Kashiwara-Schapira stack along the thickened skeleton. Pull back along the inclusion of the original skeleton. Gromov's h-principle for contact embeddings guarantees existence and uni…
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and ve…
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…
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…
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…
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…
We lift a Hamiltonian loop on a symplectic manifold to a Hamiltonian loop on the symplectic one-point blow up of a symplectic manifold. Then we use Weinstein's morphism to show that the lifted Hamiltonian loop has infinite order on the fundamental group of the group of Hamiltonian diffeomorphisms of the blown up manifold.
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.
For any asymptotically dynamically convex contact manifold $Y$, we show that $SH_*(W)=0$ is a property independent of the choice of topologically simple (i.e.\ $c_1(W)=0$ and $π_{1}(Y)\rightarrow π_1(W)$ is injective) Liouville filling $W$. In particular, if $Y$ is the boundary of a flexible Weinstein domain, then any topologically simple Liouville filling $W$ has vanishing symplectic homology. As a consequence, we answer a question of Lazarev partially: a contact manifold $Y$ admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of $…
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.
We prove that the fundamental group of the group of Hamiltonian diffeomorphisms of the symplectic manifold that is obtain by blowing up a submanifold contains an element of infinite order. We prove this using Weinstein's morphism and by constructing explicitly such loop of Hamiltonian diffeomorphisms.
How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds -- the assertion that Reeb orbits exist and are detected by symplectic homology. More precisely, I show that for polarizably Weinstein fillable contact manifolds, said property is equivalent to the failure of Hochschild homology to commute with certain colimits of representation categories of tree quivers. So as to be intelligible to algebraists, I try to include or black-box as much of the geometric background as possible.
Folded symplectic forms in contact topology
本文建立了折叠辛形式与接触拓扑中凸超曲面理论的关系，并利用该理论强化了Cannas da Silva的存在性结果，同时将Baykur的4维折叠Weinstein结构与Lefschetz纤维化存在性结果推广至所有偶数维。
Stabilized symplectic embeddings of higher-dimensional ellipsoids
本文通过研究高维辛椭球体的拉格朗日容量，提出了一种基于弦拓扑学的方法，以确定其嵌入容量的下界。
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors
该研究通过几何方法证明2n维Weinstein流形/扇形的缠绕Fukaya范畴由Liouville场n阶临界点不稳定流形生成，证得Seidel关于开闭映射为同构的猜想，并扩展了证明设定。
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.
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