Symplectic Geometry

Fundamentals for Symplectic $\mathcal{A}$-modules

Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere.

Quantitative $h$-principle in symplectic geometry

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 $6$.

Some quantitative results in $C^0$ symplectic geometry

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-…

Basic Riemannian Geometry and Sobolev Estimates used in Symplectic Topology

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.

The symplectic geometry of a new kind of Siegel upper half space of order 2 (I)

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.

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。

Quantitative $h$-principle for isotropic embeddings and applications to $C^0$-symplectic geometry

We prove here a quantitative $h$-principle statement that applies to isotropic embeddings of discs. We then apply it to get $C^0$-flexibility and rigidity results in symplectic geometry. On the flexible side, we prove that a symplectic homeomorphism might take a symplectic disc to a smooth isotropic one. We also get a $C^0$-rigidity result for the action of a symplectic homeomorphism on the reduction of a coisotropic submanifold.

拉格朗日余切在定量辛几何示例中的一些有趣应用

Some cute applications of Lagrangian cobordisms towards examples in quantitative symplectic geometry

该研究利用拉格朗日余切给出构造改进辛挤压问题示例，还证明拉格朗日同痕的子流形是拉格朗日余切的，为定量辛几何研究提供新成果。

Functoriality in categorical symplectic geometry

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim-Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya $A_\infty$-categories, and the second au…

Toric degenerations in symplectic geometry

A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. Harada and Kaveh described how one incorporates a symplectic structure into this process, providing a very useful tool for solving certain problems in symplectic geometry. Below we present applications of this method to questions about the Gromov width, and cohomological rigidity problems.

关于局部共形辛几何中精确拉格朗日子流形的投影

On the projection of exact Lagrangians in locally conformally symplectic geometry

本文构造局部共形辛流形余切丛中的精确拉格朗日例子，给出投影诱导简单同伦等价的条件，并揭示喷射空间接触几何与lcs几何的联系及本质刘维尔弦的作用。

余辛几何中的哈密顿群作用

Hamiltonian group actions in cosymplectic geometry

研究余辛流形上哈密顿群作用，通过辛加厚过程，将其提升为辛流形上哈密顿辛作用，建立相关理论桥梁，得出凸性定理等余辛类似结果，还应用于b-辛流形临界超曲面的哈密顿几何。

Dusa McDuff and symplectic geometry

I describe some of McDuff's contributions to symplectic geometry, with a focus on symplectic embedding problems.

BRST model applied to symplectic geometry

Two local macros are included (gothic.sty and fleqn.sty)

Positive divisors in symplectic geometry

In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.

When symplectic topology meets Banach space geometry

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.

Geometry of symplectic intersections

In this paper we survey several intersection and non-intersection phenomena appearing in the realm of symplectic topology. We discuss their implications and finally outline some new relations of the subject to algebraic geometry.

Applications of Grothendieck's inequality to linear symplectic geometry

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.

Lagrangian submanifolds in affine symplectic geometry

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.

Quantitative symplectic geometry

While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. We also describe several new relations between certain symplectic capaci…

辛几何与几何表示论中哈密顿约简的一些表现形式

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

The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory groups and symplectic geometry. We shall concentrate here on the Heisenberg group, although it is transparent that almost everything can be done on a general Carnot-Caratheodory group. Such generalisations will be the subject of a forthcoming note.

Symplectic and Poisson derived geometry and deformation quantization

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.

Complex of twistor operators in symplectic spin geometry

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricc…

Geometry of symplectic log Calabi-Yau pairs

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

Compact Kähler threefolds with non-nef canonical bundle and symplectic geometry

We show that the nefness of the canonical bundle of compact Kähler threefolds is invariant under deformed symplectic diffeomorphisms.

Symplectic geometry of Cartan-Hartogs domains

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.

Mirror of Atiyah flop in symplectic geometry and stability conditions

We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.

Symplectic geometry of rationally connected threefolds

We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we prove that it is symplectic rationally connected, i.e. there is a non-zero Gromov-Witten invariant with two insertions being the class of a point. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.

Symplectic and Poisson geometry of the information geometry's Frobenius manifold

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.

From symplectic cohomology to Lagrangian enumerative geometry

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.

Symplectic geometry and rationally connected 4-folds

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected 4-fold whose second Betti number is 2 is rationally connected.

导出辛几何

Derived symplectic geometry

本文受邀为《数学物理百科全书》第2版撰写，简述了研究导出栈上辛结构的导出辛几何，并通过Casson不变量的案例研究概述了这一年轻领域。

Poisson Lie algebroid与generalized complex geometry上的shifted symplectic structure

Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry

本文在同伦意义下将广义复流形重新表述为全纯symplectic formal stack，发展shifted symplectic机制证明coisotropic intersection继承shifted Poisson structure，并研究了广义复brane。

The geometry of symplectic pairs

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.

带有边界的曲面的辛几何

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。

辛几何、泊松几何及其超越

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。

Symplectic, Poisson, and contact geometry on scattering manifolds

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 Pre-Symplectic Structures: a Dirac Geometry Approach

We explain the geometric origin of the $L_{\infty}$-algebra controlling deformations of pre-symplectic structures.

Symplectic geometry of semisimple orbits

We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold. Moreover, we generalize this result to hyperbolic orbits in a real semisimple Lie algebra.

The log symplectic geometry of Poisson slices

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.

Geometry of the symplectic Stiefel manifold endowed with the Euclidean metric

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expr…

First steps into the world of systolic inequalities: From Riemannian to symplectic geometry

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.

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational $J$-holomorphic curves in a given homology class passing through a given real configuration of points.

Kahler-Sasaki geometry of toric symplectic cones in action-angle coordinates

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…

Holomorphic symplectic geometry and orbifold singularities

Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.

Equivariant symplectic geometry of cotangent bundles, II

We examine the structure of the cotangent bundle $T^{*}X$ of an algebraic variety $X$ acted on by a reductive group $G$ from the viewpoint of equivariant symplectic geometry. In particular, we construct an equivariant symplectic covering of $T^{*}X$ by the cotangent bundle of a certain variety of horospheres in $X$, and integrate the invariant collective motion on $T^{*}X$. These results are based on a "local structure theorem" describing the action of a certain parabolic in $G$ on an open subset of $X$, which is interesting by itself.

Symplectic Birational 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.

Local geometry of symplectic divisors with applications to contact torus bundles

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.

辛映射与超Kähler动量映射几何

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。

Symplectic Geometry and Connectivity of Spaces of Frames

Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper we focus on the frame homotopy conjecture posed in 2002 and recently proved by Cahill, Mixon, and Strawn, which says that the space of finite unit norm tight frames is connected. We give a simple symplectic proof of a doub…

Estimated transversality in symplectic geometry and projective maps

The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as applications to the construction of maps with generic singularities from compact symplectic manifolds to projective spaces.

刚体中的开放问题与定量辛几何

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。

The projective symplectic geometry of higher order variational problems: minimality conditions

We associate curves of isotropic, Lagrangian and coisotropic subspaces to higher order, one parameter variational problems. Minimality and conjugacy properties of extremals are described in terms of these curves.