Tian-Jun Li, Ai-Ko Liu
Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.
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In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.
Comparing Kahler cone and symplectic cone of one-point blowup of Enriques surface
本文研究Enriques曲面单点爆破的Kähler锥与辛锥的差异，证明其存在非Kähler辛形式，通过代数几何不变量定量比较两者区别，推进了$p_g=0$的Kähler曲面上辛通用复结构相关问题的研究。
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of…
The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplectic cone. We prove this conjecture in the case of the spin surfaces E(2m) using inflation and the action of self-diffeomorphisms of the elliptic surface. An additional obstruction appears in the non-spin case.
We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and complex surfaces.
Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone
本文研究移位辛结构在相对情形和非几何堆叠中的推广，提出移位余切丛和AKSZ程序，并定义移位拉格朗日映射的变形到正则锥。
In this note, we show that the Horn cone associated with symplectic eigenvalues admits the same inequalities as the classical Horn cone, except that the equality corresponding to Tr(C) = Tr(A)+Tr(B) is replaced by the inequality corresponding to Tr(C) $\ge$ Tr(A)+Tr(B).
Symplectic Classes on Elliptic Surfaces with positive Euler Number
本文研究了正欧拉数椭圆面的辛锥，探讨其辛类的结构与性质。
We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their dual resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. In a joint work with Starkston, we previously showed that if the weight of each vertex in the graph is at most -5, the contact structure has a unique symplectic filling (up to symplectic deformation and blow-up). The proof was based on a symplectic analog of de Jong-van Straten's description of smoothings of thes…
This survey paper addresses uniqueness questions for symplectic forms on closed manifolds, explains what is known in several examples, and reviews some open problems.
We show, in this note, that on any symplectic supermanifold, even or odd, there exist an infinite dimensional affine space of symmetric connections, compatible to the symplectic form.
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…
On Auroux's Luttinger-surgery uniqueness question
构造奇异辛四流形对，其拓扑不变量相同但辛小平维数不同，因辛小平维数在Luttinger手术下不变，从而否定Auroux的唯一性问题。
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…
Given a Lagrangian sphere in a symplectic 4-manifold $(M, ω)$ with $b^+=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $κ$ of $(M, ω)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic and ternary, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniq…
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to a weakly convex good cone by the cutting construction. (Note that these toric contact manifolds can not be constructed by Delzant construction.) We further prove there are no toric Sasakian structures on these contact manifolds. From this, contact toric manifolds of toric K-cont…
Infinite connected components of the space of symplectic forms on ruled surfaces
本文在直纹曲面上构造了无限族两两不同痕的微分同胚辛形式，解决了闭辛四维流形上辛结构在同痕意义下的唯一性问题。
By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.
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.
Diffeomorphism type of symplectic fillings of unit cotangent bundles
该研究针对单位余切丛的辛填充问题，证明了含高维环面在内的若干单位余切丛的辛无球填充在微分同胚意义下具有唯一性，推进了辛拓扑相关领域的研究。
We continue our previous work to prove that for any non-minimal ruled surface $(M,ω)$, the stability under symplectic deformations of $π_0, π_1$ of $Symp(M,ω)$ is guided by embedded $J$-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio $μ$ between the section and fiber goes to infinity, there is a topological colimit of $Symp(M,ω_μ).$ Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in $Symp(M,ω) \cap \rm Diff_0(M),$ where $…
Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished prepr…
Hypersurfaces containing involutive cones in projective symplectic spaces
该研究刻画了射影辛空间中包含对合锥的超曲面，通过关联空间和 Segre 类积分计算关联次数，并给出了双有理性的充分判据及特殊维数下的闭式公式。
A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and S…
In this paper we show that the classical results on the existence and uniqueness of moment maps in symplectic geometry generalize directly to weak homotopy moment maps in multisym- plectic geometry. In particular, we show that their existence and uniqueness is governed by a Lie algebra cohomology complex which reduces to the Chevalley-Eilenberg complex in the symplectic setup
Given a compact symplectic manifold M with the Hamiltonian action of a torus T, let zero be a regular value of the moment map, and M_0 the symplectic reduction at zero. Denote by κ_0 the Kirwan map H^*_T(M)-> H^*(M_0). For an equivariant cohomology class η\in H^*_T(M) we present new localization formulas which express \int_{M_0} κ_0(η) as sums of certain integrals over the connected components of the fixed point set M^T. To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect t…
Algebraic capacities as tropical polynomials over the reduced $c_1$-positive symplectic cone
本文用驯化辛形式与近复结构重新表述代数容量，证明其仅依赖上同调类，引入热带多项式描述容量，并给出光滑环面曲面上代数容量与ECH容量对应的纯辛证明。
We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.
In this note, we show a uniqueness result of homogeneous quasimorphisms defined on the universal cover of the symplectic linear group.
Symplectic fillings of unit cotangent bundles of spheres and applications
该研究证明了奇数维球面的单位余切丛的辛填充在微分同胚意义下唯一，并探讨了其在辛cobordism和闭合辛流形中的应用。
The goal of this paper is to classify symplectic toric stratified spaces with isolated singularities. This extends a result of Burns, Guillemin, and Lerman which carries out this classification in the compact connected case. In making this classification, it is necessary to classify symplectic toric cones. Via a well-known equivalence between symplectic toric cones and contact toric manifolds, this allows for the classification of contact toric manifolds as well, extending Lerman's classification of compact connected contact toric manifolds.
Given a symplectic surface $(Σ, ω)$ of genus $g \ge 4$, we show that the free group with two generators embeds into every asymptotic cone of $(\mathrm{Ham}(Σ, ω), d_\mathrm{H})$, where $d_\mathrm{H}$ is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.
The symplectic form associated to a singular Poisson algebra
本文研究奇异泊松代数上辛形式的存在性问题，以双锥为例通过Lie-Rinehart代数的de Rham复形和Gröbner基计算构造出辛形式，并将构造推广到范畴商、简单锥及线性辛轨形。
We present a new and simpler proof of the fact that any Lagrangian $\mathbb{R}P^2$ in $T^*\mathbb{R}P^2$ is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians in $T^*S^2$, using surgery to turn Lagrangian spheres into symplectic ones. The main novel contribution is a detailed proof of the folklore fact that the complement of a symplectic quadric in $\mathbb{C}P^2$ can be identified with the unit cotangent disc bundle of $\mathbb{R}P^2$.
In this paper, we study the curve cone of an almost complex $4$-manifold which is tamed by a symplectic form. In particular, we prove the cone theorem as in Mori theory for all such manifolds using the Seiberg-Witten theory. For small rational surfaces and minimal ruled surfaces, we study the configuration of negative curves. We define abstract configuration of negative curves, which records the homological and intersection information of curves. Combinatorial blowdown is the main tool to study these configurations. As an application of our investigation of the curve cone, we prove the Nakai-M…
The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this quantity is bounded from above by $π/n$ for a Lagrangian torus inside the $2n$-dimensional unit disc equipped with the standard symplectic form. A Lagrangian torus for which this upper bound is attained is called extremal. We show that an extremal Lagrangian torus inside the four-dimensional unit disc is contained in the boundary $\partial D^4=S^3$, and is hence …
We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as cotangent bundle of closed manifolds...) and we derive some consequences to C^0-symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth Hamiltonians whose flows converge to the identity for the spectral (or Hofer's) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the C^0-rigidity o…
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun struct…
We investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish between two examples based on those found in \cite{maydanskiy,maydanskiyseidel}, even though their classical symplectic invariants such as symplectic cohomology vanish. We also exhibit, for any $n$, an exact symplectic manifold with $n$ distinct but exotic symplectic structures, which again cannot be distinguished by symplectic cohomology.
In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.
Topological symplectic manifolds and bi-Lipschitz structures
本文证明拓扑辛流形自然具有双Lipschitz结构，并首次展示拓扑辛结构的非存在性和非唯一性。
In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold $(M,J,ω)$ and a symplectic form $σ$ on $M$ which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form $ω$. As a consequence, we can give an answer to the question posed by Khesin and McDuff as follows. According to symplectic Hodge theory, any symplectic form $ω$ on a smooth manifold $M$ defines \textit{symplectic harmonic forms} on $M$…
Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $Ξ$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of $Ξ$-symplectic complexity of $\bf X$. This notion generalizes the notion of complexity defined in [Vin86]. We prove several properties of this notion, and relate it to the notion of $Ξ$-complexity defined in [AG] motivated by its relation with representation theory.
We introduce an analogue of the inflation technique of Lalonde-McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic four-manifolds. We consider the implications of this construction for the symplectic cones of Kähler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudoholomorphic curves via a perturbation.
The spaces of Kähler and holomorphically tamed symplectic forms on closed 4-manifolds
研究闭 4 维流形上凯勒形式空间的相关性质，受猜想启发还研究全纯驯服辛形式，提出平行问题并关联其与凯勒型辛形式问题，探讨唯一性、连通性和开放性等。
We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…
On the cohomology ring of symplectic fillings
本文通过球丛扭曲的辛上同调和Gysin正合序列，证明了渐近动态凸流形填充上同调环的部分唯一性，并推广到柔性可填充接触流形的实同伦型唯一性。
The spread construction and non-uniqueness of visible Lagrangians
研究 Hirzebruch 曲面中可见拉格朗日量的性质，通过新的展开构造证明其包含拉格朗日环面和克莱因瓶，且其拓扑不由矩映射像决定。
Let $G$ be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group $G^\vee$ and the Poisson-Lie dual group $G^*$. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein-Kazhdan potential on the double Bruhat cell $G^{\vee; w_0, e} \subset G^\vee$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on the partial tropicalization of $K^* \subset G^*$ (the Poisson-Lie dual of the compact form $K \subset G…
We introduce the notion of relational symplectic groupoid as a way to integrate Poisson manifolds in general, following the construction through the Poisson sigma model (PSM) given by Cattaneo and Felder. We extend such construction to the infinite dimensional setting, where we are able to describe the relational symplectic groupoid in terms of immersed Lagrangian Bannach submanifolds. This corresponds to a groupoid object in the "extended symplectic category" and it can be related to the ususal version of symplectic groupoids via reduction. We prove the existence of such an object for any Poi…
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