Jack Morava
The cobordism ring of symplectic manifolds defined by V.L. Ginzburg is shown to be isomorphic to the Pontrjagin ring of complex-oriented manifolds with free circle actions. This suggests an interpretation of the formal group law of complex cobordism, in terms of a composition-law on semiclassical expansions. An appendix discusses related questions about cobordism of toric varieties.
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In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are ``symplectic cobordism equivalent.''
We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.
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。
We discuss the existence and non-existence of cobordisms between symplectic surface bundles over the circle.
A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.
We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded two-spheres of square zero. This, together with standard gauge theory, is used to show that any contact three-manifold of non-zero torsion (in the sense of Giroux) cannot be strongly symplectically fillable. John Etnyre pointed out to the author that the same argument together wi…
Lagrangian cobordisms and K-theory of symplectic bielliptic surfaces
本文研究辛双椭圆曲面上弱正合拉格朗日G-膜的拉格朗日配边群，计算了热带拉格朗日生成的子群，并利用同调镜像对称性证明Fukaya范畴的Grothendieck群与配边群计算一致，进而提出热带拉格朗日生成整个配边群的猜想。
We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling $X$ of a virtually overtwisted contact structure on $L(p,q)$ has another symplectic structure that fills the universally tight contact structure on $L(p,q)$. Moreover, we show that the Stein filling of $L(p,q)$ with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.
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 prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions.
We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddin…
This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients considered here are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. (A companion paper examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus.) Let X be a symplectic manifold, with a Hamiltonian action of a compact torus T. The main topological result of this paper describes an explicit cobordism that exists between a symplectic quotient of X by T, and a c…
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy inva…
Given two open books with equal pages we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From …
On cobordism groups of Lagrangian immersions
本文用由辛流形构造的 Thom 谱的稳定同伦群计算拉格朗日浸入的配边群，推广了 Eliashberg 的结果，并给出了闭曲面和单调流形情形下的计算与部分描述。
We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfa…
Algebraic Lagrangian cobordisms, flux and the Lagrangian Ceresa cycle
本文引入代数拉格朗日配边等价关系，并借助热带与辛通量证明拉格朗日Ceresa圈在对应配边群中非挠，从而给出Ceresa圈非挠性的辛镜像类比。
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relati…
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $Λ_+$ and $Λ_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $Λ_+$ to that of $Λ_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely c…
We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positiv…
Given a symplectic manifold $(M^{2n},ω)$ we study Lagrangian cobordisms $V\subset E$ where $E$ is the total space of a Lefschetz fibration having $M$ as generic fiber. We prove a generation result for these cobordisms in the appropriate derived Fukaya category. As a corollary, we analyze the relations among the Lagrangian submanifolds $L\subset M$ that are induced by these cobordisms. This leads to a unified treatment - and a generalization - of the two types of relations among Lagrangian submanifolds of $M$ that were previously identified in the literature: those associated to Dehn twists tha…
We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes. We apply the construction to give constraints on the algebraic topology of fillings of double covers of the 3-sphere branched over certain transverse quasipositive knots.
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.
Floer theory for Lagrangian cobordisms was developed by Biran and Cornea to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.
R Kerr's Ricci-flat Lorentz 4-manifold was shown by B Carter in 1968 to support a classically completely integrable system of geodesics; here we interpret this system in terms of a locally conformally symplectic structure, which we identify (§2.2) using characteristic classes defined by Kerr-Schild Cartesian coordinates (regarded as a map to the Cayley-Penrose compactification of Minkowski space). This leads to the definition of an interesting cobordism category of contact 3-manifolds and 4-dimensional locally conformally symplectic cobordisms between them.
Ellipsoidal superpotentials and stationary descendants
该研究计算任意维辛椭球的稳态引力派生类，推导带椭球端辛流形穿孔曲线计数的新递推公式，构建借仿射空间标准复结构显式计算的框架，并开启对作为椭球辛配边基本单元的无穷小辛配边的研究。
We compute the Lagrangian cobordism group of the standard symplectic 2-torus and prove that it is isomorphic to the Grothendieck group of its derived Fukaya category. The proofs use homological mirror symmetry for the 2-torus.
Given an exact Lagrangian $L$ of an exact symplectic manifold $(M,dλ)$, Cornea and Shelukhin recently introduced a remarkable "cobordism metric" $d_c$ on the exact Lagrangian cobordism class $\mathcal{L}(L)$ of $L$. In this note we show that the Liouville flow of $(M,dλ)$ induces a flow on $(\mathcal{L}(L),d_c)$ which expands cobordism distances. In particular we deduce that $(\mathcal{L}(L),d_c)$ has infinite diameter whenever the Liouville flow is complete. We also discuss (Hamiltonian and Lagrangian) Hofer-geometric versions of our result. The proof only uses elementary differential geometr…
Lagrangian cobordism of positroid links
本研究针对复格拉斯曼流形正半单形层的偏序，证明其可通过精确拉格朗日余挠实现辛解释，建立代数几何层结构与辛拓扑余挠的联系。
We introduce the concept of $k$-(semi)-dilation for Liouville domains, which is a generalization of symplectic dilation defined by Seidel-Solomon. We prove that the existence of $k$-(semi)-dilation is a property independent of fillings for asymptotically dynamically convex (ADC) manifolds. We construct examples with $k$-dilations, but not $k-1$-dilations for all $k\ge 0$. We extract invariants taking value in $\mathbb{N} \cup \{\infty\}$ for Liouville domains and ADC contact manifolds, which are called the order of (semi)-dilation. The order of (semi)-dilation serves as embedding and cobordism…
Unobstructed Lagrangian cobordism groups of surfaces
本文研究了亏格 $g \geq 2$ 的闭辛曲面的无阻碍浸入拉格朗日配边群，基于 Abouzaid 和 Perrier 的工作计算了该群，并证明其同构于曲面的导出 Fukaya 范畴的 Grothendieck 群，证明中利用二维拓扑技术构造了不界定特定全纯多边形的配边。
Fourier transforms and a filtration on the Lagrangian cobordism group of tori
本文针对极化热带仿射环面，证明对应辛流形的纤维拉格朗日配边群存在有限长度自然几何滤子，构造对偶辛环面Fukaya范畴间的傅里叶变换并结合同调镜像对称关联Mukai的傅里叶变换，揭示该滤子与阿贝尔簇Chow群的Bloch滤子的镜像对应关系。
Liouville hypersurfaces and connect sum cobordisms
本文提出 Liouville 超曲面与 Liouville 连通和操作，推广了 Weinstein 手柄附着与切触手术，并利用其构造了高维奇异切触结构，揭示了特定球面 Dehn 扭转的辛非平凡性及流形的不可填充性。
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。
Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…
We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic cu…
Two natural symplectic constructions, the Lagrangian suspension and Seidel's quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M,ω) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid Π(Ham(M)) on a cobordism category recently introduced in \cite{Bi-Co:cob2} and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in \cite{Bi-Co:cob2} that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsváth-Szabó invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depend…
Let $(X,ω)$ be a closed symplectic manifold. A loop $ϕ: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $π: P_ϕ \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $π$, Lalonde, McDuff and Polterovich proved that if $ϕ$ lifts to the Hamiltonian group $\mathrm{Ham}(X,ω)$, then the rational cohomology of $P_ϕ$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_ϕ$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. …
We introduce the Kodaira dimension of contact 3-manifolds and establish some basic properties. In particular, contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. On the other hand, we also prove that, given any contact 3-manifold, there is a lower bound of $2χ+3σ$ for all its minimal symplectic fillings. This is motivated by the bound of Stipsicz for Stein fillings. Finally, we discuss various aspects of exact self cobordisms of fillable 3-manifolds.
We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon, to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincare polynomial and the Euler characteristic (equivalent to those of Kirwan) can be expressed in the same recursive manner.
We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4…
This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.
Bulk-deformations, Floer complex bordism, and Grothendieck-Riemann-Roch
本文通过计算Liouville manifold的Floer同调不变量，结合Grothendieck-Riemann-Roch定理，研究了复数bordism与辛cohomology之间的关系，并给出了非基变的判据。
An origami manifold is a manifold equipped with a closed 2-form which is symplectic except on a hypersurface where it is like the pullback of a symplectic form by a folding map and its kernel fibrates with oriented circle fibers over a compact base. We can move back and forth between origami and symplectic manifolds using cutting (unfolding) and radial blow-up (folding), modulo compatibility conditions. We prove an origami convexity theorem for hamiltonian torus actions, classify toric origami manifolds by polyhedral objects resembling paper origami and discuss examples. We also prove a cobord…
Mazur manifolds and symplectic structures
利用Heegaard Floer同调的cobordism映射，阻碍了Akbulut-Kirby流形上存在辛结构的可能性，并表明存在具有确定交形式的exotic对。
We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of…
In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.
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.
We describe an operation which modifies a Lagrangian submanifold $L$ in a symplectic manifold $(M, ω)$ such as to produce a new immersed Lagrangian submanifold $L'$, which as a smooth manifold is obtained by surgery along a framed sphere in $L$. Intuitively, this can be described as collapsing an isotropic disc with boundary on $L$ to a point. The inverse operation generalizes classical Lagrangian surgery. We also describe corresponding immersed Lagrangian cobordisms between $L$ and $L'$ . After removal of their singular locus, we obtain examples of embedded Lagrangian cobordisms with precisel…
We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group G. In each case, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth s-cobordism and hence are topologically equivalent for good groups.
The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory. They form the technical backbone of the proof that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. One interesting feature of the theory is that existence of pseudoholomorphic curves gives certain restricti…
A landscape of contact manifolds via rational SFT
本文基于有理辛场论的$BL_\infty$形式定义了从精确辛配边范畴到全序集的层级函子，开发了计算工具并证明其满射性，进而用于研究精确配边存在性及探讨相关几何实例。
On symplectic fillings of spinal open book decompositions I: Geometric constructions
本文介绍了脊柱开放书与接触结构及辛或Stein结构的关系，提出新的辛cobordism构造脊柱移除手术，扩展了Eliashberg、Gay-Stipsicz和第三作者的构造，并应用该手术得到新的非强（有时非弱）辛填充的接触流形例子。
A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed.
For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describes well-defined counts of pseudoholomorphic curves in the complement of the zero-set of such "near-symplectic" for…
Bordism and resolution of singularities
本文通过改编奇点消解算法，证明复bordism到复导出orbifold及等变bordism的同调理论映射可分裂，进而为任意闭辛流形定义复cobordism取值的Gromov-Witten不变量，并约束Hamilton纤维丛空间拓扑。
Algebraic planar torsion in contact manifolds
该研究通过辛场论的函子性质，在强偏微分和手术偏微分下生成有限的代数平面扭量，并提供了新的例子家族，证实了相关猜想并揭示了高维中紧致但非弱可填充接触结构的普遍性。
Finiteness and boundedness of positive monotone Hamiltonian GKM$_3$ spaces
针对正单调紧辛流形上的GKM$_3$作用，证明了在固定维数和欧拉示性数下，复配边类有限，矩映射像可嵌入有界盒子，且陈数满足定量界。
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