Symplectic geography in dimension 8

On geography of symplectic fillings of contact branched covers

In this paper, we determine the Euler characteristics and signatures of the exact symplectic fillings of the contact double, 3-fold or 4-fold cyclic covers of the standard contact 3-sphere branched over certain transverse quasi-positive links. These links include all quasi-positive knots with crossing numbers smaller than 11 and all quasi-positive links with crossing numbers smaller than 12 and nonzero nullity.

自旋辛4-流形的几何问题的斜率不等式

Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds

该研究基于Roulleau与Urzúa的复曲面成果，构造了自旋辛4-流形的无限多互不同胚光滑结构，并指出其结果接近辛Bogomolov-Miyaoka-Yau不等式给出的最优边界。

非形式辛流形与切触流形的地理分布

Geography of non-formal symplectic and contact manifolds

本文研究非形式紧定向流形何时可实现为辛或切触流形，证明在相应维数与第一 Betti 数条件下，非形式例子的存在蕴含同性质辛或切触流形的存在。

On the geography of simply connected nonspin symplectic $4$-manifolds with nonnegative signature

In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 25$, and the families of simply connected irreducible nonspin symplectic $4$-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in \cite{AP3, AHP}. In particular, we construct (i) an infinitely many irreducible symplectic and non-symplectic $4$-manifolds that…

Geography of surface bundles over surfaces

We construct symplectic surface bundles over surfaces with positive signatures for all but 18 possible pairs of fiber and base genera. Meanwhile, we determine the commutator lengths of a few new mapping classes.

Geography of Genus 2 Lefschetz fibrations

Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz f…

A formula for the Chern classes of symplectic blow-ups

It is shown that the formula for the Chern classes (in the Chow ring) of blow-ups of algebraic varieties, due to Porteous and Lascu-Scott, also holds (in the cohomology ring) for blow-ups of symplectic and complex manifolds. This was used by the second-named author in her solution of the geography problem for 8-dimensional symplectic manifolds. The proof equally applies to real blow-ups of arbitrary manifolds and yields the corresponding blow-up formula for the Stiefel-Whitney classes. In the course of the argument the topological analogue of Grothendieck's `formule clef' in intersection theor…

Minimality and symplectic sums

Let X_1, X_2 be symplectic 4-manifolds containing symplectic surfaces F_1,F_2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X_1 and X_2 along the F_k. Using relative Gromov--Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the X_k contains a (-1)-sphere disjoint from F_k; or one of the X_k admits a ruling with F_k as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefsche…

Many closed symplectic manifolds have infinite Hofer-Zehnder capacity

We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manif…

二维等价辛子流形在toric流形中的研究

Two-dimensional equivariant symplectic submanifolds in toric manifolds

本文通过结合凸几何与局部模型，研究二维等价辛子流形在toric流形中的判定准则。

Holomorphic spheres and four-dimensional symplectic pairs

We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with non-negative self-intersection, following the strategy of McDuff's classification of rational and ruled symplectic four manifolds.

Recent progress on symplectic embedding problems in four dimensions

We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. The new theory of "ECH capacities" gives general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.

Symplectic packings in dimension 4 and singular curves

The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff's inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of $P^2$ by 6, 7 or 8 balls.

Symplectic structures from Lefschetz pencils in high dimensions

A symplectic structure is canonically constructed on any manifold endowed with a topological linear k-system whose fibers carry suitable symplectic data. As a consequence, the classification theory for Lefschetz pencils in the context of symplectic topology is analogous to the corresponding theory arising in differential topology.

高维中辛容量的一致性研究

On the agreement of symplectic capacities in high dimension

本研究将Gutt-Hutchings-Ramos关于归一化辛容量对四维单调环面域取值一致的定理推广到任意维，通过构造L形域到无穷圆柱的辛嵌入，正面解决了Gutt-Pereira-Ramos的相关猜想。

关于K-接触非Sasakian Smale-Barden流形的构造

On the construction of K-contact non-Sasakian Smale-Barden manifolds

本文澄清并推广了构造首个单连通紧致5维K-接触非Sasakian Smale-Barden流形的方法，确定了所需的辛曲面数量，并推广了辛4-流形中具有横截交的辛曲面构造以分析地理问题。

Exotically knotted disks and complex curves

This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some di…

Symplectic isotopy classes of ellipsoids and polydisks in dimension greater than four

In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such nonisotopic embeddings can never be extended to the same ellipsoid.

Symplectic harmonicity and generalized coeffective cohomologies

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by C.-J. Tsai, L.-S. Tseng and S.-T. Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms $ω_t$ such that the dimensions of these symplectic cohomology groups vary with respect to $t$. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifol…

Reduction,Induction and Ricci flat symplectic connections

In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold $(M,ω)$ of dimension $2n$, we define induction as a way to construct a symplectic manifold $(P,μ)$ of dimension $2n+2$. Given any symplectic connection $\nabla$ on $(M,ω)$, we define an induced connection $\nabla^P$ which is a Ricci-flat symplectic connection on $(P,μ)$.

Some optimal embeddings of symplectic ellipsoids

We construct symplectic embeddings of ellipsoids of dimension $2n \ge 6$ into the product of a 4-ball or 4-dimensional cube with Euclidean space. A sequence of these embeddings can be shown to be optimal.

Singular polarizations and symplectic embeddings

We prove in this paper that any 4-dimensional symplectic manifold is essentially made of finitely many symplectic ellipsoids. The key tool is a singular analogue of Donaldson's symplectic hypersurfaces in irrational symplectic manifolds.

Toward a topological characterization of symplectic manifolds

A topological condition is given, characterizing which closed manifolds in dimensions < 8 (and conjecturally in general) admit symplectic structures. The condition is the existence of a certain fibration-like structure called a hyperpencil. A deformation class of hyperpencils on a manifold X of any even dimension is shown to determine an isotopy class of symplectic structures on X. This provides an inverse (at least in dimensions < 8) to Donaldson's program for constructing linear systems on symplectic manifolds. It follows that (at least in dimensions < 8) the set of deformation classes of hy…

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.

Hamiltonian $S^1$ actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds

The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010 paper Cho, Hwang, and Suh proved in the 6-dimensional case that if we have $b_2^+=1$ at a reduced space at a regular level $λ$ of the circle valued moment map, then the action is Hamiltonian. In this paper, we will use this to prove that certain 6-dimensional symplectic actions which ar…

An 8-dimensional non-formal simply connected symplectic manifold

A non-formal simply connected compact symplectic manifold of dimension 8 is constructed.

Symplectic actions of 2-tori on 4-manifolds

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of the torus action and of the symplectic form. We construct explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants. We also classify, up to equivariant symplectomorphisms, symplectic actions of (2n-2)-dimensional tori on 2n-dimensional s…

Topological properties of punctual Hilbert schemes of symplectic 4-dimensional manifolds

This paper has been replaced by the papers arXiv:1001.0114 and arXiv:1001.0119

Four dimensions from two in symplectic topology

This paper introduces two-dimensional diagrams that are slight generalizations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as well as methods for visualizing other manifolds, including the K3 surface, and for visualizing certain surgeries. Most of the techniques extend to higher dimensions.

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions III

In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian $S^1$-actions. We also show that every such manifold is $S^1$-equivariantly symplectomorphic to some Käahler Fano manifold with a certain holomorphic Hamiltonian circle action.

Kodaira dimension and symplectic sums

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T^2-bundles over T^2 each having b_1=2.

Symplectic actions of non-Hamiltonian type

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is $n$-dimensional and the symplectic manifold is $2n$-dimensional. In this case the $n$-dimensional orbits are Lagrangian, so it is natural to wonder whether there are interesting classes of symplectic actions with Lagrangian orbits, and that are not Hamiltonian. It turns out that there are many such classes which contain for example the Kodaira variety, and which can be…

A symplectic Kovacic's algorithm in dimension 4

Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of …

Symplectic maps to projective spaces and symplectic invariants

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any compact symplectic manifold by a series of words in braid groups and a word in a symmetric group.

Existence of conformal symplectic foliations on closed manifolds

We consider the existence of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5. Our main theorem, based on a recent result by Bertelson-Meigniez, states that in dimension at least 7 any almost contact structure is homotopic to a conformal symplectic foliation. In dimension 5 we construct explicit conformal symplectic foliations on every closed, simply-connected, almost contact manifold, as well as honest symplectic foliations on a large subset of them. Lastly, via round-connected sums, we obtain, on closed manifolds, examples of conforma…

The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

This paper defines a symplectic form on the infinite dimensional Fréchet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated by the geometric invariants of the curves on the base manifold and relates them to the equations of mathematical physics.

Symplectic manifolds with disconnected contact type boundary in dimension 4n

We give examples of compact symplectic manifolds with disconnected contact type boundary in dimension $4n$ for any $n\geq 1$. The example is given by a subset of the tangent bundle of a compact quotient of the complex hyperbolic space endowed with the canonical symplectic form plus a generalized magnetic field and its boundary is given by two hypersurfaces of constant kinetic energy. In particular, when $n=1$, it is obtained by the tangent bundle of a surface of genus greater than 1 endowed with the canonical symplectic form plus the pullback of the Kähler form by the canonical projection.

Virtual Betti numbers and the symplectic Kodaira dimension of fibered 4-manifolds

We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion symplectic canonical class, nor is of Kodaira dimension zero.

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.

四维辛曲率流的静态解

Static solutions to symplectic curvature flow in dimension four

该研究针对四维辛曲率流，推导静态解的全纯数据局部正规形并证明完备静态解为凯勒-爱因斯坦度量，同时通过外微分系统分析与嘉当-凯勒定理证明孤子的局部存在性与一般性。

来自超椭圆Lefschetz纤维化的6维辛子流形

Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations

本研究利用4-流形上的超椭圆Lefschetz纤维化，构造出含无穷多同调等价但同伦不等价且无复结构的余维2辛子流形的闭单连通辛6-流形，并推广至高维，还在附录证明了ℂℙ³中低次余维2辛子流形的微分同胚性。

Four-dimensional symplectic cobordisms containing three-handles

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…

Symplectic blowing down in dimension Six

We establish a blowing down criterion in the context of birational symplectic geometry in dimension 6.

辛流形的单位余切丛的填充及其应用

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。

Kodaira Dimension of Fiber Sums along Spheres

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic -4-sphere.

Semi-free Hamiltonian circle actions on 6 dimensional symplectic manifolds

Assume $(M, ω)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim$ $H^2(M)<3$. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.

辛log柯西-维特里维斯维度-∞，希尔兹-朱恩字符串和加权投影平面

Symplectic log Kodaira dimension $-\infty$, Hirzebruch--Jung strings and weighted projective planes

本文研究加权投影平面CP(a,b,c)的辛极小解析，从断开辛除子的视角出发，引入例外间隙概念，建立某些三希尔兹-朱恩字符串配置的托里利型定理，并证明辛仿射刻度的存在性与加权投影平面极小解析的联系。

Homogeneous symplectic 4-manifolds and finite dimensional Lie algebras of symplectic vector fields on the symplectic 4-space

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras $\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded transitive finite-dimensional subalgebras of the full prolongations $\mathfrak{p}_1^{(\infty)}$ and $\mathfrak{p}_2^{(\infty)}$, where $\mathfrak{p}_1$ and $\mathfr…

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions II

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the maximal and the minimal fixed component are both two dimensional, then $(M,ω_M)$ is $S^1$-equivariantly symplectomorphic to some Kähler Fano manifold $(X, ω_X, J)$ equipped with a certain holomorphic Hamiltonian $S^1$-action. We also give a complete list of all such Fano manifolds together with an explicit description of the corresponding $S^1$-actions.

Topological complexity of symplectic manifolds

We prove that the topological complexity of every symplectically atoroidal manifold is equal to twice its dimension. This is the analogue for topological complexity of a result of Rudyak and Oprea, who showed that the Lusternik--Schnirelmann category of a symplectically aspherical manifold equals its dimension. Symplectically hyperbolic manifolds are symplectically atoroidal, as are symplectically aspherical manifolds whose fundamental group does not contain free abelian subgroups of rank two. Thus we obtain many new calculations of topological complexity, including iterated surface bundles an…

Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that…

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.

Complex symplectic structures on Lie algebras

We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize this construction to the nilpotent case and apply complex symplectic oxidation to classify eight-dimensional nilpotent complex symplectic Lie algebras.

Symplectic Divisorial Capping in Dimension 4

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $ω$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever…