On existence of nonformal simply connected symplectic manifolds

Simply connected minimal symplectic 4-manifolds with signature less than --1

For each pair $(e,σ)$ of integers satisfying $2e+3σ\ge 0$, $σ\leq -2$, and $e+σ\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $σ$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).

Non-simply connected symplectic fillings of lens spaces

We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The Fibonacci numbers make a cameo appearance.

Simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{\mathbf CP}^2}$ admits an exotic smooth structure.

辛哈达玛问题

The symplectic Hadamard question

研究单连通辛流形，通过证明其与\(R^{2n}\)辛同胚，肯定回答了麦克杜夫 - 萨拉蒙的“辛哈达玛问题”。

Cohomological properties of ruled symplectic structures

This survey presents some recent results by the authors and Polterovich on the topological properties of ruled symplectic manifolds. The bundle M \to P \to B that is associated with a ruled manifold has the group of Hamiltonian symplectomorphisms of M as structure group if the base is simply connected. Thus information about Hamiltonian bundles gives stability results for ruled structures as well as obstructions to their existence.

Simply Connected Symplectic Calabi-Yau 6-Manifolds

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.

Geography of spin symplectic 4-manifolds with abelian fundamental group

In this paper, results of J. Park and of B.D Park and Szabo on simply connected symplectic 4-manifolds are re-proven and extended to non-simply connected manifolds using Luttinger surgeries.

On certain geometric and homotopy properties of closed symplectic manifolds

The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we disc…

Constructions of small symplectic 4-manifolds using Luttinger surgery

In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but …

A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit Kähler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.

Symplectic four-manifolds and conformal blocks

We apply ideas from conformal field theory to study symplectic four-manifolds, by using modular functors to "linearise" Lefschetz fibrations. In Chern-Simons theory this leads to the study of parabolic vector bundles of conformal blocks. Motivated by the Hard Lefschetz theorem, we show the bundles of SU(2) conformal blocks associated to Kaehler surfaces are Brill-Noether special, although the associated flat connexions may be irreducible if the surface is simply-connected and not spin.

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…

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…

The geometric structure of symplectic contraction

We show that the symplectic contraction map of Hilgert-Manon-Martens -- a symplectic version of Popov's horospherical contraction -- is simply the quotient of a Hamiltonian manifold $M$ by a "stratified null foliation" that is determined by the group action and moment map. We also show that the quotient differential structure on the symplectic contraction of $M$ supports a Poisson bracket. We end by proving a very general description of the topology of fibers of Gelfand-Zeitlin systems on multiplicity free Hamiltonian $U(n)$ and $SO(n)$ manifolds.

Smooth surfaces with non-simply-connected complements

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.

Manifolds which are complex and symplectic but not Kähler

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kähler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected $8$-dimensional compact manifold of [\textsc{M. Fernández and V. Muñoz}, \emph{An 8-dimensional non-formal simply connected symplectic manifold}, Ann. of Math. (2) \textbf{167}, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does no…

Rabinowitz Floer homology and symplectic homology

The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,ξ)$ into a symplectic manifold $(W,ω)$. They depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz-Floer homology $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, where $ST^*L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact…

Hard Lefschetz property of symplectic structures on compact Kaehler manifolds

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 $M$…

Symmetric symplectic spaces with Ricci-type curvature

We determine the isomorphism classes of symmetric symplectic manifolds of dimension at least 4 which are connected, simply-connected and have a curvature tensor which has only one non-vanishing irreducible component -- the Ricci tensor.

Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of …

The Yamabe invariant of a class of symplectic manifolds

We compute the Yamabe invariant for a class of symplectic 4-manifolds of general type obtained by taking the rational blowdown of Kahler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal symplectic manifold for which we compute the Yamabe invariant.

The Isotopy Problem for Symplectic 4-manifolds

The purpose of this paper is to present some results on the existence of homologous, nonisotopic symplectic or lagrangian surfaces embedded in a simply connected symplectic 4-dimensional manifold.

Tori in symplectic 4-manifolds

We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.

Examples of symplectic 4-manifolds with positive signature

We construct a new family {K_n} of simply connected symplectic 4-manifolds with the property c_1^2(K_n)/chi(K_n) -> 9 (as n goes to infinity).

A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Let (M,ω) be a four dimensional compact connected symplectic manifold. We prove that (M,ω) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl(M,ω) is finite. Our proof is "soft". The proof uses the fact that for symplectic blow-ups of \CP^2 the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we describe results of McDuff that give a properness result for a general compact symplectic four-manifold, u…

Complex and Symplectic Structures on Panelled Web 4-Manifolds

We analyze the symplectic and complex structures on the panelled web 4-manifolds. In particular, we give infinite family of examples of almost complex but not symplectic and not complex 4-manifolds in the non-simply connected case.

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。

关于辛填充的上同调环

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。

Symplectic fillings of asymptotically dynamically convex manifolds I

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…

Symplectic nilmanifolds with a symplectic non-free $Z_3$-action

This paper expands some of the issues of the paper math.SG/0506449. We introduce a new technique to produce symplectic manifolds, by taking a symplectic non-free action of a finite group on a symplectic manifold and resolving symplectically the singularities of the quotient. We also introduce a new type of Massey product to detect non-formality. This allows to show the first example of a non-formal simply connected compact symplectic manifold of dimension 8.

On the extrinsic topology of Lagrangian submanifolds

We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone Lagrangian submanifold vanishes in the homology of the ambient symplectic manifold. Combining this with spectral invariants we provide a new mechanism for proving Lagrangian intersection results e.g. entailing that any two simply connected Lagrangian submanifold in $\CP^n\times\CP^n$ must intersect.

Geometrically simply connected 4-manifolds and stable cohomotopy Seiberg-Witten invariants

We show that every positive definite closed 4-manifold with $b_2^+>1$ and without 1-handles has a vanishing stable cohomotopy Seiberg-Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented 4-manifold with $b_2^+\not\equiv 1$ and $b_2^-\not\equiv 1\pmod{4}$ and without 1-handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the 1-handle condition, we prove these results under more general conditions which are much easier to verify.

Symplectic surfaces in a fixed homology class

The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there are many instances when K can be representated by infinitely many non-isotopic symplectic tori.

On the geography of symplectic 4-manifolds with divisible canonical class

In this article we consider a version of the geography question for simply-connected symplectic 4-manifolds that takes into account the divisibility of the canonical class as an additional parameter. We also find new examples of 4-manifolds admitting several symplectic structures, inequivalent under deformation and self-diffeomorphisms of the manifold.

Symplectic Tori in Homotopy E(1)'s

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic 4-manifold E(1)_K, obtained by knot surgery on the rational elliptic surface E(1) with the left-handed trefoil knot K. E(1)_K has the simplest homotopy type among simply-connected symplectic 4-manifolds known to exhibit such property.

On moduli spaces of symplectic forms

We prove that, for any n, there are simply-connected four-manifolds which admit n-tuples of symplectic forms whose first Chern classes have pairwise different divisibilities in integral cohomology. It follows that the moduli space of symplectic forms modulo diffeomorphisms on such a manifold has at least n connected components.

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.

Rational surfaces and symplectic 4-manifolds with one basic class

We present constructions of simply connected symplectic 4-manifolds which have (up to sign) one basic class and which fill up the geographical region between the half-Noether and Noether lines.

On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups

We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.

Diffeomorphism of simply connected algebraic surfaces

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation equivalent, and proving their diffeomorphism, we give a counterexample to a weaker form of the speculation DEF = DIFF of R. Friedman and J. Morgan, i.e., in the case where (by M. Freedman's theorem) the topological type is completely determined by the numerical invariants of the surface. We hope that the methods of proof may turn out to be quite useful to s…

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…

Inequivalent contact structures on Boothby-Wang 5-manifolds

We consider contact structures on simply-connected 5-manifolds which arise as circle bundles over simply-connected symplectic 4-manifolds and show that invariants from contact homology are related to the divisibility of the canonical class of the symplectic structure. As an application we find new examples of inequivalent contact structures in the same equivalence class of almost contact structures with non-zero first Chern class.

Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures

We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9χ_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic $4$-manifolds constructed in \cite{AP2, AS}. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1)CP^{2}\#(2n-1)\bar{CP}^{2}$ for each integer $n \geq 9$, (ii) the families of simply connecte…

Webs of Lagrangian Tori in Projective Symplectic Manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville's. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt's theory of subnormal subgroups.

Formality and the Lefschetz property in symplectic and cosymplectic geometry

We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the $b_1=1$ case (in the coKähler/cosymplectic situation).

A note on embeddings of 3-manifolds in symplectic 4-manifolds

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…

Geography of symplectic 4-manifolds with Kodaira dimension one

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples.

Homogeneous Symplectic Spaces and Central Extensions

We summarise recent work (arXiv:2203.07405 [math.SG]) on the classical result of Kirillov that any simply-connected homogeneous symplectic space of a connected group $G$ is a hamiltonian $\widehat{G}$-space for a one-dimensional central extension $\widehat{G}$ of $G$, and is thus (by a result of Kostant) a cover of a coadjoint orbit of $\widehat{G}$. We emphasise that existing proofs in the literature assume that $G$ is simply-connected and that this assumption can be removed by application of a theorem of Neeb. We also interpret Neeb's theorem as relating the integrability of one-dimensional …

Symplectic groupoids for cluster manifolds

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and Mărcuţ. These groupoid charts and their analogues for the symplectic double and blow-up groupoids are glued by lifting the cluster mutations to …

The diversity of symplectic Calabi-Yau six-manifolds

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to Kähler manifolds with c_1 = 0. The construction begins with a certain orientable four-dimensional hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the re…

A K-contact simply connected 5-manifold with no semi-regular Sasakian structure

We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus…

A simply connected surface of general type with p_g=0 and K^2=3

Motivated by a recent result of Y. Lee and the second author[7], we construct a simply connected minimal complex surface of general type with p_g=0 and K^2=3 using a rational blow-down surgery and Q-Gorenstein smoothing theory. In a similar fashion, we also construct a new simply connected symplectic 4-manifold with b_2^+=1 and K^2=4.

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。

The growth rate of symplectic homology and affine varieties

We will show that the cotangent bundle of a manifold whose free loopspace homology grows exponentially is not symplectomorphic to any smooth affine variety. We will also show that the unit cotangent bundle of such a manifold is not Stein fillable by a Stein domain whose completion is symplectomorphic to a smooth affine variety. For instance, these results hold for end connect sums of simply connected manifolds whose cohomology with coefficients in some field has at least two generators. We use an invariant called the growth rate of symplectic homology to prove this result.