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Applications of the full Kostant–Toda lattice and hyper-functions to unitary representations of the Heisenberg groups J. Symplectic Geom. (IF 0.7) Pub Date : 2023-12-22 Kaoru Ikeda
We consider a new orbit method for unitary representations which determines the explicit values of the multiplicities of the irreducible components of unitary representations of the connected Lie groups. We provide the polarized symplectic affine space on which the Lie group acts. This polarization is obtained by the Hamiltonian flows of the full Kostant–Toda lattice. The Hamiltonian flows of the ordinary
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Generalizations of planar contact manifolds to higher dimensions J. Symplectic Geom. (IF 0.7) Pub Date : 2023-12-22 Bahar Acu, John B. Etnyre, Burak Ozbagci
Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions.We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer $m \gt 3$, any finitely presented group can be realized
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On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications) J. Symplectic Geom. (IF 0.7) Pub Date : 2023-12-22 Urs Frauenfelder, Agustin Moreno
In this article, we will introduce a collection of tools aimed at studying periodic orbits of Hamiltonian systems, their (linear) stability, and their bifurcations. We will provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by anti-symplectic involutions (e.g. the circular restricted three-body problem). Such cylinders induce continuous
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$T$-duality for transitive Courant algebroids J. Symplectic Geom. (IF 0.7) Pub Date : 2023-12-22 Vicente Cortés, Liana David
We develop a theory of $T$-duality for transitive Courant algebroids. We show that $T$-duality between transitive Courant algebroids $E \to M$ and $\tilde{E} \to \tilde{M}$ induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles $\mathbb{S}_E$ and $\mathbb{S}_\tilde{E}$ intertwining the canonical Dirac generating operators. The map is shown to induce an isomorphism
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Toric generalized Kähler structures. II J. Symplectic Geom. (IF 0.7) Pub Date : 2023-09-28 Yicao Wang
Anti-diagonal toric generalized Kähler (GK) structures of symplectic type on a compact toric symplectic manifold were investigated in [$\href{https://doi.org/10.48550/arXiv.1811.06848}{18}$]. In this article, we consider general toric GK structures of symplectic type, without requiring them to be anti-diagonal. Such a structure is characterized by a triple $(\tau,C,F)$ where $\tau$ is a strictly convex
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Simplified SFT moduli spaces for Legendrian links J. Symplectic Geom. (IF 0.7) Pub Date : 2023-09-28 Russell Avdek
We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ to $\mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^3, \xi_{std})$. We allow our domains, $\dot{\Sigma}$ , to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$
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Arboreal models and their stability J. Symplectic Geom. (IF 0.7) Pub Date : 2023-09-28 Daniel Álvarez-Gavela, Yakov Eliashberg, David Nadler
The main result of this paper is the uniqueness of local arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. A parametric version implies that the space of germs of symplectomorphisms that preserve the local model is weakly homotopy equivalent to the space of automorphisms of the corresponding
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Realising perfect derived categories of Auslander algebras of type $\mathbb{A}$ as Fukaya–Seidel categories J. Symplectic Geom. (IF 0.7) Pub Date : 2023-09-28 Ilaria Di Dedda
We prove that the Fukaya–Seidel categories of a certain family of Lefschetz fibrations on $\mathbb{C}^2$ are equivalent to the perfect derived categories of Auslander algebras of Dynkin type $\mathbb{A}$. We give an explicit equivalence between these categories and the partially wrapped Fukaya categories considered in [$\href{https://doi.org/10.1017/fms.2021.2}{9}$]. We provide a complete description
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Maurer–Cartan deformation of Lagrangians J. Symplectic Geom. (IF 0.7) Pub Date : 2023-07-27 Hansol Hong
The Maurer–Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer–Cartan algebra with the 0‑th cohomology of the Koszul dual dga of $CF(L,L;\Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer–Cartan algebra of $L$ and a suitable
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Cohomologies of complex manifolds with symplectic $(1,1)$-forms J. Symplectic Geom. (IF 0.7) Pub Date : 2023-07-27 Adriano Tomassini, Xu Wang
$\def\partialol{\bar{\partial}}$Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $\omega$. Then we have a natural double complex $\partialol+ \partialol^\Lambda$, where $\partialol^\Lambda$ denotes the symplectic adjoint of the $\partialol$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic
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First steps in twisted Rabinowitz–Floer homology J. Symplectic Geom. (IF 0.7) Pub Date : 2023-07-27 Yannis Bähni
Rabinowitz–Floer homology is the Morse–Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz–Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric
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Lagrangian cobordisms between enriched knot diagrams J. Symplectic Geom. (IF 0.7) Pub Date : 2023-07-27 Ipsita Datta
In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched knot diagrams for generic smooth links. The existence of Lagrangian cobordisms gives a well-defined transitive relation on equivalence classes of enriched knot diagrams
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Genus-one complex quantum Chern–Simons theory J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-26 Jørgen Ellegaard Andersen, Alessandro Malusà, Gabriele Rembado
We consider the geometric quantisation of Chern–Simons theory for closed genus-one surfaces and semisimple complex algebraic groups. First we introduce the natural complexified analogue of the Hitchin connection in Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach
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Poisson maps between character varieties: gluing and capping J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-26 Indranil Biswas, Jacques Hurtubise, Lisa C. Jeffrey, Sean Lawton
Let $G$ be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between $G$-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when $G = \mathrm{SL} (2, \mathbb{C})$. We demonstrate this algorithm by explicitly
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Every real $3$-manifold is real contact J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-26 Merve Cengiz, Ferit Öztürk
A real $3$-manifold is a smooth $3$-manifold together with an orientation preserving smooth involution, which is called a real structure. A real contact $3$-manifold is a real $3$-manifold with a contact distribution that is antisymmetric with respect to the real structure. We show that every real $3$-manifold can be obtained via surgery along invariant knots starting from the standard real $3$ and
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Polyhedral approximation by Lagrangian and isotropic tori J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-26 Yann Rollin
We prove that every smoothly immersed $2$-torus of $\mathbb{R}^4$ can be approximated, in the $C^0$-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the $C^1$-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic $2$-tori of $\mathbb{R}^{2n}$
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On the minimal symplectic area of Lagrangians J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-26 Zhengyi Zhou
We show that the minimal symplectic area of Lagrangian submanifolds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a $k$-semi-dilation, then the minimal symplectic area is universally bounded for $K(\pi,1)$-Lagrangians. As a corollary, we show that the Arnol’d chord conjecture holds for the following four cases: (1) $Y$ admits
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Caustics of Lagrangian homotopy spheres with stably trivial Gauss map J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Daniel Álvarez-Gavela, David Darrow
For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $\pi_n U_n / O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h‑principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably
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Contact $(+1)$-surgeries on rational homology $3$-spheres J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Fan Ding, Youlin Li, Zhongtao Wu
In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology $3$-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(\pm 1)$-surgeries along Legendrian links in the standard contact $3$-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian twocomponent
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Nonexistence of exact Lagrangian tori in affine conic bundles over $\mathbb{C}^n$ J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Yin Li
Let $M \subset \mathbb{C}^{n+1}$ be a smooth affine hypersurface defined by the equation $xy + p(z_1, \dotsm , z_{n-1}) = 1$, where $p$ is a Brieskorn–Pham polynomial and $n \geq 2$. We prove that if $L \subset M$ is a closed, orientable, exact Lagrangian submanifold, then $L$ cannot be a $K(\pi,1)$ space. The key point of the proof is to establish a version of homological mirror symmetry for the wrapped
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A Poisson bracket on the space of Poisson structures J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Thomas Machon
Let $M$ be a smooth, closed, orientable manifold and $\mathcal{P}(M)$ the set of Poisson structures on $M$. We construct a Poisson bracket for a class of admissible functions on $\mathcal{P}(M)$, depending on a choice of volume form for $M$. The Hamiltonian flow of the bracket acts on $\mathcal{P}(M)$ by volume-preserving diffeomorphisms of $M$, corresponding to exact gauge transformations. Fixed points
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Epsilon-non-squeezing and $C^0$-rigidity of epsilon-symplectic embeddings J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Stefan Müller
An embedding $\varphi : (M_1, \omega_1) \to (M_2, \omega_2)$ (of symplectic manifolds of the same dimension) is called $\epsilon$-symplectic if the difference $\varphi^\ast \omega_2 - \omega_1$ is $\epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $\epsilon$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the
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A max inequality for spectral invariants of disjointly supported Hamiltonians J. Symplectic Geom. (IF 0.7) Pub Date : 2023-04-24 Shira Tanny
We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by Humilière, Le Roux and Seyfaddini. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich’s Poisson bracket invariant and to Entov and Polterovich’s notion of superheavy sets.
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Self-crossing stable generalized complex structures J. Symplectic Geom. (IF 0.7) Pub Date : 2023-03-16 Gil R. Cavalcanti, Ralph L. Klaasse, Aldo Witte
We extend the notion of (smooth) stable generalized complex structures to allow for an anticanonical section with normal self-crossing singularities. This weakening not only allows for a number of natural examples in higher dimensions but also sheds some light into the smooth case in dimension four: in this dimension there is a natural connected sum construction for these structures as well as a smoothing
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Twisted cyclic group actions on Fukaya categories and mirror symmetry J. Symplectic Geom. (IF 0.7) Pub Date : 2023-03-16 Chi Hong Chow, Naichung Conan Leung
Let $(X, \omega)$ be a compact symplectic manifold whose first Chern class $c_1(X)$ is divisible by a positive integer $n$. We construct a twisted $\mathbb{Z}_{2n}$-action on its Fukaya category $Fuk(X)$ and a $\mathbb{Z}_n$-action on the local models of its moduli of Lagrangian branes. We show that this action is compatible with the gluing functions for different local models.
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Generating systems and representability for symplectic capacities J. Symplectic Geom. (IF 0.7) Pub Date : 2023-03-16 Dušan Joksimović, Fabian Ziltener
K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the
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Reductive subalgebras of semisimple Lie algebras and Poisson commutativity J. Symplectic Geom. (IF 0.7) Pub Date : 2023-03-16 Dmitri I. Panyushev, Oksana S. Yakimova
Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h} \subset \mathfrak{g}$ a reductive subalgebra such that the orthogonal complement $\mathfrak{h}^\bot$ is a complementary $\mathfrak{h}$-submodule of $\mathfrak{g}$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra $\mathcal{S} (\mathfrak{g})$ by taking the subalgebra $\mathcal{Z}$ generated
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Differential forms, Fukaya $A_\infty$ algebras, and Gromov–Witten axioms J. Symplectic Geom. (IF 0.7) Pub Date : 2023-03-16 Jake P. Solomon, Sara B. Tukachinsky
Consider the differential forms $A^\ast (L)$ on a Lagrangian submanifold $L \subset X$. Following ideas of Fukaya–Oh–Ohta–Ono, we construct a family of cyclic unital curved $A_\infty$ structures on $A^\ast (L)$, parameterized by the cohomology of $X$ relative to $L$. The family of $A_\infty$ structures satisfies properties analogous to the axioms of Gromov–Witten theory. Our construction is canonical
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Chekanov-Eliashberg $\mathrm{dg}$-algebras for singular Legendrians J. Symplectic Geom. (IF 0.7) Pub Date : 2023-02-28 Johan Asplund, Tobias Ekholm
The Chekanov–Eliashberg $\mathrm{dg}$-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams [22]. It also leads to a proof of a conjectured isomorphism [17, 25]
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Splitting formulas for the local real Gromov–Witten invariants J. Symplectic Geom. (IF 0.7) Pub Date : 2023-02-28 Penka Georgieva, Eleny-Nicoleta Ionel
Motivated by the real version of the Gopakumar–Vafa conjecture for $3$-folds, the authors introduced in [GI] the notion of local real Gromov–Witten invariants associated to local $3$-folds over Real curves. This article is devoted to the proof of a splitting formula for these invariants under target degenerations. It is used in [GI] to show that the invariants give rise to a $2$-dimensional Klein TQFT
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Contact categories of disks J. Symplectic Geom. (IF 0.7) Pub Date : 2023-02-28 Ko Honda, Yin Tian
In the first part of the paper we associate a pre-additive category $\mathcal{C}(\Sigma)$ to a closed oriented surface $\Sigma$, called the contact category and constructed from contact structures on $\Sigma \times [0, 1]$. There are also $\mathcal{C}(\Sigma, F)$, where $\Sigma$ is a compact oriented surface with boundary and $F \subset \partial\Sigma$ is a finite oriented set of points which bounds
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Augmentations are sheaves for Legendrian graphs J. Symplectic Geom. (IF 0.7) Pub Date : 2022-12-23 Byung Hee An, Youngjin Bae, Tao Su
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_\infty$-category, which lifts the set of augmentations of the associated Chekanov–Eliashberg DGA, and a DG category of constructible sheaves on the front plane, with micro-support at contact infinity controlled
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Symplectic homology of fiberwise convex sets and homology of loop spaces J. Symplectic Geom. (IF 0.7) Pub Date : 2022-12-23 Kei Irie
For any nonempty, compact and fiberwise convex set $K$ in $T^{\ast} \mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove
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Iso-contact embeddings of manifolds in co-dimension $2$ J. Symplectic Geom. (IF 0.7) Pub Date : 2022-12-23 Dishant M. Pancholi, Suhas Pandit
The purpose of this article is to study co-dimension $2$ iso‑contact embeddings of closed contact manifolds.We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso‑contact embeds in a contact manifold $(N^{2n+1}, \xi_N)$, provided $M$ contact embeds in $(N, \xi_N)$ with trivial normal bundle and the contact structure induced on $M$ via this embedding is overtwisted and homotopic as an
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Removing parametrized rays symplectically J. Symplectic Geom. (IF 0.7) Pub Date : 2022-12-23 Bernd Stratmann
Let $(M, \omega)$ be a symplectic manifold. Let $[0,\infty) \times Q \subset \mathbb{R} \times Q$ be considered as parametrized rays $[0,\infty)$ and let $\varphi : [-1,\infty) \times Q \to M$ be an injective, proper, continuous map immersive on $(-1,\infty) \times Q$. If for the standard vector field $\frac{\partial}{\partial t}$ on $\mathbb{R}$ and any further vector field $\nu$ tangent to $(-1,\infty)
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On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C} \mathrm{P}^d$ via generating functions J. Symplectic Geom. (IF 0.7) Pub Date : 2022-10-21 Simon Allais
Inspired by the techniques of Givental and Théret, we provide a proof with generating functions of a recent result of Ginzburg–Gürel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C} \mathrm{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely
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Ruling invariants for Legendrian graphs J. Symplectic Geom. (IF 0.7) Pub Date : 2022-10-21 Byung Hee An, Youngjin Bae, Tamás Kálmán
We define ruling invariants for even-valent Legendrian graphs in standard contact three-space. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is fourvalent, then the ungraded ruling polynomial appears in Kauffman–Vogel’s graph
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On a systolic inequality for closed magnetic geodesics on surfaces J. Symplectic Geom. (IF 0.7) Pub Date : 2022-10-21 Gabriele Benedetti, Jungsoo Kang
We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close to a Zoll one or large enough.
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The $\log$ symplectic geometry of Poisson slices J. Symplectic Geom. (IF 0.7) Pub Date : 2022-10-21 Peter Crooks, Markus Röser
Let $\mathfrak{g}$ be a complex semisimple Lie algebra with adjoint group $G$. An $\mathfrak{sl}_2$-triple $\tau=(\xi,h,\eta)\in\mathfrak{g}^{\oplus 3}$ and a Poisson Hamiltonian $G$-variety $X$ together yield a distinguished Poisson transversal $X_{\tau}:=\nu^{-1}(\mathcal{S}_{\tau})$, where $\nu:X\longrightarrow\mathfrak{g}$ is the moment map and $\mathcal{S}_{\tau}:= \xi+\mathfrak{g}_{\eta}$ is
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Monopoles and foliations without holonomy-invariant transverse measure J. Symplectic Geom. (IF 0.7) Pub Date : 2022-10-21 Boyu Zhang
This article proves a uniform exponential decay estimate for Seiberg–Witten equations on non-compact $4$‑manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost Kähler (AFAK) structures by Kronheimer and Mrowka [17]. As an application, we construct an invariant for smooth foliations without holonomy-invariant transverse measure,
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Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions J. Symplectic Geom. (IF 0.7) Pub Date : 2022-06-08 Benjamin Delarue, Pablo Ramacher
We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden–Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional
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Knot homologies in monopole and instanton theories via sutures J. Symplectic Geom. (IF 0.7) Pub Date : 2022-06-08 Zhenkun Li
In this paper we construct possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot $K \subset Y$ and a base point $p \in K$, we associate the minus versions, $\underline{\mathrm{KHM}}^- (Y,K,p)$ and $\underline{\mathrm{KHI}}^- (Y,K,p)$, to the triple $(Y,K,p)$. We prove that a Seifert surface of $K$ induces a $\mathbb{Z}$-grading, and there
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Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds J. Symplectic Geom. (IF 0.7) Pub Date : 2022-06-08 Brian Tervil
We prove a version of Sandon’s conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern
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An unoriented skein relation via bordered–sutured Floer homology J. Symplectic Geom. (IF 0.7) Pub Date : 2022-06-08 David Shea Vela-Vick, C.-M. Michael Wong
We show that the bordered–sutured Floer invariant of the complement of a tangle in an arbitrary $3$-manifold $Y$, with minimal conditions on the bordered–sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu [Man07] for links in $S^3$. We give a theoretical proof of this result by adapting holomorphic polygon counts to the bordered–sutured setting
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Generalized chain surgeries and applications J. Symplectic Geom. (IF 0.7) Pub Date : 2022-05-27 Anar Akhmedov, Çağrı Karakurt, Sümeyra Sakallı
We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this
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Distributions associated to almost complex structures on symplectic manifolds J. Symplectic Geom. (IF 0.7) Pub Date : 2022-05-27 Michel Cahen, Maxime Gérard, Simone Gutt, Manar Hayyani
We look at methods to select triples $(M,\omega,J)$ consisting of a symplectic manifold $(M,\omega)$ endowed with a compatible positive almost complex structure $J$, in terms of the Nijenhuis tensor $N^J$ associated to $J$. We study the image distribution $\operatorname{Im} N^J$, which is the span at each point of the values of $N^J$, measuring the non integrability of $J$ by the dimension of $\operatorname{Im}
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Non-degeneracy of the Hofer norm for Poisson structures J. Symplectic Geom. (IF 0.7) Pub Date : 2022-05-27 Dušan Joksimović, Ioan Mărcuţ
We remark that, as in the symplectic case, the Hofer norm on the Hamiltonian group of a Poisson manifold is non-degenerate. The proof follows from the symplectic case after reducing the problem to a symplectic leaf.
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The $\mathbb{Z} /p \mathbb{Z}$-equivariant product-isomorphism in fixed point Floer cohomology J. Symplectic Geom. (IF 0.7) Pub Date : 2022-05-27 Egor Shelukhin, Jingyu Zhao
[THIS ARTICLE IS TEMPORARILY UNAVAILABLE]
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Squared Dehn twists and deformed symplectic invariants J. Symplectic Geom. (IF 0.7) Pub Date : 2022-05-27 Kyler Siegel
We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in [27].
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Fiber Floer cohomology and conormal stops J. Symplectic Geom. (IF 0.7) Pub Date : 2021-12-08 Johan Asplund
Let $S$ be a closed orientable spin manifold. Let $K \subset S$ be a submanifold and denote its complement by $M_K$. In this paper we prove that there exists an isomorphism between partially wrapped Floer cochains of a cotangent fiber stopped by the unit conormal $\Lambda_K$ and chains of a Morse theoretic model of the based loop space of $M_K$, which intertwines the $A_\infty$-structure with the Pontryagin
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Liouville hypersurfaces and connect sum cobordisms J. Symplectic Geom. (IF 0.7) Pub Date : 2021-12-08 Russell Avdek
The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a contact manifold $(M,\xi)$ gives an exact—and in many cases, Weinstein—cobordism whose
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Lagrangian torus invariants using $ECH = SWF$ J. Symplectic Geom. (IF 0.7) Pub Date : 2021-12-08 Chris Gerig
We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a $3$‑torus, associated with Lagrangian tori in symplectic $4$‑manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg–Witten) invariants of various torus surgeries.We then recover a result of Morgan–Mrowka–Szabó on product formulas
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Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells J. Symplectic Geom. (IF 0.7) Pub Date : 2021-12-08 Victor Mouquin
Given a standard complex semisimple Poisson Lie group $(G,\pi_\mathrm{st})$, generalised double Bruhat cells $G^\mathbf{u,v}$ and generalised Bruhat cells $\mathcal{O}^\mathbf{u}$ equipped with naturally defined holomorphic Poisson structures, where $\mathbf{u,v}$ are finite sequences of Weyl group elements, were defined and studied by Jiang-Hua Lu and the author. We prove in this paper that $G^\mathbf{u
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Formally integrable complex structures on higher dimensional knot spaces J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Domenico Fiorenza, Hông Vân Lê
Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let $\operatorname{Imm}_f (S,M)$ the space of all free immersions $\varphi : S \to M$ and let $B^{+}_{i,f} (S,M)$ the quotient space $\operatorname{Imm}_f (S,M) / \operatorname{Diff}^{+} (S)$, where $\operatorname{Diff}^{+} (S)$ denotes the group of orientation preserving diffeomorphisms of $S$
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Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Kei Irie
We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d \lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based
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Real and complex hedgehogs, their symplectic area, curvature and evolutes J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Yves Martinez-Maure
Classical (real) hedgehogs can be regarded as the geometrical realizations of formal differences of convex bodies in $\mathbb{R}^{n+1}$. Like convex bodies, hedgehogs can be identified with their support functions. Adopting a projective viewpoint, we prove that any holomorphic function $h : \mathbb{C}^n \to \mathbb{C}$ can be regarded as the ‘support function’ of a complex hedgehog $\mathcal{H}_h$
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Asymptotic behavior of exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{C}P^2$ as $a+b+c \to \infty$ J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Weonmo Lee, Yong-Geun Oh, Renato Vianna
In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{C}P^2$ associated to Markov triples $(a,b,c)$ described in [Via16]. We first prove that the Gromov capacity of the complement $\mathbb{C}P^2 \setminus T_{a,b,c}$ is greater than or equal to $\frac{1}{3}$ of the area of the complex line for all Markov triple $(a,b,c)$. We then
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Functorial LCH for immersed Lagrangian cobordisms J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Yu Pan, Dan Rutherford
For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in [16], to a class of immersed exact Lagrangian cobordisms by considering their Legendrian lifts as conical Legendrian cobordisms. To a conical Legendrian cobordism $\Sigma$ from $\Lambda_{-}$ to $\Lambda_{+}$, we
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Picard groups of $b$-symplectic manifolds J. Symplectic Geom. (IF 0.7) Pub Date : 2021-07-21 Joel Villatoro
We compute the Picard group of a stable $b$-symplectic manifold $M$ by introducing a collection of discrete invariants $\mathfrak{Gr}$ which classify $M$ up to Morita equivalence.
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Polyfold regularization of constrained moduli spaces J. Symplectic Geom. (IF 0.7) Pub Date : 2021-03-01 Benjamin Filippenko
We introduce tame sc‑Fredholm sections and slices of sc-Fredholm sections. A slice is a notion of subpolyfold that is compatible with the sc‑Fredholm section and has finite locally constant codimension. We prove that the subspace of a tame polyfold that satisfies a transverse sc-smooth constraint in a finite dimensional smooth manifold is a slice of any tame sc‑Fredholm section compatible with the