Journal of Symplectic Geometry Articles (Project Euclid)

Spectral measures on toric varieties and the asymptotic expansion of Tian-Yau-Zelditch

Thu, 05 Aug 2010 15:41 EDT

Rosa Sena-Dias

Source: J. Symplectic Geom., Volume 8, Number 2, 119--142.

Abstract:
We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety.We show how this extended result together with the Tian–Yau–Zelditch asymptotic expansion can be used to deduce Abreu’s formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data.

Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids

Thu, 05 Aug 2010 15:41 EDT

Ari Stern

Source: J. Symplectic Geom., Volume 8, Number 2, 225--238.

Abstract:
We present a discrete analog of the recently introduced Hamilton– Pontryagin variational principle in Lagrangian mechanics. This unifies two, previously disparate approaches to discrete Lagrangian mechanics: either using the discrete Lagrangian to define a finite version of Hamilton’s action principle, or treating it as a symplectic generating function. This is demonstrated for a discrete Lagrangian defined on an arbitrary Lie groupoid; the often encountered special case of the pair groupoid (or Cartesian square) is also given as a worked example.

Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

Tue, 07 Sep 2010 09:19 EDT

Yanir A. Rubinstein, Steve Zelditch

Source: J. Symplectic Geom., Volume 8, Number 3, 239--265.

Abstract:
We generalize the results of Song–Zelditch on geodesics in spaces of Kähler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kähler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the $C2$ topology by harmonic maps into the spaces of Bergman metrics. In particular, Wess–Zumino–Witten (WZW) maps, or equivalently solutions of a homogeneous Monge–Ampère equation on the product of the manifold with a Riemann surface with $S1$ boundary admit such approximations. We also show that the Eells–Sampson flow on the space of Kähler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform, and hence it exists for all time and converges.

Compactness for holomorphic curves with switching Lagrangian boundary conditions

Tue, 07 Sep 2010 09:19 EDT

K. Cieliebak, T. Ekholm, J. Latschev

Source: J. Symplectic Geom., Volume 8, Number 3, 267--298.

Abstract:
We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As an important consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.

Integrals of equivariant forms over noncompact symplectic manifolds

Tue, 07 Sep 2010 09:19 EDT

Matvei Libine

Source: J. Symplectic Geom., Volume 8, Number 3, 299--321.

Local Floer homology and the action gap

Tue, 07 Sep 2010 09:19 EDT

Viktor L. Ginzburg, Başak Z. Gürel

Source: J. Symplectic Geom., Volume 8, Number 3, 323--357.

Abstract:
In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. We prove that an isolated fixed point of a diffeomorphism remains isolated for the so-called admissible iterations and that the local Floer homology groups of a Hamiltonian diffeomorphism for such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically degenerate maxima), which plays an important role in the proof of the Conley conjecture. Finally, we apply these results to show that for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian diffeomorphism with isolated fixed points the minimal action gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem is a generalization of the Conley conjecture.

Symmetry of a symplectic toric manifold

Fri, 03 Dec 2010 09:52 EST

Mikiya Masuda

Source: J. Symplectic Geom., Volume 8, Number 4, 359--380.

Positivity of equivariant Schubert classes through moment map degeneration

Fri, 03 Dec 2010 09:52 EST

Catalin Zara

Source: J. Symplectic Geom., Volume 8, Number 4, 381--402.

Complexifications of Morse functions and the directed Donaldson-Fukaya category

Fri, 03 Dec 2010 09:52 EST

Joseph Johns

Source: J. Symplectic Geom., Volume 8, Number 4, 403--500.

On the algebraic independence of Hamiltonian characteristic classes

Fri, 01 Jul 2011 14:46 EDT

Światoslaw Gal, Jarek Kędra, Aleksy Tralle

Source: J. Symplectic Geom., Volume 9, Number 1, 1--10.

Abstract:
We prove that Hamiltonian characteristic classes defined as fibre integrals of powers of the coupling class are algebraically independent for generic coadjoint orbits.

Immersions in a manifold with a pair of symplectic forms

Fri, 01 Jul 2011 14:46 EDT

Mahuya Datta

Source: J. Symplectic Geom., Volume 9, Number 1, 11--32.

Legendrian contact homology and nondestabilizability

Fri, 01 Jul 2011 14:46 EDT

Clayton Shonkwiler, David Shea Vela-Vick

Source: J. Symplectic Geom., Volume 9, Number 1, 33--44.

Abstract:
We provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston–Bennequin invariant is not maximal.

Symplectic mapping class groups of some Stein and rational surfaces

Fri, 01 Jul 2011 14:46 EDT

Jonathan David Evans

Source: J. Symplectic Geom., Volume 9, Number 1, 45--82.

Abstract:
In this paper we compute the homotopy groups of the symplectomorphism groups of the three-, four- and five-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $RP^2$ and of $C^∗ ×C$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on n-strands.

Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions

Fri, 01 Jul 2011 14:46 EDT

Garrett Alston

Source: J. Symplectic Geom., Volume 9, Number 1, 83--106.

The Calabi invariant for some groups of homeomorphisms

Fri, 01 Jul 2011 14:46 EDT

Vincent Humilière

Source: J. Symplectic Geom., Volume 9, Number 1, 107--117.

Abstract:
We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The proof is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Oh and Muller).

Two surfaces in $D^4$ bounded by the same knot

Fri, 01 Jul 2011 14:47 EDT

Andrew Geng

Source: J. Symplectic Geom., Volume 9, Number 2, 119--122.

Abstract:
We exhibit two symplectic surfaces embedded in the 4-ball which bound the same transverse knot, have the same topology (as abstract surfaces), and are distinguished by the fundamental groups of their complements.

Equivariant homology for generating functions and orderability of lens spaces

Fri, 01 Jul 2011 14:47 EDT

Sheila Sandon

Source: J. Symplectic Geom., Volume 9, Number 2, 123--146.

Abstract:
In her PhD thesis, Milin developed a $Z_k$-equivariant version of the contact homology groups constructed in Geometry of contact transformations and domains: orderability vs squeezing, "Geom. Topol." 10 (2006), 1635–1747 and used it to prove a $Z_k$-equivariant contact non-squeezing theorem. In this article, we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in Contact homology, capacity and non-squeezing in $R^2n × S^1$ via generating functions, "Ann. Inst. Fourier (Grenoble)" 61 (2011), 145–185. As Milin showed, this result implies orderability of lens spaces.

Negative inflation and stability in symplectomorphism groups of ruled surfaces

Fri, 01 Jul 2011 14:47 EDT

Olguţa Buşe

Source: J. Symplectic Geom., Volume 9, Number 2, 147--160.

Tamed to compatible: symplectic forms via moduli space integration

Fri, 01 Jul 2011 14:47 EDT

Clifford Henry Taubes

Source: J. Symplectic Geom., Volume 9, Number 2, 161--250.

Abstract:
Fix a compact 4-dimensional manifold with self-dual second Betti number one and with a given symplectic form. This article proves the following: The Frêchet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.

Maslov index formulas for Whitney $n$-gons

Fri, 01 Jul 2011 14:47 EDT

Sucharit Sarkar

Source: J. Symplectic Geom., Volume 9, Number 2, 251--270.

Abstract:
In this short article, we find an explicit formula for Maslov index of Whitney $n$-gons joining intersections points of $n$ half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an $n$-gon with the fat diagonal in the symmetric product.

Non-displaceable contact embeddings and infinitely many leaf-wise intersections

Mon, 11 Jul 2011 08:55 EDT

Peter Albers, Mark McLean

Source: J. Symplectic Geom., Volume 9, Number 3, 271--284.

Abstract:
We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leafwise intersection points. Moreover, any Stein filling of dimension at least six has infinite-dimensional symplectic homology.

Differentiable stacks and gerbes

Mon, 11 Jul 2011 08:55 EDT

Kai Behrend, Ping Xu

Source: J. Symplectic Geom., Volume 9, Number 3, 285--341.

Abstract:
We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern–Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier–Douady classes in terms of analog of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.

Toric geometry of convex quadrilaterals

Mon, 11 Jul 2011 08:55 EDT

Eveline Legendre

Source: J. Symplectic Geom., Volume 9, Number 3, 343--385.

Abstract:
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler–Einstein and toric Sasaki–Einstein metrics constructed. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler–Einstein ones, and show that for a toric orbi-surface with four fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

Polynomial Poisson structures on affine solvmanifolds

Mon, 11 Jul 2011 08:55 EDT

Alberto Medina, Mohamed Boucetta

Source: J. Symplectic Geom., Volume 9, Number 3, 387--401.

On cohomological decomposition of almost-complex manifolds and deformations

Mon, 11 Jul 2011 08:55 EDT

Daniele Angella, Adriano Tomassini

Source: J. Symplectic Geom., Volume 9, Number 3, 403--428.

Cohomologically symplectic solvmanifolds are symplectic

Tue, 28 Feb 2012 09:57 EST

Hisashi Kasuya

Source: J. Symplectic Geom., Volume 9, Number 4, 429--434.

Abstract:
We consider aspherical manifolds with torsion-free virtually polycyclic fundamental groups, constructed by Baues. We prove that if those manifolds are cohomologically symplectic then they are symplectic. As a corollary we show that cohomologically symplectic solvmanifolds are symplectic.

On the extistence of symplectic realizations

Tue, 28 Feb 2012 09:57 EST

Marius Crainic, Ioan Mǎrcuţ

Source: J. Symplectic Geom., Volume 9, Number 4, 435--444.

Abstract:
We present a new proof of the existence of symplectic realizations of a Poisson manifold $(M,\pi)$. The proof consists in the construction of a symplectic form on an open neighborhood of the zero-section of the cotangent bundle of $M$, defined by an explicit global formula.

Extended flux maps on surfaces and the contracted Johnson homomorphism

Tue, 28 Feb 2012 09:57 EST

Matthew B. Day

Source: J. Symplectic Geom., Volume 9, Number 4, 445--482.

Abstract:
On a closed symplectic surface $\Sigma$ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group $\operatorname{Symp}(\Sigma)$ to the cohomology group $H^1(\Sigma;\mathbb{R})$ that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry.

Floer trajectories with immersed nodes and scale-dependent gluing

Tue, 28 Feb 2012 09:57 EST

Yong-Guen Oh, Ke Zhu

Source: J. Symplectic Geom., Volume 9, Number 4, 483--636.

Abstract:
Development of pseudo-holomorphic curves and Floer homology in symplectic topology has led to moduli spaces of pseudo-holomorphic curves consisting of both “smooth elements” and “spiked elements”, where the latter are combinations of $J$-holomorphic curves (or Floer trajectories) and gradient flow line segments. In many cases the “spiked elements” naturally arise under adiabatic degeneration of “smooth elements” which gradually go through thick–thin decomposition . The reversed process, the recovering problem of the “smooth elements” from “spiked elements” is recently of much interest. In this paper, we define an enhanced compactification of the moduli space of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the one-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of nonzero length will be treated elsewhere. An immediate application of our result is a complete proof of the isomorphism property of the PSS map: a proof of this isomorphism property was outlined by Piunikhin–Salamon–Schwarz in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in "Lagrangian intersection Floer theory-anomaly and obstruction," in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves ‘adiabatic’ parameters or rescaling of the targets.

Degenerate maxima in Hamiltonian systems

Tue, 27 Mar 2012 08:57 EDT

Mike Chance

Source: J. Symplectic Geom., Volume 10, Number 1, 1--16.

Abstract:
In this paper, we explore loops of non-autonomous Hamiltonian diffeomorphisms with degenerate fixed maxima. We show that such loops can not have totally degenerate fixed global maxima. This has applications for the Hofer geometry of the group of Hamiltonians for certain symplectic four manifolds and also gives criteria for certain four manifolds to be uniruled.

Non-symplectic actions on complex projective spaces

Tue, 27 Mar 2012 08:57 EDT

Marek Kaluba, Wojciech Politarczyk

Source: J. Symplectic Geom., Volume 10, Number 1, 17--26.

Abstract:
We construct smooth actions of arbitrary compact Lie groups on complex projective spaces, such that the corresponding transformations arising from the group action do not preserve any symplectic structure on the complex projective space.

On the wrapped Fukaya category and based loops

Tue, 27 Mar 2012 08:57 EDT

Mohammed Abouzaid

Source: J. Symplectic Geom., Volume 10, Number 1, 27--79.

Abstract:
Given an exact relatively Pin Lagrangian embedding $Q \subset M$, we construct an $A^∞$ restriction functor from the wrapped Fukaya category of $M$ to the category of modules on the differential graded algebra of chains over the based loop space of $Q$. If $M$ is the cotangent bundle of $Q$, this functor induces an $A^∞$ equivalence between the wrapped Floer cohomology of a cotangent fibre and the chains over the based loop space of $Q$, extending a result proved by Abbondandolo and Schwarz at the level of homology.

Periodic Floer homology and Seiberg-Witten-Floer cohomology

Tue, 27 Mar 2012 08:57 EDT

Yi-Jen Lee, Clifford Henry Taubes

Source: J. Symplectic Geom., Volume 10, Number 1, 81--164.

Abstract:
Various Seiberg–Witten–Floer cohomologies are defined for a closed, oriented three-manifold; and if it is the mapping torus of an areapreserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg–Witten–Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.

On Legendrian surgeries between lens spaces

Thu, 07 Jun 2012 15:13 EDT

Olga Plamenevskaya

Source: J. Symplectic Geom., Volume 10, Number 2, 165--182.

Abstract:
We obtain some obstructions to existence of Legendrian surgeries between tight lens spaces. We also study Legendrian surgeries between overtwisted contact manifolds.

The Conley conjecture for irrational symplectic manifolds

Thu, 07 Jun 2012 15:13 EDT

Doris Hein

Source: J. Symplectic Geom., Volume 10, Number 2, 183--202.

Abstract:
We prove a generalization of the Conley conjecture: every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, this removes the rationality condition from similar theorems by Ginzburg and Gürel. The proof in the irrational case involves several new ingredients including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. We also develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition, where one of the summands only depends on the behavior of the Hamiltonian in a fixed open set.

Tamed Symplectic forms and Strong Kahler with torsion metrics

Thu, 07 Jun 2012 15:13 EDT

Nicola Enrietti, Anna Fino, Luigi Vezzoni

Source: J. Symplectic Geom., Volume 10, Number 2, 203--223.

Abstract:
Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial\bar{\partial}$-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if $M$ is a nilmanifold (not a torus) endowed with an invariant complex structure $J$, then $(M,J)$ does not admit any symplectic form taming $J$. Moreover, we show that if a nilmanifold $M$ endowed with an invariant complex structure $J$ admits an SKT metric, then $M$ is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

Symplectic reduction of quasi-morphisms and quasi-states

Thu, 07 Jun 2012 15:13 EDT

Matthew Strom Borman

Source: J. Symplectic Geom., Volume 10, Number 2, 225--246.

Abstract:
We prove that quasi-morphisms and quasi-states on a closed rational symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.

A product formula for Gromov-Witten invariants

Thu, 07 Jun 2012 15:13 EDT

Clément Hyvrier

Source: J. Symplectic Geom., Volume 10, Number 2, 247--324.

Abstract:
We establish a product formula for Gromov–Witten invariants for closed relatively semi-positive Hamiltonian fibrations, with connected fiber, and over any connected symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked points in the total space of the Hamiltonian fibration to the corresponding moduli space of pseudo-holomorphic maps with marked points in the base. We use this induced map to recover the product formula by means of integration. Finally, we give applications to c-splitting and symplectic uniruledness.

Fukaya $A_\infty$-structures associated to Lefschetz fibrations. I

Tue, 16 Oct 2012 09:01 EDT

Paul Seidel

Source: J. Symplectic Geom., Volume 10, Number 3, 325--388.

Abstract:
We consider the Fukaya-Floer $A_\infty$-structures arising from a basis of thimbles in a Lefschetz fibration. Particular attention is paid to the relation between these structures in the total space of the fibration, and their counterparts in the fibre.

Spin-quantization commutes with reduction

Tue, 16 Oct 2012 09:01 EDT

Paul-Emile Paradan

Source: J. Symplectic Geom., Volume 10, Number 3, 389--422.

Abstract:
In this paper, we prove that the “quantization commutes with reduction” phenomenon of Guillemin and Sternberg applies in the context of the metaplectic correction.

$U$-action on perturbed Heegaard Floer homology

Tue, 16 Oct 2012 09:01 EDT

Zhongtao Wu

Source: J. Symplectic Geom., Volume 10, Number 3, 423--445.

Abstract:
This paper has two purposes. First, as a continuation of "Perturbed Floer homology of some fibered three manifolds," we apply a similar method to compute the perturbed $HF^+$ for some special classes of fibered three-manifolds in the second highest $spin^c$-structures, including the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of curves. Second, we establish an adjunction inequality for the perturbed Heegaard Floer homology, which indicates a potential connection between the $U$-action on the homology group and the Thurston norm of a three-manifold. As an application, we find the $U$-action on the perturbed $HF^+$ of the above classes of fibered three-manifolds is trivial.

Classification of $\mathbb{Q}$-trivial Bott manifolds

Tue, 16 Oct 2012 09:01 EDT

Suyoung Choi, Mikiya Masuda

Source: J. Symplectic Geom., Volume 10, Number 3, 447--461.

Abstract:
A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\mathbb{C}P^1$-bundle starting with a point, where each $\mathbb{C}P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A $\mathbb{Q}$- trivial Bott manifold of dimension $2n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(\mathbb{C}P^1)^n$ with $\mathbb{Q}$-coefficients. We find all diffeomorphism types of $\mathbb{Q}$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $\mathbb{Z}$-coefficients. As a consequence, the number of diffeomorphism classes of $\mathbb{Q}$-trivial Bott manifolds of dimension $2n$ is equal to the number of partitions of $n$. We even show that any cohomology ring isomorphism between two $\mathbb{Q}$-trivial Bott manifolds is induced by a diffeomorphism.

On the regularization of the Kepler problem

Tue, 16 Oct 2012 09:01 EDT

Gert Heckman, Tim de Laat

Source: J. Symplectic Geom., Volume 10, Number 3, 463--473.

Abstract:
In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in $\mathbb{R}^n$ for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in $\mathbb{R}^{n+1}$, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat. In this paper, we derive the Ligon–Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.

A geometric refinement of a theorem of Chekanov

Tue, 16 Oct 2012 09:01 EDT

Françis Charette

Source: J. Symplectic Geom., Volume 10, Number 3, 475--491.

Abstract:
We prove a conjecture of Barraud and Cornea in the monotone setting, refining a result of Chekanov on the Hofer distance between two Hamiltonian isotopic Lagrangian submanifolds.

Symplectic Genric Complex Structures on Four-Manifolds with $b_+ = 1$

Wed, 02 Jan 2013 14:03 EST

Paolo Cascini, Dmitri Panov

Source: J. Symplectic Geom., Volume 10, Number 4, 493--502.

Abstract:
We study symplectic structures on Kähler surfaces with $p_g = 0$. We give an example of a projective surface which admits a symplectic structure which is not compatible with any Kähler metric.

Real points of coarse moduli schemes of vector bundles on a real algebraic curve

Wed, 02 Jan 2013 14:03 EST

Florent Schaffhauser

Source: J. Symplectic Geom., Volume 10, Number 4, 503--534.

Abstract:
We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semistable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a complex projective variety. We relate our point of view to previous work by Biswas et al. , and we use this to study the $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-action $[\mathcal{E}] \mapsto [\overline{\sigma^*E}]$ on moduli varieties of stable holomorphic bundles on a complex curve with given real structure $\sigma$. We show in particular a Harnack-type theorem, bounding the number of connected components of the fixed-point set of that action by $2^g + 1$, where $g$ is the genus of the curve. In fact, taking into account all the topological invariants of $\sigma$, we give an exact count of the number of connected components, thus generalizing to rank $r \gt 1$ the results of Gross and Harris on the Picard scheme of a real algebraic curve.

Noncommutative integrability and action-angle variables in contct geometry

Wed, 02 Jan 2013 14:03 EST

Božidar Jovanović

Source: J. Symplectic Geom., Volume 10, Number 4, 535--561.

Abstract:
We introduce a notion of the noncommutative integrability within a framework of contact geometry.

L∞-algebras and higher analogues of Dirac sturctures and Courant albegroids

Wed, 02 Jan 2013 14:03 EST

Marco Zambon

Source: J. Symplectic Geom., Volume 10, Number 4, 563--599.

Abstract:
We define a higher analogue of Dirac structures on a manifold $M$. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on $M$, and are equivalent to (and simpler to handle than) the multi-Dirac structures recently introduced in the context of field theory by Vankerschaver et al. We associate an $L_\infty$-algebra of observables to every higher Dirac structure, extending work of Baez et al. on multisymplectic forms. Further, applying a recent result of Getzler, we associate an $L_\infty$-algebra to any manifold endowed with a closed differential form $H$, via a higher analogue of split Courant algebroid twisted by $H$. Finally, we study the relations between the $L_\infty$-algebras appearing above.

On the growth rate of Leaf-Wise intersections

Wed, 02 Jan 2013 14:03 EST

Leonardo Macarini, Will J. Merry, Gabriel P. Paternain

Source: J. Symplectic Geom., Volume 10, Number 4, 601--653.

Abstract:
We define a new variant of Rabinowitz Floer homology that is particularly well suited to studying the growth rate of leaf-wise intersections. We prove that for closed manifolds $M$ whose loop space $\Lambda M$ is "complicated", if $\Sigma \subseteq T^*M$Σ⊆ T*M is a non-degenerate fibrewise starshaped hypersurface and $\varphi \in \mathrm{Ham}_c (T^*M,\omega)$ is a generic Hamiltonian diffeomorphism then the number of leaf-wise intersection points of $\varphi$ in $\Sigma$ grows exponentially in time. Concrete examples of such manifolds are $(S^2 \times S^2)\#(S^2\#S^2)$, $\mathbb{T}^4\#\mathbb{C}P^2$, or any surface of genus greater than one.

On regular Courant algebroids

Fri, 01 Mar 2013 09:05 EST

Zhuo Chen, Mathieu Stiénon, Ping Xu

Source: J. Symplectic Geom., Volume 11, Number 1, 1--24.

Abstract:
For any regular Courant algebroid, we construct a characteristic class à la Chern–Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Ševera class in $H^3_{dR}(M)$. On the other hand, when the Courant algebroid is a quadratic Lie algebra $\mathfrak{g}$, it coincides with the class of the Cartan 3-form in $H^3(\mathfrak{g})$. We also give a complete classification of regular Courant algebroids and discuss its relation to the characteristic class.

A new bound on the size of symplectic 4-manifolds with prescribed fundamental group

Fri, 01 Mar 2013 09:05 EST

Jonathan T. Yazinski

Source: J. Symplectic Geom., Volume 11, Number 1, 25--36.

Abstract:
Given any finitely presented group with $g$ generators and $r$ relations, we produce a symplectic 4-manifold of Euler characteristic $10+4(g+r)$ and signature $−2$. This is an improvement on the result in S. Baldridge and P. Kirk, On symplectic 4-manifolds with prescribed fundamental group , and our construction utilizes a construction in R. Fintushel, B. Doug Park and R. J. Stern, Reverse engineering small 4-manifolds .

Uniqueness of generating Hamiltonians for topological Hamiltonian flows

Fri, 01 Mar 2013 09:05 EST

Lev Buhovsky, Sobhan Seyfaddini

Source: J. Symplectic Geom., Volume 11, Number 1, 37--52.

Abstract:
We prove that a topological Hamiltonian flow as defined by Oh and Müller, has a unique $L^{1,\infty}$ generating topological Hamiltonian function. This answers a question raised by Oh and Müller, and improves a result of Viterbo.

2-plectic geometry, Courant algebroids, and categorified prequantization

Fri, 01 Mar 2013 09:05 EST

Christopher L. Rogers

Source: J. Symplectic Geom., Volume 11, Number 1, 53--91.

Abstract:
A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry one finds the higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to “coherent chain homotopy”. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid. This Courant algebroid is the 2-plectic analogue of a transitive Lie algebroid over a symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1-forms. Furthermore, we prove that it is quasi-isomorphic to a central extension of the (trivial) Lie 2-algebra of Hamiltonian vector fields, and therefore is the higher analogue of the well-known Kostant–Souriau central extension in symplectic geometry. We interpret all of these results within the context of a categorified prequantization procedure for 2-plectic manifolds. In doing so, we describe how $U(1)$-gerbes, equipped with a connection and curving, and Courant algebroids are the 2-plectic analogues of principal $U(1)$ bundles equipped with a connection and their associated Atiyah Lie algebroids.

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Fri, 01 Mar 2013 09:05 EST

Kyousuke Uchino

Source: J. Symplectic Geom., Volume 11, Number 1, 93--108.

Abstract:
Given a Lie algebra, there uniquely exists a Poisson algebra that is called a Lie–Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday–Poisson algebras. In the super/graded cases, the Loday–Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten–Nijenhuis bracket. It will be shown that the Loday–Poisson algebras form a special subclass of Aguiar’s dual-prePoisson algebras. We also study a problem of deformation quantization over the Loday–Poisson algebra. It will be shown that the polynomial Loday–Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday’s associative dialgebra.

Polarizations and symplectic isotopies

Fri, 01 Mar 2013 09:05 EST

Emmanuel Opshtein

Source: J. Symplectic Geom., Volume 11, Number 1, 109--133.

Abstract:
The aim of this paper is to explain a link between symplectic isotopies of open objects such as balls and flexibility properties of symplectic hypersurfaces. We get connectedness results for spaces of symplectic ellipsoids or maximal packings of $\mathbb{P}^2$.

The annulus property of simple holomorphic discs

Fri, 01 Mar 2013 09:05 EST

Kai Zehmisch

Source: J. Symplectic Geom., Volume 11, Number 1, 135--161.

Abstract:
We show that any simple holomorphic disc admits the annulus property, i.e., each interior point is surrounded by an arbitrary small annulus consisting entirely of injective points. As an application we show that interior singularities of holomorphic discs can be resolved after slight perturbation of the almost complex structure. Moreover, for boundary points the analogue notion, the half-annulus property, is introduced and studied in detail.