Journal of Symplectic Geometry Articles (Project Euclid)
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The latest articles from Journal of Symplectic Geometry on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTFri, 03 Dec 2010 09:52 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Spectral measures on toric varieties and the asymptotic expansion of Tian-Yau-Zelditch
http://projecteuclid.org/euclid.jsg/1279199212
<strong>Rosa Sena-Dias</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 8, Number 2, 119--142.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1279199212_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn the growth rate of Leaf-Wise intersectionshttp://projecteuclid.org/euclid.jsg/1357153430<strong>Leonardo Macarini</strong>, <strong>Will J. Merry</strong>, <strong>Gabriel P. Paternain</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 10, Number 4, 601--653.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1357153430_Wed, 02 Jan 2013 14:03 ESTWed, 02 Jan 2013 14:03 ESTOn regular Courant algebroidshttp://projecteuclid.org/euclid.jsg/1362146730<strong>Zhuo Chen</strong>, <strong>Mathieu Stiénon</strong>, <strong>Ping Xu</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1362146730_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTA new bound on the size of symplectic 4-manifolds with prescribed fundamental grouphttp://projecteuclid.org/euclid.jsg/1362146731<strong>Jonathan T. Yazinski</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 25--36.</p><p><strong>Abstract:</strong><br/>
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 .
</p>projecteuclid.org/euclid.jsg/1362146731_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTUniqueness of generating Hamiltonians for topological Hamiltonian flowshttp://projecteuclid.org/euclid.jsg/1362146732<strong>Lev Buhovsky</strong>, <strong>Sobhan Seyfaddini</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 37--52.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1362146732_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 EST2-plectic geometry, Courant algebroids, and categorified prequantizationhttp://projecteuclid.org/euclid.jsg/1362146733<strong>Christopher L. Rogers</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 53--91.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1362146733_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTNoncommutative Poisson brackets on Loday algebras and related deformation quantizationhttp://projecteuclid.org/euclid.jsg/1362146734<strong>Kyousuke Uchino</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 93--108.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1362146734_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTPolarizations and symplectic isotopieshttp://projecteuclid.org/euclid.jsg/1362146735<strong>Emmanuel Opshtein</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 109--133.</p><p><strong>Abstract:</strong><br/>
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 $\mathbb{P}^2$.
</p>projecteuclid.org/euclid.jsg/1362146735_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTThe annulus property of simple holomorphic discshttp://projecteuclid.org/euclid.jsg/1362146736<strong>Kai Zehmisch</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 1, 135--161.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.jsg/1362146736_Fri, 01 Mar 2013 09:05 ESTFri, 01 Mar 2013 09:05 ESTSymplectic rigidity and weak commutativityhttp://projecteuclid.org/euclid.jsg/1384202060<strong>Franco Cardin</strong>, <strong>Simone Vazzoler</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 163--166.</p><p><strong>Abstract:</strong><br/>
We present a new and simple proof of Eliashberg-Gromov's theorem based on the notion of $C^0$-commutativity introduced by Cardin and Viterbo.
</p>projecteuclid.org/euclid.jsg/1384202060_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTThe contact homology of Legendrian knots with maximal Thurston-Bennequin invarianthttp://projecteuclid.org/euclid.jsg/1384202061<strong>Steven Sivek</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 167--178.</p><p><strong>Abstract:</strong><br/>
We show that there exists a Legendrian knot with maximal
Thurston–Bennequin invariant whose contact homology is trivial. We
also provide another Legendrian knot which has the same knot type
and classical invariants but nonvanishing contact homology.
</p>projecteuclid.org/euclid.jsg/1384202061_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTNew techniques for obtaining Schubert-type formulas for Hamiltonian manifoldshttp://projecteuclid.org/euclid.jsg/1384202062<strong>Silvia Sabatini</strong>, <strong>Susan Tolman</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 179--230.</p><p><strong>Abstract:</strong><br/>
In Towards Generalizing Schubert Calculus in the Symplectic Category, Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. The main goal of this
paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $\alpha_p$ in the equivariant
cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these
classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing
the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for
calculating the restrictions of a canonical class $\alpha_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula
is manifestly positive and integral in certain important special cases.
</p>projecteuclid.org/euclid.jsg/1384202062_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTDisplacement of polydisks and Lagrangian Floer theoryhttp://projecteuclid.org/euclid.jsg/1384202063<strong>Kenji Fukaya</strong>, <strong>Yong-Geun Oh</strong>, <strong>Hiroshi Ohta</strong>, <strong>Kaoru Ono</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 231--268.</p><p><strong>Abstract:</strong><br/>
There are two purposes of the present paper. One is to correct an
error in the proof of Theorem 6.1.25 in Lagrangian intersection Floer theory anomaly and obstruction, Part I, and Part II, from which Theorem J
follows. In the course of doing so, we also obtain a new lower bound of the displacement energy of polydisks in general dimension.
</p>projecteuclid.org/euclid.jsg/1384202063_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTPacking numbers of rational ruled four-manifoldshttp://projecteuclid.org/euclid.jsg/1384202064<strong>Olguta Buse</strong>, <strong>Martin Pinsonnault</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 269--316.</p><p><strong>Abstract:</strong><br/>
We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic four-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths,
the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \mathbb{N}$, embeds
in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the embedded contact homology capacities give sharp inequalities
for embedding ellipsoids into polydisks.
</p>projecteuclid.org/euclid.jsg/1384202064_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTCorrigendum for scaling limits for equivariant Szegö kernelshttp://projecteuclid.org/euclid.jsg/1384202065<strong>Roberto Paoletti</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 2, 317--318.</p>projecteuclid.org/euclid.jsg/1384202065_Mon, 11 Nov 2013 15:34 ESTMon, 11 Nov 2013 15:34 ESTSymplectic microgeometry III: monoidshttp://projecteuclid.org/euclid.jsg/1384282839<strong>Alberto S. Cattaneo</strong>, <strong>Benoit Dherin</strong>, <strong>Alan Weinstein</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 319--341.</p><p><strong>Abstract:</strong><br/>
We show that the category of Poisson manifolds and Poisson
maps, the category of symplectic microgroupoids and Lagrangian submicrogroupoids
(as morphisms), and the category of monoids and
monoid morphisms in the microsymplectic category are equivalent symmetric
monoidal categories.
</p>projecteuclid.org/euclid.jsg/1384282839_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTOn exotic monotone Lagrangian tori in $\mathbb{CP}^2$ and $\mathbb{S}_2 \times \mathbb{S}_2$http://projecteuclid.org/euclid.jsg/1384282840<strong>Agnès Gadbled</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 343--361.</p><p><strong>Abstract:</strong><br/>
In this note, we prove that two constructions of exotic monotone Lagrangian tori, namely the one by Chekanov and Schlenk and the one obtained by the circle bundle construction of Biran, are
Hamiltonian isotopic in $\mathbb{CP}^2$ and $\mathbb{S}_2 \times \mathbb{S}_2$.
</p>projecteuclid.org/euclid.jsg/1384282840_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTConvex plumbings and Lefschetz fibrationshttp://projecteuclid.org/euclid.jsg/1384282841<strong>David Gay</strong>, <strong>Thomas E. Mark</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 363--375.</p><p><strong>Abstract:</strong><br/>
We show that under appropriate hypotheses, a plumbing of symplectic
surfaces in a symplectic 4-manifold admits strongly convex neighborhoods.
Moreover the neighborhoods are Lefschetz fibered with an
easily described open book on the boundary supporting the induced
contact structure. We point out some applications to cut-and-paste
constructions of symplectic 4-manifolds.
</p>projecteuclid.org/euclid.jsg/1384282841_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTAffine Yang-Mills-Higgs metricshttp://projecteuclid.org/euclid.jsg/1384282842<strong>Indranil Biswas</strong>, <strong>John Loftin</strong>, <strong>Matthias Stemmler</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 377--404.</p><p><strong>Abstract:</strong><br/>
Let $(E,\varphi)$ be a flat Higgs bundle on a compact special affine manifold $M$ equipped with an affine Gauduchon metric. We prove that $(E,\varphi)$ is polystable if and only if it admits an affine Yang-Mills-Higgs metric.
</p>projecteuclid.org/euclid.jsg/1384282842_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTTopological recursion relations in non-equivariant cylindrical contact homologyhttp://projecteuclid.org/euclid.jsg/1384282843<strong>Oliver Fabert</strong>, <strong>Paolo Rossi</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 405--448.</p><p><strong>Abstract:</strong><br/>
It was pointed out by Eliashberg in his ICM 2006 plenary talk that
the integrable systems of rational Gromov–Witten theory very naturally
appear in the rich algebraic formalism of symplectic field theory
(SFT). Carefully generalizing the definition of gravitational descendants
from Gromov–Witten theory to SFT, one can assign to every contact
manifold a Hamiltonian system with symmetries on SFT homology
and the question of its integrability arises. While we have shown
how the well-known string, dilaton and divisor equations translate from
Gromov–Witten theory to SFT, the next step is to show how genus-zero
topological recursion translates to SFT. Compatible with the example
of SFT of closed geodesics, it turns out that the corresponding localization
theorem requires a non-equivariant version of SFT, which is generated
by parameterized instead of unparameterized closed Reeb orbits.
Since this non-equivariant version is so far only defined for cylindrical
contact homology, we restrict ourselves to this special case. As an
important result we show that, as in rational Gromov–Witten theory,
all descendant invariants can be computed from primary invariants, i.e.,
without descendants.
</p>projecteuclid.org/euclid.jsg/1384282843_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTActon-index relationss for perfect Hamiltonian diffeomorphismshttp://projecteuclid.org/euclid.jsg/1384282844<strong>Mike Chance</strong>, <strong>Viktor Ginzburg</strong>, <strong>Başak Gürel</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 449--474.</p><p><strong>Abstract:</strong><br/>
We show that the actions and indexes of fixed points of a Hamiltonian
diffeomorphism with finitely many periodic points must satisfy
certain relations, provided that the quantum cohomology of the ambient
manifold meets an algebraic requirement satisfied for projective
spaces, Grassmannians and many other manifolds. We also refine a
previous result on the Conley conjecture for negative monotone symplectic
manifolds, due to the second and third authors, and show that
a Hamiltonian diffeomorphism of such a manifold must have simple
periodic orbits of arbitrarily large period whenever its fixed points are
isolated.
</p>projecteuclid.org/euclid.jsg/1384282844_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTOn the Hofer geometry for weakly exact Lagrangian submanifoldshttp://projecteuclid.org/euclid.jsg/1384282845<strong>Frol Zapolsky</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 475--488.</p><p><strong>Abstract:</strong><br/>
We use spectral invariants in Lagrangian Floer theory in order to
show that there exist isometric embeddings of normed linear spaces
(finite or infinite-dimensional, depending on the case) into the space
of Hamiltonian deformations of certain weakly exact Lagrangian submanifolds
in tame symplectic manifolds. In addition to providing a new
class of examples in which the Lagrangian Hofer metric can be computed
explicitly, we refine and generalize some known results about it.
</p>projecteuclid.org/euclid.jsg/1384282845_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTA note on $C^0$ rigidity of Hamiltonian isotopieshttp://projecteuclid.org/euclid.jsg/1384282846<strong>Sobhan Seyfaddini</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 489--496.</p><p><strong>Abstract:</strong><br/>
We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian if the corresponding sequence of generating Hamiltonians converge in $L^{(1,\infty)}$ topology.
</p>projecteuclid.org/euclid.jsg/1384282846_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTThe Koszul complex of a moment maphttp://projecteuclid.org/euclid.jsg/1384282847<strong>Hans-Christian Herbig</strong>, <strong>Gerald W. Schwarz</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 3, 497--508.</p><p><strong>Abstract:</strong><br/>
Let $K \to \operatorname{U}(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho \colon V \to {\mathfrak k}^*$. We have the Koszul complex
$\mathcal{K}(\rho, \mathcal{C}^\infty(V))$ of the component functions $\rho_1, \dots, \rho_k$ of $\rho$. Let $G=K_{\mathbb {C}}$, the complexification of $K$. We show that the Koszul complex is a resolution
of the smooth functions on $\rho ^{-1}(0)$ if and only if $G \to \operatorname{GL}(V)$ is $1$-large, a concept introduced in [11,12]. Now let $M$ be a symplectic manifold with a Hamiltonian action
of $K$. Let $\rho$ be a moment mapping and consider the Koszul complex given by the component functions of $\rho$. We show that the Koszul complex is a resolution of the smooth functions
on $Z= \rho ^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is $1$-large.
</p>projecteuclid.org/euclid.jsg/1384282847_Tue, 12 Nov 2013 14:00 ESTTue, 12 Nov 2013 14:00 ESTA symplectically non-squeezable small set and the regular coisotropic capacityhttp://projecteuclid.org/euclid.jsg/1384783390<strong>Jan Swoboda</strong>, <strong>Fabian Ziltener</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 4, 509--523.</p><p><strong>Abstract:</strong><br/>
We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $\mathbb{R}^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed
into the standard open symplectic cylinder. The second main result is a lower bound on the $d$th regular coisotropic capacity, which is sharp up to a factor of $3$. For an open subset of a geometrically
bounded, aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was
considered by Audin and Polterovich.
</p>projecteuclid.org/euclid.jsg/1384783390_Mon, 18 Nov 2013 09:03 ESTMon, 18 Nov 2013 09:03 ESTCapping off open books and the Ozsváth-Szabó contact invarianthttp://projecteuclid.org/euclid.jsg/1384783391<strong>John A. Baldwin</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 4, 525--561.</p><p><strong>Abstract:</strong><br/>
We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $\mathbb{R}^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into
the standard open symplectic cylinder. The second main result is a lower bound on the $d$th regular coisotropic capacity, which is sharp up to a factor of $3$. For an open subset of a geometrically bounded,
aspherical symplectic manifold, this capacity is a lower bound on its displacement energy. The proofs of the results involve a certain Lagrangian submanifold of linear space, which was considered by
Audin and Polterovich.
</p>projecteuclid.org/euclid.jsg/1384783391_Mon, 18 Nov 2013 09:03 ESTMon, 18 Nov 2013 09:03 ESTA proof of the classification theorem of overtwisted contact structures via convex surface theoryhttp://projecteuclid.org/euclid.jsg/1384783392<strong>Yang Huang</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 4, 563--601.</p><p><strong>Abstract:</strong><br/>
In Contact 3-manifolds, twenty years since J. Martinet’s work, Eliashberg proved that two overtwisted contact structures on a
closed oriented 3-manifold are isotopic through contact structures if and
only if they are homotopic as 2-plane fields. We provide an alternative
proof of this theorem using the convex surface theory and bypasses.
</p>projecteuclid.org/euclid.jsg/1384783392_Mon, 18 Nov 2013 09:03 ESTMon, 18 Nov 2013 09:03 ESTDegeneration of Kähler structures and half-form quantization of toric varietieshttp://projecteuclid.org/euclid.jsg/1384783393<strong>William D. Kirwin</strong>, <strong>José M. Mourão</strong>, <strong>João P. Nunes</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 4, 603--643.</p><p><strong>Abstract:</strong><br/>
We study the half-form Kähler quantization of a smooth symplectic toric manifold $(X,\omega)$, such that $[ \omega/ 2\pi]- c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}} )$ and is non-negative.
We define the half-form corrected quantization of $(X,\omega)$ to be given by holomorphic sections of a certain Hermitian line bundle $L\to X$ with Chern class $[ \omega/ 2\pi]- c_{1}(X)/2$. These
sections then correspond to integral points of a "corrected" polytope $P_{L}$ with integral vertices. For a suitably translated moment polytope $P_{X}$ for $(X,\omega)$, we have that $P_{L}\subset P_{X}$ is
obtained from $P_{X}$ by a one-half inward-pointing normal shift along the boundary.
We use our results on the half-form corrected Kähler quantization to motivate a definition of half-form corrected quantization in the singular real toric polarization. Using families of complex structures studied
in Baier-Florentino-Mourão-Nunes, which include the degeneration of Kähler polarizations to the vertical polarization, we show that, under this degeneration, the half-form corrected $L^{2}$-normalized
monomial holomorphic sections converge to Dirac-delta-distributional sections supported on the fibers over the integral points of $P_{L}$, which correspond to corrected Bohr-Sommerfeld fibers.
This result and the limit of the corrected connection, with curvature singularities along the boundary of $P_X$, justifies the direct definition we give for the corrected quantization in the singular real toric polarization.
We show that the space of quantum states for this definition coincides with the space obtained via degeneration of the Kähler quantization.
We also show that the BKS pairing between Kähler polarizations is not unitary in general. On the other hand, the unitary connection induced by this pairing is flat.
</p>projecteuclid.org/euclid.jsg/1384783393_Mon, 18 Nov 2013 09:03 ESTMon, 18 Nov 2013 09:03 ESTHomoclinic points and Floer homologyhttp://projecteuclid.org/euclid.jsg/1384783394<strong>Sonja Hohloch</strong><p><strong>Source: </strong>J. Symplectic Geom., Volume 11, Number 4, 645--701.</p><p><strong>Abstract:</strong><br/>
A new relation between homoclinic points and Lagrangian Floer
homology is presented: in dimension two, we construct a Floer homology
generated by primary homoclinic points.We compute two examples
and prove an invariance theorem. Moreover, we establish a link to the
(absolute) flux and growth of symplectomorphisms.
</p>projecteuclid.org/euclid.jsg/1384783394_Mon, 18 Nov 2013 09:03 ESTMon, 18 Nov 2013 09:03 ESTDeformation quantization and irrational numbershttp://projecteuclid.org/euclid.jsg/1409317362<strong>Eli Hawkins</strong>, <strong>Alan Haynes</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 1--22.</p><p><strong>Abstract:</strong><br/>
Diophantine approximation is the problem of approximating a real
number by rational numbers. We propose a version of this in which
the numerators are approximately related to the denominators by a
Laurent polynomial. Our definition is motivated by the problem of
constructing strict deformation quantizations of symplectic manifolds.
We show that this type of approximation exists for any real number and
also investigate what happens if the number is rational or a quadratic
irrational.
</p>projecteuclid.org/euclid.jsg/1409317362_20140829090245Fri, 29 Aug 2014 09:02 EDTLocalization and specialization for Hamiltonian torus actionshttp://projecteuclid.org/euclid.jsg/1409317363<strong>Milena Pabiniak</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 23--47.</p><p><strong>Abstract:</strong><br/>
We consider a Hamiltonian action of $n$-dimensional torus, $T^n$, on a compact symplectic manifold $(M,\omega)$ with $d$ isolated fixed points. For every fixed point $p$, there exists (although not unique)
a class $a_p \in H^*_{T}(M; \mathbb{Q})$ such that the collection $\{a_p\}$, over all fixed points, forms a basis for $H^*_{T}(M; \mathbb{Q})$ as an $H^*(BT; \mathbb{Q})$ module. The map induced by the
inclusion, $\iota^*:H^*_{T}(M; \mathbb{Q}) \rightarrow H^*_{T}(M^{T}; \mathbb{Q})= \oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n] $ is injective. We use such classes $\{a_p\}$ to give necessary and sufficient
conditions for $f=(f_1, \ldots ,f_d)$ in $\oplus_{j=1}^{d}\mathbb{Q}[x_1, \ldots, x_n]$ to be in the image of $\iota^*$, i.e., to represent an equivariant cohomology class on $M$. In the case when $T$ is a circle
and present these conditions explicitly. We explain how to combine this $1$-dimensional solution with Chang-Skjelbred Lemma in order to obtain the result for a torus $T$ of any dimension. Moreover, for
a Goresky-Kottwitz-MacPherson (GKM) $T$-manifold $M$ our techniques give combinatorial description of $H^*_{K}(M; \mathbb{Q})$, for a generic subgroup $K \hookrightarrow T$, even if $M$ is not a GKM $K$-manifold.
</p>projecteuclid.org/euclid.jsg/1409317363_20140829090245Fri, 29 Aug 2014 09:02 EDTThe group of contact diffeomorphisms for compact contact manifoldshttp://projecteuclid.org/euclid.jsg/1409317364<strong>John Bland</strong>, <strong>Tom Duchamp</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 49--104.</p><p><strong>Abstract:</strong><br/>
For a compact contact manifold $M^{2n + 1}$, it is shown that the anisotropic Folland-Stein function spaces $\Gamma^{s} (M), s \geq (2n + 4)$ form an algebra. The notion of anisotropic regularity is extended to define
the space of $\Gamma^{s}$-contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyse the action
of the group of contact diffeomorphisms on the space of CR structures on a compact, three-dimensional manifold.
</p>projecteuclid.org/euclid.jsg/1409317364_20140829090245Fri, 29 Aug 2014 09:02 EDTThe Hamiltonian geometry of the space of unitary connections with symplectic curvaturehttp://projecteuclid.org/euclid.jsg/1409317365<strong>Joel Fine</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 105--123.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{L} \to \mathcal{M}$ be a Hermitian line bundle over a compact manifold. Write $\mathcal{S}$ for the space of all unitary connections in $\mathcal{L}$ whose curvatures define symplectic forms on
$\mathcal{M}$ and $\mathcal{G}$ for the identity component of the group of unitary bundle isometries of $\mathcal{L}$, which acts on $\mathcal{S}$ by pullback. The main observation of this note is that
$\mathcal{S}$ carries a $\mathcal{G}$-invariant symplectic structure, there is a moment map for the $\mathcal{G}$-action and that this embeds the components of $\mathcal{S}$ as $\mathcal{G}$-coadjoint orbits.
Restricting to the subgroup of $\mathcal{G}$ which covers the identity on $\mathcal{M}$, we see that prescribing the volume form of a symplectic structure can be seen as finding a zero of a moment map.
When $\mathcal{M}$ is a Kähler manifold, this gives a moment-map interpretation of the Calabi conjecture. We also describe some directions for future research based upon the picture outlined here.
</p>projecteuclid.org/euclid.jsg/1409317365_20140829090245Fri, 29 Aug 2014 09:02 EDTAnalytic test configurations and geodesic rayshttp://projecteuclid.org/euclid.jsg/1409317366<strong>Julius Ross</strong>, <strong>David Witt Nyström</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 125--169.</p><p><strong>Abstract:</strong><br/>
Starting with the data of a curve of singularity types, we use the Legendre transform to construct weak geodesic rays in the space of locally bounded metrics on an ample line bundle $L$ over a compact manifold. Using this
we associate weak geodesics to suitable filtrations of the algebra of sections of $L$. In particular this works for the natural filtration coming from an algebraic test configuration, and we show how this recovers the weak
geodesic ray of Phong-Sturm.
</p>projecteuclid.org/euclid.jsg/1409317366_20140829090245Fri, 29 Aug 2014 09:02 EDTThe Duistermaat-Heckman formula and the cohomology of moduli spaces of polygonshttp://projecteuclid.org/euclid.jsg/1409317367<strong>Alessia Mandini</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 1, 171--213.</p><p><strong>Abstract:</strong><br/>
We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb{R}^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes
in $\mathbb{C}^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group $U(n)$. We prove that the first Chern classes of the $n$ line bundles associated with the
fibration ($r$-level set) $\to M(r)$ generate the cohomology ring $H*(M(r),\mathbb{C})$. By applying the Duistermaat-Heckman Theorem, we then deduce the relations on these generators from the piece-wise polynomial
function that describes the volume of $M(r)$. We also give an explicit description of the birational map between $M(r)$ and $M(r ')$ when the lengths vectors $r$ and $r '$ are in different chambers of the moment polytope.
This wallcrossing analysis is the key step to prove that the Chern classes above are generators of $H*(M(r))$, (This is well-known when $M(r)$ is toric, and by wall-crossing we prove that it holds also when $M(r)$ is not toric).
</p>projecteuclid.org/euclid.jsg/1409317367_20140829090245Fri, 29 Aug 2014 09:02 EDTSymplectical manifolds and cohomological decompositionhttp://projecteuclid.org/euclid.jsg/1409317929<strong>Daniele Angella</strong>, <strong>Adriano Tomassini</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 215--236.</p><p><strong>Abstract:</strong><br/>
Given a closed symplectic manifold, we study when the Lefschetz decomposition induced by the $\mathfrak{sl}(2 ; \mathbb{R})$-representation yields a decomposition of the de Rham cohomology. In particular, this holds
always true for the second de Rham cohomology group, or if the symplectic manifold satisfies the Hard Lefschetz Condition.
</p>projecteuclid.org/euclid.jsg/1409317929_20140829091210Fri, 29 Aug 2014 09:12 EDTA bordered Legendrian contact algebrahttp://projecteuclid.org/euclid.jsg/1409317930<strong>John G. Harper</strong>, <strong>Michael G. Sullivan</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 237--255.</p><p><strong>Abstract:</strong><br/>
In A bordered Chekanov–Eliashberg algebra , Sivek proves a "van Kampen" decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard
contact $\mathbb{R}^3$. We prove an analogous result for the holomorphic curve version of the Legendrian contact algebra of certain Legendrians submanifolds in standard contact $J^1(M)$. This includes all one- and
two-dimensional Legendrians, and some higher-dimensional ones. We present various applications including a Mayer-Vietoris sequence for linearized contact homology similar to
A bordered Chekanov–Eliashberg algebra and a connect sum formula for the augmentation variety introduced in L. Ng, Framed knot contact homology . The main tool is the theory of gradient flow trees
developed in T. Ekholm, Morse flow trees and Legendrian contact homology in 1-jet spaces .
</p>projecteuclid.org/euclid.jsg/1409317930_20140829091210Fri, 29 Aug 2014 09:12 EDTRemoval of singularities and Gromov compactness for symplectic vorticeshttp://projecteuclid.org/euclid.jsg/1409317931<strong>Andreas Ott</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 257--311.</p><p><strong>Abstract:</strong><br/>
We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends
results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in
their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.
</p>projecteuclid.org/euclid.jsg/1409317931_20140829091210Fri, 29 Aug 2014 09:12 EDTOn the anti-diagonal filtration for the Heegard Floer chain complex of a branched double-coverhttp://projecteuclid.org/euclid.jsg/1409317932<strong>Eamonn Tweedy</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 313--363.</p><p><strong>Abstract:</strong><br/>
Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group $Kh_{\rm symp,~inv}(K)$ for a knot $K \subset S^{3}$, as well as a spectral sequence converging to the Heegaard Floer homology
group $\widehat{HF}(\Sigma (K) \# (S^2 \times S^1))$ with $E^1$-page isomorphic to a factor of $Kh_{\rm symp,~inv}(K)$. There the authors proved that $Kh_{\rm symp,~inv}$ is a knot invariant. We show here that the
higher pages of their spectral sequence are knot invariants also.
</p>projecteuclid.org/euclid.jsg/1409317932_20140829091210Fri, 29 Aug 2014 09:12 EDTThe closure of the symplectic cone of elliptic surfaceshttp://projecteuclid.org/euclid.jsg/1409317933<strong>M. J. D. Hamilton</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 365--377.</p><p><strong>Abstract:</strong><br/>
The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kähler surface. We consider the case of
the elliptic surfaces $E(n)$ and focus on a slightly weaker conjecture for the closure of the symplectic cone. We prove this conjecture in the case of the spin surfaces $E(2m)$ using inflation and the action of
self-diffeomorphisms of the elliptic surface. An additional obstruction appears in the non-spin case.
</p>projecteuclid.org/euclid.jsg/1409317933_20140829091210Fri, 29 Aug 2014 09:12 EDTOpen books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristichttp://projecteuclid.org/euclid.jsg/1409317934<strong>River Chiang</strong>, <strong>Fan Ding</strong>, <strong>Otto van Koert</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 2, 379--426.</p><p><strong>Abstract:</strong><br/>
We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic
of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fibered Dehn twists are not symplectically isotopic to the identity relative to the boundary. This complements
results of Biran and Giroux.
</p>projecteuclid.org/euclid.jsg/1409317934_20140829091210Fri, 29 Aug 2014 09:12 EDTSquare roots of Hamiltonian diffeomorphismshttp://projecteuclid.org/euclid.jsg/1409319456<strong>Peter Albers</strong>, <strong>Urs Frauenfelder</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 427--434.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove that on any closed symplectic manifold there exists an arbitrarily $C^\infty$-small Hamiltonian diffeomorphism not admitting a square root.
</p>projecteuclid.org/euclid.jsg/1409319456_20140829093738Fri, 29 Aug 2014 09:37 EDTGeneralized reduction and pure spinorshttp://projecteuclid.org/euclid.jsg/1409319457<strong>T. Drummond</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 435--471.</p><p><strong>Abstract:</strong><br/>
We study reduction of Dirac structures as developed by H. Bursztyn et al. from the point of view of pure spinors. We describe explicitly the pure spinor line bundle of the reduced Dirac structure. We also obtain results on
reduction of generalized Calabi-Yau structures.
</p>projecteuclid.org/euclid.jsg/1409319457_20140829093738Fri, 29 Aug 2014 09:37 EDTConvergence of Kähler to real polarizations on flag manifolds via toric degenerationshttp://projecteuclid.org/euclid.jsg/1409319458<strong>Mark D. Hamilton</strong>, <strong>Hiroshi Konno</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 473--509.</p><p><strong>Abstract:</strong><br/>
In this paper, we construct a family of complex structures on a
complex flag manifold that converge to the real polarization coming
from the Gelfand–Cetlin integrable system, in the sense that holomorphic
sections of a prequantum line bundle converge to delta-function
sections supported on the Bohr–Sommerfeld fibers. Our construction
is based on a toric degeneration of flag varieties and a deformation of
Kähler structure on toric varieties by symplectic potentials.
</p>projecteuclid.org/euclid.jsg/1409319458_20140829093738Fri, 29 Aug 2014 09:37 EDTSymplectic homology of disc cotangent bundles of domains in Euclidean spacehttp://projecteuclid.org/euclid.jsg/1409319459<strong>Kei Irie</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 511--552.</p><p><strong>Abstract:</strong><br/>
Let $V$ be a bounded domain with smooth boundary in $\mathbb{R}^n$, and $D^*V$ denote its disc cotangent bundle. We compute symplectic homology of $D^*V$, in terms of relative homology of loop spaces on the closure
of $V$. We use this result to show that the Floer-Hofer-Wysocki capacity of $D^*V$ is between $2r(V)$ and $2(n + 1)r(V)$, where $r(V)$ denotes the inradius of $V$. As an application, we study periodic billiard
trajectories on $V$.
</p>projecteuclid.org/euclid.jsg/1409319459_20140829093738Fri, 29 Aug 2014 09:37 EDTBilinearized Legendrian contact homology and the augmentation categoryhttp://projecteuclid.org/euclid.jsg/1409319460<strong>Frédéric Bourgeois</strong>, <strong>Baptiste Chantraine</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 553--583.</p><p><strong>Abstract:</strong><br/>
In this paper, we construct an $\mathcal{A}_{\infty}$-category associated to a Legendrian submanifold of a jet space. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(\Lambda)$ and the homology
of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearized Legendrian contact homology. Those are constructed as a generalization of linearized Legendrian contact homology
using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order composition maps in the category and generalizes the idea of G. Civan, P. Koprowski,
J. Etnyre, J.M. Sabloff and A. Walker, Product structures for Legendrian contact homology , where an $\mathcal{A}_{\infty}$-algebra was constructed from one augmentation. This category allows us to define a notion
of equivalence of augmentations when the coefficient ring is a field regardless of its characteristic. We use simple examples to show that bilinearized cohomology groups are efficient to distinguish those equivalences
classes.We also generalize the duality exact sequence from T. Ekholm, J. Etnyre and M. Sullivan in our context, and interpret geometrically the bilinearized homology in terms of the Floer homology of Lagrangian fillings
(following T. Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology ).
</p>projecteuclid.org/euclid.jsg/1409319460_20140829093738Fri, 29 Aug 2014 09:37 EDTSmoothings of singularities and symplectic surgeryhttp://projecteuclid.org/euclid.jsg/1409319461<strong>Heesang Park</strong>, <strong>András I. Stipsicz</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 585--597.</p><p><strong>Abstract:</strong><br/>
Suppose that $C$ is a connected configuration of two-dimensional symplectic submanifolds in a symplectic 4-manifold with negative definite intersection graph $\Gamma_C$. Let $(S, 0)$ be a normal surface singularity
with resolution graph $\Gamma_C$ and suppose that $W_S$ is a smoothing of $(S, 0)$. We show that if we replace an appropriate neighborhood of $C$ with $W_S$, then the resulting 4-manifold admits a symplectic
structure. The operation generalizes the rational blow-down operation of Fintushel-Stern, and therefore our result extends Symington's theorem about symplectic rational blow-downs.
</p>projecteuclid.org/euclid.jsg/1409319461_20140829093738Fri, 29 Aug 2014 09:37 EDTBypass attachments and homotopy classes of 2-plane fields in contact topologyhttp://projecteuclid.org/euclid.jsg/1409319462<strong>Yang Huang</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 599--617.</p><p><strong>Abstract:</strong><br/>
We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a three-dimensional contact manifold with convex boundary,
we show that the bypass triangle attachment changes the homotopy class of the contact structure relative to the boundary, and the difference is measured by the homotopy group $\pi_3(S^2)$.
</p>projecteuclid.org/euclid.jsg/1409319462_20140829093738Fri, 29 Aug 2014 09:37 EDTHofer Geometry and cotangent fibershttp://projecteuclid.org/euclid.jsg/1409319463<strong>Michael Usher</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 3, 619--656.</p><p><strong>Abstract:</strong><br/>
For a class of Riemannian manifolds that include products of
arbitrary compact manifolds with manifolds of nonpositive sectional
curvature on the one hand, or with certain positive-curvature examples
such as spheres of dimension at least 3 and compact semisimple
Lie groups on the other, we show that the Hamiltonian diffeomorphism
group of the cotangent bundle contains as subgroups infinitedimensional
normed vector spaces that are bi-Lipschitz embedded with
respect to Hofer’s metric; moreover these subgroups can be taken to
consist of diffeomorphisms supported in an arbitrary neighborhood of
the zero section. In fact, the orbit of a fiber of the cotangent bundle with
respect to any of these subgroups is quasi-isometrically embedded with
respect to the induced Hofer metric on the orbit of the fiber under the
whole group. The diffeomorphisms in these subgroups are obtained from
reparametrizations of the geodesic flow. Our proofs involve a study of
the Hamiltonian-perturbed Floer complex of a pair of cotangent fibers
(or, more generally, of a conormal bundle together with a cotangent
fiber). Although the homology of this complex vanishes, an analysis of
its boundary depth yields the lower bounds on the Lagrangian Hofer
metric required for our main results.
</p>projecteuclid.org/euclid.jsg/1409319463_20140829093738Fri, 29 Aug 2014 09:37 EDTA remark on the Reeb flow for sphereshttp://projecteuclid.org/euclid.jsg/1433196059<strong>Roger Casals</strong>, <strong>Francisco Presas</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 657--671.</p><p><strong>Abstract:</strong><br/>
>We prove the non-triviality of the Reeb flow for the standard contact spheres $\mathbb{S}^{2n+1}, n \neq 3$, inside the fundamental group of their contactomorphism group. The argument uses the existence of homotopically non-trivial $2$-spheres in the space of contact structures of a $3$-Sasakian manifold.
</p>projecteuclid.org/euclid.jsg/1433196059_20150601180109Mon, 01 Jun 2015 18:01 EDTInfinitely many small exotic Stein fillingshttp://projecteuclid.org/euclid.jsg/1433196060<strong>Selman Akbulut</strong>, <strong>Kouichi Yasui</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 673--684.</p><p><strong>Abstract:</strong><br/>
We show that there exist infinitely many simply connected compact Stein 4-manifolds with $b_2 = 2$ such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact $3$-manifold on their boundaries. We also describe their handlebody pictures.
</p>projecteuclid.org/euclid.jsg/1433196060_20150601180109Mon, 01 Jun 2015 18:01 EDTToric structures on bundles of projective spaceshttp://projecteuclid.org/euclid.jsg/1433196061<strong>Andrew Fanoe</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 685--724.</p><p><strong>Abstract:</strong><br/>
Recently, extending work by Karshon
et al. , Borisov and McDuff showed in
The topology of toric symplectic manifolds that a given closed symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, in
The topology of toric symplectic manifolds McDuff also showed that a product of two projective spaces $\mathbb{C}P^r \times \mathbb{C}P^s$ with any given symplectic form has a unique toric structure provided that $r, s \geq 2$. In contrast, the product $\mathbb{C}P^r \times \mathbb{C}P^1$ can be given infinitely many distinct toric structures, although only a finite number of these are compatible with each given symplectic form $\omega$. In this paper, we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M) = 2$. In particular, all such manifolds are $\mathbb{C}P^r$ bundles over $\mathbb{C}P^s$ for some $r, s$. We show that there is a unique toric structure if $r \lt s$, and also that if $r, s \geq 2$, $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,\omega)$ has a unique toric structure, such as the case where $(M,\omega)$ is monotone.
</p>projecteuclid.org/euclid.jsg/1433196061_20150601180109Mon, 01 Jun 2015 18:01 EDTLagrangian blow-ups, blow-downs, and applications to real packinghttp://projecteuclid.org/euclid.jsg/1433196062<strong>Antonio Reiser</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 725--789.</p><p><strong>Abstract:</strong><br/>
Given a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L$, we construct versions of the symplectic blow-up and blow-down which are defined relative to $L$. We further show that if $M$ admits an anti-symplectic involution $\phi$, i.e., a diffeomorphism such that $\phi^2 = \mathrm{Id}$ and $\phi^* \omega = - \omega$, and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an antisymplectic involution on the blow-up $\tilde{M}$ as well. We then derive a homological condition for real Lagrangian surfaces $L = \mathrm{Fix} (\phi)$ which determines when the topology of $L$ changes after a blowdown, and we use these constructions to study the relative packing numbers and packing stability for real symplectic four manifolds which are non-Seiberg-Witten simple.
</p>projecteuclid.org/euclid.jsg/1433196062_20150601180109Mon, 01 Jun 2015 18:01 EDTSystems of global surfaces of section for dynamically convex Reeb flows on the 3-spherehttp://projecteuclid.org/euclid.jsg/1433196063<strong>Umberto L. Hryniewicz</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 791--862.</p><p><strong>Abstract:</strong><br/>
We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section come in families, organized as open book decompositions. As an application we obtain new global surfaces of section for the Hamiltonian dynamics on strictly convex three-dimensional energy levels.
</p>projecteuclid.org/euclid.jsg/1433196063_20150601180109Mon, 01 Jun 2015 18:01 EDTOn cohomological obstructions for the existence of log-symplectic structureshttp://projecteuclid.org/euclid.jsg/1433196064<strong>Ioan Mărcuţ</strong>, <strong>Boris Osorno Torres</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 863--866.</p><p><strong>Abstract:</strong><br/>
We prove that a compact log-symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of log-symplectic structures similar to those in symplectic geometry.
</p>projecteuclid.org/euclid.jsg/1433196064_20150601180109Mon, 01 Jun 2015 18:01 EDTGKM-sheaves and nonorientable surface group representationshttp://projecteuclid.org/euclid.jsg/1433196065<strong>Thomas Baird</strong>. <p><strong>Source: </strong>Journal of Symplectic Geometry, Volume 12, Number 4, 867--921.</p><p><strong>Abstract:</strong><br/>
Let $T$ be a compact torus and $X$ a nice compact $T$-space (say a manifold or variety). We introduce a functor assigning to $X$ a
GKM-sheaf $\mathcal{F}_X$ over a
GKM-hypergraph $\Gamma_X$. Under the condition that $X$ is equivariantly formal, the ring of global sections of $\mathcal{F}_X$ are identified with the equivariant cohomology, $H^*_T (X; \mathbb{C}) \cong H^0(\mathcal{F}_X)$. We show that GKM-sheaves provide a general framework able to incorporate numerous constructions in the GKM-theory literature. In the second half of the paper we apply these ideas to study the equivariant topology of the representation variety $\mathcal{R}K := \mathrm{Hom}(\pi_1 (\Sigma),K)$ under conjugation by $K$, where $\Sigma$ is a nonorientable surface and $K$ is a compact connected Lie group. We prove that $\mathcal{R}_{SU(3)}$ is equivariantly formal for all $\Sigma$ and compute its equivariant cohomology ring. We also produce conjectural betti number formulas for some other Lie groups.
</p>projecteuclid.org/euclid.jsg/1433196065_20150601180109Mon, 01 Jun 2015 18:01 EDT