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2,900 | Formality of canonical symplectic complexes and Frobenius manifolds | math.SG | It is shown that the de Rham complex of a symplectic manifold $M$ satisfying
the hard Lefschetz condition is formal. Moreover, it is shown that the
differential Gerstenhaber-Batalin-Vilkoviski algebra associated to such a
symplectic structure gives rise, along the lines explained in the papers of
Barannikov and Kontsev... | math |
2,901 | Almost Complex Structures on $S^2\times S^2$ | math.SG | In this note we investigate the structure of the space $\Jj$ of smooth almost
complex structures on $S^2\times S^2$ that are compatible with some symplectic
form. This space has a natural stratification that changes as the cohomology
class of the form changes and whose properties are very closely connected to
the topol... | math |
2,902 | A Note on Higher Cohomology Groups of Kähler Quotients | math.SG | Consider a holomorphic torus action on a possibly non-compact K\"ahler
manifold. We show that the higher cohomology groups appearing in the geometric
quantization of the symplectic quotient are isomorphic to the invariant parts
of the corresponding cohomology groups of the original manifold. For
non-Abelian group actio... | math |
2,903 | On existence of nonformal simply connected symplectic manifolds | math.SG | Examples of nonformal simply connected symplectic manifolds are constructed. | math |
2,904 | A limit of toric symplectic forms that has no periodic Hamiltonians | math.SG | We calculate the Riemann-Roch number of some of the pentagon spaces defined
in [Klyachko,Kapovich-Millson,HK1]. Using this, we show that while the regular
pentagon space is diffeomorphic to a toric variety, even symplectomorphic to
one under arbitrarily small perturbations of its symplectic structure, it does
not admit... | math |
2,905 | On nonformal simply connected symplectic manifolds | math.SG | For any $N \geq 5$ nonformal simply connected symplectic manifolds of
dimension $2N$ are constructed. This disproves the formality conjecture for
simply connected symplectic manifolds which was introduced by Lupton and Oprea. | math |
2,906 | A Note on n-ary Poisson Brackets | math.SG | A class of n-ary Poisson structures of constant rank is indicated. Then, one
proves that the ternary Poisson brackets are exactly those which are defined by
a decomposable 3-vector field. The key point is the proof of a lemma which
tells that an n-vector $(n\geq3)$ is decomposable iff all its contractions with
up to n-... | math |
2,907 | Graded Lagrangian submanifolds | math.SG | In the usual setup, the grading on Floer homology is relative: it is unique
only up to adding a constant. "Graded Lagrangian submanifolds" are Lagrangian
submanifolds with a bit of extra structure, which fixes the ambiguity in the
grading. The idea is originally due to Kontsevich. This paper contains an
exposition of t... | math |
2,908 | Stability for holomorphic spheres and Morse theory | math.SG | In this paper we study the question of when does a closed, simply connected,
integral symplectic manifold (W,omega) have the stability property for its
spaces of based holomorphic spheres? This property states that in a stable
limit under certain gluing operators, the space of based holomorphic maps from
a sphere to X,... | math |
2,909 | A note on generating functions | math.SG | An afinne-invariant view of generating functions of symplectic
transformations of an affine symplectic space is discussed. More generally, it
works for symmetric symplectic spaces. The note is completely elementary, but
it yields some nice pictures. | math |
2,910 | Completely integrable systems: a generalization | math.SG | We present a slight generalization of the notion of completely integrable
systems to get them being integrable by quadratures. We use this generalization
to integrate dynamical systems on double Lie groups. | math |
2,911 | Lefschetz fibrations and the Hodge bundle | math.SG | Integral symplectic 4-manifolds may be described in terms of Lefschetz
fibrations. In this note we give a formula for the signature of any Lefschetz
fibration in terms of the second cohomology of the moduli space of stable
curves. As a consequence we see that the sphere in moduli space defined by any
(not necessarily h... | math |
2,912 | Classical dynamical r-matrices and homogeneous Poisson structures on $G/H$ and $K/T$ | math.SG | Let G be a finite dimensional simple complex group equipped with the standard
Poisson Lie group structure. We show that all G-homogeneous (holomorphic)
Poisson structures on $G/H$, where $H \subset G$ is a Cartan subgroup, come
from solutions to the Classical Dynamical Yang-Baxter equations which are
classified by Etin... | math |
2,913 | Construction of completely integrable systems by Poisson mappings | math.SG | Pulling back sets of functions in involution by Poisson mappings and adding
Casimir functions during the process allows to construct completely integrable
systems. Some examples are investigated in detail. | math |
2,914 | Integrable Hamiltonian systems on Lie groups: Kowalevski type | math.SG | The contributions of Sophya Kowalewski to the integrability theory of the
equations for the heavy top extend to a larger class of Hamiltonian systems on
Lie groups; this paper explains these extensions, and along the way reveals
further geometric significance of her work in the theory of elliptic curves.
Specifically, ... | math |
2,915 | The structure of a symplectic manifold on the space of loops of 7-manifold | math.SG | We define a symplectic structure on the space of non parametrized loops in
$G_2$ manifold. We also develop some basics of intersection theory of
Lagrangian submanifolds. | math |
2,916 | Remarks on the flux groups | math.SG | I study flux groups of compact symplectic manifolds. Under some topological
assumptions, I give a new estimate of the rank of flux groups and give a method
of construcion of compact symplectic aspherical manifolds. | math |
2,917 | Transversality theory, cobordisms, and invariants of symplectic quotients | math.SG | This paper gives methods for understanding invariants of symplectic
quotients. The symplectic quotients considered here are compact symplectic
manifolds (or more generally orbifolds), which arise as the symplectic
quotients of a symplectic manifold by a compact torus. (A companion paper
examines symplectic quotients by... | math |
2,918 | Symplectic quotients by a nonabelian group and by its maximal torus | math.SG | This paper examines the relationship between the symplectic quotient X//G of
a Hamiltonian G-manifold X, and the associated symplectic quotient X//T, where
T is a maximal torus, in the case in which X//G is a compact manifold or
orbifold.
The three main results are: a formula expressing the rational cohomology ring
o... | math |
2,919 | On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds | math.SG | In the present paper we study the variation of the dimensions $h_k$ of spaces
of symplectically harmonic cohomology classes (in the sense of Brylinski) on
closed symplectic manifolds. We give a description of such variation for all
6-dimensional nilmanifolds equipped with symplectic forms. In particular, it
turns out t... | math |
2,920 | On certain geometric and homotopy properties of closed symplectic manifolds | math.SG | The paper deals with relations between the Hard Lefschetz property,
(non)vanishing of Massey products and the evenness of odd-degree Betti numbers
of closed symplectic manifolds. It is known that closed symplectic manifolds
can violate all these properties (in contrast with the case of Kaehler
manifolds). However, the ... | math |
2,921 | Quantization of bending deformations of polygons in Euclidean space, hypergeometric integrals and the Gassner representation | math.SG | We quantize the bending deformations of n-gon linkages by linearizing the
bending fields at a degenerate n-gon to get a representation of the Malcev Lie
algebra of the pure braid group. This linearization yields a flat connection on
the space of n distinct points on the complex line. We show that the monodromy
is (esse... | math |
2,922 | Loops of Lagrangian submanifolds and pseudoholomorphic discs | math.SG | The main theorem of this paper asserts that the inclusion of the space of
projective Lagrangian planes into the space of Lagrangian submanifolds of
complex projective space induces an injective homomorphism of fundamental
groups. We introduce three invariants of exact loops of Lagrangian submanifolds
that are modelled ... | math |
2,923 | Convexity of multi-valued momentum map | math.SG | We extend the famous convexity theorem of Atiyah, Guillemin and Sternberg to
the case of non-Hamiltonian actions. We show that the image of a generalized
momentum map is a bounded polytope times a vector space. We prove that this
picture is stable for small perturbations of the symplectic form. We also
observe that an ... | math |
2,924 | Holomorphic Disks and the Chord Conjecture | math.SG | We prove (a weak version of) Arnold's Chord Conjecture using Gromov's
``classical'' idea in to produce holomorphic disks with boundary on a
Lagrangian submanifold. | math |
2,925 | Locally Lagrangian Symplectic and Poisson Manifolds | math.SG | We discuss symplectic manifolds where, locally, the structure is that
encountered in Lagrangian dynamics. Exemples and characteristic properties are
given. Then, we refer to the computation of the Maslov classes of a Lagrangian
submanifold. Finally, we indicate the generalization of this type of structures
to Poisson m... | math |
2,926 | Coupling Tensors and Poisson Geometry Near a Single Symplectic Leaf | math.SG | In the framework of the connection theory, a contravariant analog of the
Sternberg coupling procedure is developed for studying a natural class of
Poisson structures on fiber bundles, called coupling tensors. We show that
every Poisson structure near a closed symplectic leaf can be realized as a
coupling tensor. Our ma... | math |
2,927 | The Topological Structure of Contact and Symplectic Quotients | math.SG | We show that if a Lie group acts properly on a co-oriented contact manifold
preserving the contact structure, then the contact quotient is topologically a
stratified space (in the sense that a neighborhood of a point in the quotient
is a product of a disk with a cone on a compact stratified space). As a
corollary, we o... | math |
2,928 | Hamiltonian symplectomorphisms and the Berry phase | math.SG | On the space ${\cal L}$, of loops in the group of Hamiltonian
symplectomorphisms of a symplectic quantizable manifold, we define a closed
${\bf Z}$-valued 1-form $\Omega$. If $\Omega$ vanishes, the prequantization map
can be extended to a group representation. On ${\cal L}$ one can define an
action integral as an ${\bf... | math |
2,929 | Homotopy type of symplectomorphism groups of S^2 X S^2 | math.SG | In this paper we discuss the topology of the symplectomorphism group of a
product of two 2-dimensional spheres when the ratio of their areas lies in the
interval (1,2]. More precisely we compute the homotopy type of this
symplectomorphism group and we also show that the group contains two finite
dimensional Lie groups ... | math |
2,930 | Symplectomorphism groups and almost complex structures | math.SG | This paper studies groups of symplectomorphisms of ruled surfaces for
symplectic forms with varying cohomology class. This class is characterized by
the ratio R of the size of the base to that of the fiber. By considering
appropriate spaces of almost complex structures, we investigate how the
topological type of these ... | math |
2,931 | Symplectic Structures on Fiber Bundles | math.SG | Let $\pi: P\to B$ be a locally trivial fiber bundle over a connected CW
complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then
$\pi$ is said to be a symplectic fiber bundle if its structural group is the
group of symplectomorphisms $\Symp(M,\om)$, and is called Hamiltonian if this
group may be r... | math |
2,932 | Cohomological properties of ruled symplectic structures | math.SG | This survey presents some recent results by the authors and Polterovich on
the topological properties of ruled symplectic manifolds. The bundle M \to P
\to B that is associated with a ruled manifold has the group of Hamiltonian
symplectomorphisms of M as structure group if the base is simply connected.
Thus information... | math |
2,933 | Sets of singular restrictions of symplectic forms | math.SG | Let $\Omega$ be a non-singular syplectic form on some vector space $V$.
Denote by $S^{n}_{k}(\Omega)$ the set of all $k$-dimensional planes $s$ in $V$
such that the restriction of $\Omega$ onto $s$ is singular. For the cases when
$k=2,n-2$ a simple geometric characteristic of $S^{n}_{k}(\Omega)$ will be
described. | math |
2,934 | Lefschetz pencils and divisors in moduli space | math.SG | We study Lefschetz pencils on symplectic four-manifolds via the associated
spheres in the moduli spaces of curves, and in particular their intersections
with certain natural divisors. An invariant defined from such intersection
numbers can distinguish manifolds with torsion first Chern class. We prove that
pencils of l... | math |
2,935 | Geometric monodromy and the hyperbolic disc | math.SG | Symplectic four-manifolds give rise to Lefschetz fibrations, which are
determined by monodromy representations of free groups in mapping class groups.
We study the topology of Lefschetz fibrations by analysing the action of the
monodromy on the universal cover of a smooth fibre. We give new and simple
proofs that Lefsc... | math |
2,936 | Hofer-Zehnder capacity and length minimizing Hamiltonian paths | math.SG | We use the criteria of Lalonde and McDuff to show that a path that is
generated by a generic autonomous Hamiltonian is length minimizing with respect
to the Hofer norm among all homotopic paths provided that it induces no
non-constant closed trajectories in M. This generalizes a result of Hofer for
symplectomorphisms o... | math |
2,937 | On bilinear invariant differential operators acting on tensor fields on the symplectic manifold | math.SG | Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation
$\rho: GL(n)\longrightarrow GL(V)$. Locally, let $T(V)$ be the space of
sections of the tensor bundle with fiber $V$ over a sufficiently small open set
$U\subset M$, in other words, $T(V)$ is the space of tensor fields of type $V$
on $M$ on which... | math |
2,938 | Coherent Orientations in Symplectic Field Theory | math.SG | We study the coherent orientations of the moduli spaces of holomorphic curves
in Symplectic Field Theory, generalizing a construction due to Floer and Hofer.
In particular we examine their behavior at multiple closed Reeb orbits under
change of the asymptotic direction. The orientations are determined by a
certain choi... | math |
2,939 | Geometric variants of the Hofer norm | math.SG | This note discusses some geometrically defined seminorms on the group
$\Ham(M, \omega)$ of Hamiltonian diffeomorphisms of a closed symplectic
manifold $(M, \omega)$, giving conditions under which they are nondegenerate
and explaining their relation to the Hofer norm. As a consequence we show that
if an element in $\Ham... | math |
2,940 | Symplectically aspherical manifolds | math.SG | The main subjects of the paper is studying the fundamental groups of closed
symplectically aspherical manifolds. Motivated by some results of Gompf, we
introduce two classes of fundamental groups $\pi_1(M)$ of symplectically
aspherical manifolds $M$ with $\pi_2(M)=0$ and $\pi_2(M)\neq 0$. Relations
between these classe... | math |
2,941 | Action integrals along closed isotopies in coadjoint orbits | math.SG | Let ${\cal O}$ be the orbit of $\eta\in{\frak g}^*$ under the coadjoint
action of the compact Lie group $G$. We give two formulae for calculating the
action integral along a closed Hamiltonian isotopy on ${\cal O}$. The first one
expresses this action in terms of a particular character of the isotropy
subgroup of $\eta... | math |
2,942 | Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group | math.SG | In this paper we first apply the chain level Floer theory to the study of
Hofer's geometry of Hamiltonian diffeomorphism group in the cases without
quantum contribution: we prove that any quasi-autonomous Hamiltonian path on
weakly exact symplectic manifolds or any autonomous Hamiltonian path on
arbitrary symplectic ma... | math |
2,943 | Some title containing the words "homotopy" and "symplectic", e.g. this one | math.SG | Using a basic idea of Sullivan's rational homotopy theory, one can see a Lie
groupoid as the fundamental groupoid of its Lie algebroid. This paper studies
analogues of Lie algebroids with non-trivial higher homotopy. Using various
homotopy classes one can obtain e.g. central extensions of loop groups, or one
can integr... | math |
2,944 | A long exact sequence for symplectic Floer cohomology | math.SG | The long exact sequence describes how the Floer cohomology of two Lagrangian
submanifolds changes if one of them is modified by applying a Dehn twist. We
give a proof in the simplest case (no bubbling). The paper contains a certain
amount of material that may be of interest independently of the exact sequence:
in parti... | math |
2,945 | How to (symplectically) thread the eye of a (Lagrangian) needle | math.SG | We show that there exists no Lagrangian embeddings of the Klein bottle into
$\CC^{2}$. Using the same techniques we also give a new proof that any
Lagrangian torus in $\CC^2$ is smoothly isotopic to the Clifford torus. | math |
2,946 | Gromov-Witten invariants of symplectic quotients and adiabatic limits | math.SG | We study pseudoholomorphic curves in symplectic quotients as adiabatic limits
of solutions of a system of nonlinear first order elliptic partial differential
equations in the ambient symplectic manifold. The symplectic manifold carries a
Hamiltonian group action. The equations involve the Cauchy-Riemann operator
over a... | math |
2,947 | The equivariant cohomology of Hamiltonian $G$-spaces From Residual $S^1$ Actions | math.SG | We show that for a Hamiltonian action of a compact torus $G$ on a compact,
connected symplectic manifold $M$, the $G$-equivariant cohomology is determined
by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1
tori. This theorem allows us to compute the equivariant cohomology of certain
manifold... | math |
2,948 | Propagation in Hamiltonian dynamics and relative symplectic homology | math.SG | The main result asserts the existence of noncontractible periodic orbits for
compactly supported time dependent Hamiltonian systems on the unit cotangent
bundle of the torus or of a negatively curved manifold whenever the generating
Hamiltonian is sufficiently large over the zero section. The proof is based on
Floer ho... | math |
2,949 | Bending flows for sums of rank one matrices | math.SG | We study certain symplectic quotients of n-fold products of complex
projective m-space by the unitary group acting diagonally. After studying
nonemptiness and smoothness these quotients we construct the action-angle
variables, defined on an open dense subset of an integrable Hamiltonian system.
The semiclassical quanti... | math |
2,950 | An effective algorithm for the cohomology ring of symplectic reductions | math.SG | Let G be a compact torus acting on a compact symplectic manifold M in a
Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of
$\kappa:H_G^*(M)\to H^*(M//G)$ is generated by a small number of classes
$\alpha\in H_G^*(M)$ satisfying very explicit restriction properties. Our main
tool is the equivariant ... | math |
2,951 | Grafting Seiberg-Witten monopoles | math.SG | We demonstrate that the operation of taking disjoint unions of J-holomorphic
curves (and thus obtaining new J-holomorphic curves) has a Seiberg-Witten
counterpart. The main theorem asserts that, given two solutions (A_i, psi_i),
i=0,1 of the Seiberg-Witten equations for the Spin^c-structure W^+_{E_i}= E_i
direct sum (E... | math |
2,952 | A classification of topologically stable Poisson structures on a compact oriented surface | math.SG | Poisson structures vanishing linearly on a set of smooth closed disjoint
curves are generic in the set of all Poisson structures on a compact connected
oriented surface. We construct a complete set of invariants classifying these
structures up to an orientation-preserving Poisson isomorphism. We show that
there is a se... | math |
2,953 | Symplectically harmonic cohomology of nilmanifolds | math.SG | This paper can be considered as an extension to our paper [On symplectically
harmonic forms on six-dimensional nilmanifolds, Comment. Math. Helv. 76 (2001),
n 1, 89-109]. Also, it contains a brief survey of recent results on
symplectically harmonic cohomology. | math |
2,954 | Symplectic action around loops in $\text{Ham}(M)$ | math.SG | Let $\text{Ham(M)}$ be the group of Hamiltonian symplectomorphisms of a
quantizable, compact, symplectic manifold $(M,\omega)$. We prove the existence
of an action integral around loops in $\text{Ham(M)}$, and determine the value
of this action integral on particular loops when the manifold is a coadjoint
orbit. | math |
2,955 | The Chord Problem and a new method of filling by pseudoholomorphic curves | math.SG | In this paper we give a simple application of the filling methods developed
earlier to the chord problem in three dimensional contact geometry. | math |
2,956 | The symplectic vortex equations and invariants of Hamiltonian group actions | math.SG | In this paper we define invariants of Hamiltonian group actions for central
regular values of the moment map. The key hypotheses are that the moment map is
proper and that the ambient manifold is symplectically aspherical. The
invariants are based on the symplectic vortex equations. Applications include
an existence th... | math |
2,957 | Codimension one symplectic foliations | math.SG | We define the concept of symplectic foliation on a symplectic manifold and
provide a method of constructing many examples, by using asymptotically
holomorphic techniques. | math |
2,958 | Cohomology rings of symplectic quotients by circle actions | math.SG | In this article we are concerned with how to compute the cohomology ring of a
symplectic quotient by a circle action using the information we have about the
cohomology of the original manifold and some data at the fixed point set of the
action. Our method is based on the Tolman-Weitsman theorem which gives a
characteri... | math |
2,959 | An extension theorem in symplectic geometry | math.SG | We extend the ``Extension after Restriction Principle'' for symplectic
embeddings of bounded starlike domains to a large class of symplectic
embeddings of unbounded starlike domains. | math |
2,960 | On a question of Dusa McDuff | math.SG | Consider the $2n$-dimensional closed ball $B$ of radius 1 in the
$2n$-dimensional symplectic cylinder $Z = D \times R^{2n-2}$ over the closed
disc $D$ of radius 1. We construct for each $\epsilon >0$ a Hamiltonian
deformation $\phi$ of $B$ in $Z$ of energy less than $\epsilon$ such that the
area of each intersection of... | math |
2,961 | Lectures on Groups of Symplectomorphisms | math.SG | These notes combine material from short lecture courses given in Paris,
France, in July 2001 and in Srni, the Czech Republic, in January 2003. They
discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om)
from various points of view. Lectures 1 and 2 provide an overview of our
current knowledge of th... | math |
2,962 | Geodesic flows and contact toric manifolds | math.SG | These are notes for a course on contact manifolds and torus actions delivered
at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems
at Centre de Recerca Matem\`atica in Barcelona in July 2001. To be published by
Birkhauser. | math |
2,963 | Enumerative vs. Symplectic Invariants and Obstruction Bundles | math.SG | We give detailed descriptions of gluing pseudoholomorphic maps in symplectic
geometry, especially in the presence of an obstruction bundle. The main
motivation is to try to compare the symplectic and enumerative invariants of
algebraic manifolds. These descriptions can also be used to enumerate rational
curves with hig... | math |
2,964 | Hamiltonian structures on foliations | math.SG | We discuss hamiltonian structures of the Gelfand-Dorfman complex of
projectable vector fields and differential forms on a foliated manifold. Such a
structure defines a Poisson structure on the algebra of foliated functions, and
embeds the given foliation into a larger, generalized foliation with
presymplectic leaves. I... | math |
2,965 | The Hormander and Maslov Classes and Fomenko's Conjecture | math.SG | Some functorial properties are studied for the H\"{o}rmander classes defined
for symplectic bundles. The behaviour of the Chern first form on a Lagrangian
submanifold in an almost Hermitian manifold is also studied, and Fomenko's
conjecture about the behaviour of a Maslov class on minimal Lagrangian
submanifolds is con... | math |
2,966 | Vertical Cohomologies and Their Application to Completely Integrable Hamiltonian Systems | math.SG | Some functorial and topological properties of vertical cohomologies and their
application to completely integrable Hamiltonian systems are studied. | math |
2,967 | Braids and symplectic four-manifolds with abelian fundamental group | math.SG | We explain how a version of Floer homology can be used as an invariant of
symplectic manifolds with $b_1>0$. As a concrete example, we look at
four-manifolds produced from braids by a surgery construction. The outcome
shows that the invariant is nontrivial; however, it is an open question whether
it is stronger than th... | math |
2,968 | On the Floer homology of plumbed three-manifolds | math.SG | We calculate the Heegaard Floer homologies for three-manifolds obtained by
plumbings of spheres specified by certain graphs. Our class of graphs is
sufficiently large to describe, for example, all Seifert fibered rational
homology spheres. These calculations can be used to determine also these groups
for other three-ma... | math |
2,969 | The residue formula and the Tolman-Weitsman theorem | math.SG | We give a simple direct proof (for the case of Hamiltonian circle actions
with isolated fixed points) that Tolman and Weitsman's description of the
kernel of the Kirwan map (in other words the sum of those equivariant
cohomology classes vanishing on one side of a collection of hyperplanes) is
equivalent to the characte... | math |
2,970 | Symplectic surfaces and generic J-holomorphic structures on 4-manifolds | math.SG | It is a well known fact that every embedded symplectic surface $\Sigma$ in a
symplectic 4-manifold $(X^4,\omega)$ can be made $J$-holomorphic for some
almost-complex structure $J$ compatible with $\omega$. In this paper we
investigate when such a $J$ can be chosen from a generic set of almost-complex
structures. As an ... | math |
2,971 | A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds | math.SG | We compare Hofer's geometries on two spaces associated with a closed
symplectic manifold M. The first space is the group of Hamiltonian
diffeomorphisms. The second space L consists of all Lagrangian submanifolds of
$M \times M$ which are exact Lagrangian isotopic to the diagonal. We show that
in the case of a closed sy... | math |
2,972 | A geometric proof of Conn's linearization theorem for analytic Poisson structures | math.SG | We give a geometric proof of Conn's linearization theorem for analytic
Poisson structures, without using the fast convergence method. | math |
2,973 | A de Rham theorem for symplectic quotients | math.SG | We introduce a de Rham model for stratified spaces arising from symplectic
reduction. It turns out that the reduced symplectic form and its powers give
rise to well-defined cohomology classes, even on a singular symplectic
quotient. | math |
2,974 | Examples for nonequivalence of symplectic capacities | math.SG | We construct an open bounded star-shaped set in R^4 whose cylindrical
capacity is strictly bigger than its proper displacement energy. | math |
2,975 | Real loci of symplectic reductions | math.SG | Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action
of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an
anti-symplectic involution $\sigma$ compatible with the $T$-action. The real
locus of $M$ is the fixed point set $M^\sigma$ of $\sigma$. Duistermaat
introduced real ... | math |
2,976 | Symplectic conifold transitions | math.SG | We introduce a symplectic surgery in six dimensions which collapses
Lagrangian three-spheres and replaces them by symplectic two-spheres. Under
mirror symmetry it corresponds to an operation on complex 3-folds studied by
Clemens, Friedman and Tian. We describe several examples which show that there
are either many more... | math |
2,977 | Toward a topological characterization of symplectic manifolds | math.SG | A topological condition is given, characterizing which closed manifolds in
dimensions < 8 (and conjecturally in general) admit symplectic structures. The
condition is the existence of a certain fibration-like structure called a
hyperpencil. A deformation class of hyperpencils on a manifold X of any even
dimension is sh... | math |
2,978 | Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair | math.SG | Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost
complex structure, and $L$ be a Lagrangian submanifold. The stable
compactification of the moduli space of parametrized $J$-holomorphic curves in
$X$ with boundary in $L$ (with prescribed topological data) is compact and
Hausdorff in Gromov's $C^... | math |
2,979 | On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces | math.SG | We study the connectedness of the moduli space of gauge equivalence classes
of flat G-connections on a compact orientable surface or a compact
nonorientable surface for a class of compact connected Lie groups. This class
includes all the compact, connected, simply connected Lie groups, and some
non-semisimple classical... | math |
2,980 | Spectral invariants and length minimizing property of Hamiltonian paths | math.SG | In this paper we provide a criterion for the quasi-autonomous Hamiltonian
path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds
$(M,\omega)$ to be length minimizing in its homotopy class in terms of the
spectral invariants $\rho(G;1)$ that the author has recently constructed
(math.SG/0206092). As an appl... | math |
2,981 | Cohomological Splitting of Coadjoint Orbits | math.SG | The rational cohomology of a coadjoint orbit ${\cal O}$ is expressed as
tensor product of the cohomology of other coadjoint orbits ${\cal O}_k$, with $
\hbox{dim} {\cal O}_k< \hbox{dim} {\cal O}$. | math |
2,982 | Complexity one Hamiltonian SU(2) and SO(3) actions | math.SG | We consider compact connected six dimensional symplectic manifolds with
Hamiltonian SU(2) or SO(3) actions with cyclic principal stabilizers. We
classify such manifolds up to equivariant symplectomorphisms. | math |
2,983 | A remark on the c--splitting conjecture | math.SG | Let $M$ be a closed symplectic manifold and suppose $M\to P\to B$ is a
Hamiltonian fibration. Lalonde and McDuff raised the question whether one
always has $H^*(P;\mathbb Q)=H^*(M;\mathbb Q)\otimes H^*(B;\mathbb Q)$ as
vector spaces. This is known as the c--splitting conjecture. They showed, that
this indeed holds when... | math |
2,984 | Symplectic four-manifolds and conformal blocks | math.SG | We apply ideas from conformal field theory to study symplectic
four-manifolds, by using modular functors to "linearise" Lefschetz fibrations.
In Chern-Simons theory this leads to the study of parabolic vector bundles of
conformal blocks. Motivated by the Hard Lefschetz theorem, we show the bundles
of SU(2) conformal bl... | math |
2,985 | Distinguishing the Chambers of the Moment Polytope | math.SG | Let M be a compact manifold with a Hamiltonian T action and moment map Phi.
The restriction map in equivariant cohomology from M to a level set Phi^{-1}(p)
is a surjection, and we denote the kernel by I_p. When T has isolated fixed
points, we show that I_p distinguishes the chambers of the moment polytope for
M. In par... | math |
2,986 | Cup-length estimate for Lagrangian intersections | math.SG | In this paper we consider the Arnold conjecture on the Lagrangian
intersections of some closed Lagrangian submanifold of a closed symplectic
manifold with its image of a Hamiltonian diffeomorphism. We prove that if the
Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a
topology number defined by... | math |
2,987 | On Generalized Moment Maps for Symplectic Compact Group Actions | math.SG | A generalized moment map is proposed for arbitrary symplectic actions of
compact connected Lie groups on closed symplectic manifolds, in the spirit of
the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian
circle actions. We study equivariance properties of generalized moments, show
that they al... | math |
2,988 | On the holomorphicity of genus two Lefschetz fibrations | math.SG | We prove that any genus-2 Lefschetz fibration without reducible fibers and
with ``transitive monodromy'' is holomorphic. The latter condition comprises
all cases where the number of singular fibers is not congruent to 0 modulo 40.
An auxiliary statement of independent interest is the holomorphicity of
symplectic surfac... | math |
2,989 | On the Homotopy of Symplectomorphism Groups of Homogeneous Spaces | math.SG | Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group
$G$. Under certain hypotheses we prove that $#(\pi_1(\text{Ham}({\cal O})))\geq
#(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian
symplectomorphisms of ${\cal O}$. | math |
2,990 | Almost Homogeneous Poisson Spaces | math.SG | We prove that any holomorphic Poisson manifold has an open symplectic leaf
which is a pseudo-K\"ahler submanifold, and we define an obstruction to study
the equivariance of momentum map for tangential Poisson action. Some properties
of almost homogeneous Poisson manifolds are studied and we show that any
compact symple... | math |
2,991 | GKM theory for torus actions with non-isolated fixed points | math.SG | Let $M^{2d}$ be a compact symplectic manifold and $T$ a compact
$n$-dimensional torus. A Hamiltonian action, $\tau$, of $T$ on $M$ is a GKM
action if, for every $p \in M^T$, the isotropy representation of $T$ on $T_pM$
has pair-wise linearly independent weights. For such an action the projection
of the set of zero and ... | math |
2,992 | Contact 3-manifolds with infinitely many Stein fillings | math.SG | Infinitely many contact 3-manifolds each admitting infinitely many, pairwise
non-diffeomorphic Stein fillings are constructed. We use Lefschetz fibrations
in our constructions and compute their first homologies to distinguish the
fillings. | math |
2,993 | Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions | math.SG | In a previous paper, the author introduced a Floer-theoretic torsion
invariant I_F, which roughly takes the form of a product of a power series
counting perturbed pseudo-holomorphic tori, and the Reidemeister torsion of the
symplectic Floer complex. We pointed out the formal resemblance of I_F with a
generating functio... | math |
2,994 | Large Radius Limit and SYZ Fibrations of Hyper-Kahler Manifolds | math.SG | In this paper the relations between the existence of Lagrangian fibration of
Hyper-K\"{a}hler manifolds and the existence of the Large Radius Limit is
established. It is proved that if the the rank of the second homology group of
a Hyper-K\"{a}hler manifold N of complex dimension $2n\geq4$ is at least 5,
then there exi... | math |
2,995 | Lectures on four-dimensional Dehn twists | math.SG | These are notes from the 2003 C.I.M.E. summer school "symplectic 4-manifolds
and algebraic surfaces". They cover the same material as the author's (by now
ancient) Ph.D. thesis. | math |
2,996 | Noncentral extensions as anomalies in classical dynamical systems | math.SG | A two cocycle is associated to any action of a Lie group on a symplectic
manifold. This allows to enlarge the concept of anomaly in classical dynamical
systems considered by F. Toppan [in J. Nonlinear Math. Phys. 8, no.3 (2001)
518-533] so as to encompass some extensions of Lie algebras related to
noncanonical actions. | math |
2,997 | Proofs On Arnold Chord Conjecture And Weinstein Conjecture In M\times C | math.SG | We give new proofs on Arnold Chord Conjecture and Weinstein Conjecture in
M\times C which generalizes the previous works. | math |
2,998 | A Proof On Weinstein Conjecture On Cotangent Bundles Of Open Manifold | math.SG | We give an proof on the Weinstein conjecture on the cotangent bundles of open
manifolds. Its proof is based on Gromov's nonlinear Fredholm alternative. | math |
2,999 | A Proof On Arnold Chord Conjecture In Cotangent Bundles | math.SG | We prove the Arnold chord conjecture on cotangent bundles of open manifold by
Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves. | math |
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