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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 Kontsevich [alg-geom/9710032] and Manin [math/9801006], to the
structure of a Frobenius manifold on the de Rham cohomology of $M$.
|
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 topology of the group of symplectomorphisms of $S^2\times S^2$.
By globalizing standard gluing constructions in the theory of stable maps, we
show that the strata of $\Jj$ are Fr\'echet manifolds of finite codimension,
and that the normal link of each stratum is a finite dimensional stratified
space. The topology of these links turns out to be surprisingly intricate, and
we work out certain cases. Our arguments apply also to other ruled surfaces,
though they give complete information only for bundles over $S^2$ and $T^2$.
|
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 actions on compact K\"ahler manifolds, this result was proved
recently by Teleman and by Braverman. Our approach is applying the holomorphic
instanton complex to the prequantum line bundles over the symplectic cuts. We
also settle a conjecture of Zhang and the present author on the exact sequence
of higher cohomology groups in the context of symplectic cutting.
|
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 a symplectic circle action. In particular, within the cohomology
classes of symplectic structures, the subset admitting a circle action is not
closed.
|
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-2 covectors are decomposable.
|
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 the theory. Several applications are given, amongst them:
(1) topological restrictions on Lagrangian submanifolds of projective space,
(2) the existence of "symplectically knotted" Lagrangian spheres on a K3
surface, (3) a result about the symplectic monodromy of weighted homogeneous
hypersurface singularities.
Revised version: minor modifications, journal reference added.
|
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, becomes homotopy equivalent to the space of all continuous maps,
lim_{->} Hol_{x_0}(P^1,X) = Omega^2 X.
This limit will be viewed as a kind of stabilization of Hol_{x_0}(P^1,X). We
conjecture that this stability holds if and only if an evaluation map E:
lim_{->} Hol_{x_0}(P^1,X) -> X is a quasifibration. In this paper we will prove
that in the presence of this quasifibration condition, then the stability
property holds if and only if the Morse theoretic flow category (defined in
[4]) of the symplectic action functional on the Z-cover of the loop space, L~X,
defined by the symplectic form, has a classifying space that realizes the
homotopy type of L~X. We conjecture that in the presence of this quasifibration
condition, this Morse theoretic condition always holds. We will prove this in
the case of X a homogeneous space, thereby giving an alternate proof of the
stability theorem for holomorphic spheres for a projective homogeneous variety
originally due to Gravesen [7].
|
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 holomorphic) Lefschetz fibration has positive "symplectic
volume"; it evaluates positively with the Kahler class. Some other applications
of the signature formula and some more general results for genus two fibrations
are discussed.
|
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 Etingof and Varchenko. A similar result holds for the maximal
compact subgroup K, and we get a family of K-homogeneous Poisson structures on
$K/T$, where $T = K \cap H$ is a maximal torus of K. This family exhausts all
K-homogeneous Poisson structures on $K/T$ up to isomorphisms. We study some
Poisson geometrical properties of members of this family such as their
symplectic leaves, their modular classes, and the moment maps for the T-action.
|
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, in this paper we shall be concerned with the solutions of the
following differential system in six variables h_1,h_2,h_3,H_1,H_2,H_3
dH_1/dt = H_2 H_3 (1/c_3 - 1/c_2) + h_2 a_3 - h_3 a_2,
dH_2/dt = H_1 H_3 (1/c_1 - 1/c_3) + h_3 a_1 - h_1 a_3,
dH_3/dt = H_1 H_2 (1/c_2 - 1/c_1) + h_1 a_2 - h_2 a_1,
dh_1/dt = h_2 H_3/c_3 - h_3 H_2/c_2 + k (H_2 a_3 - H_3 a_2),
dh_2/dt = h_3 H_1/c_1 - h_1 H_3/c_3 + k (H_3 a_1 - H_1 a_3),
dh_3/dt = h_1 H_2/c_2 - h_2 H_1/c_1 + k (H_1 a_2 - H_2 a_1),
in which a_1,a_2,a_3,c_1,c_2,c_3 and k are constants.
|
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 a nonabelian group, showing how to reduce to
the maximal torus.)
Let X be a symplectic manifold, with a Hamiltonian action of a compact torus
T. The main topological result of this paper describes an explicit cobordism
that exists between a symplectic quotient of X by T, and a collection of
iterated projective bundles over components of the set of T-fixed-points.
The characteristic classes of these bundles can be determined explicitly, and
another result uses this to give formulae for integrals of cohomology classes
over the symplectic quotient, in terms of data localized at the T-fixed points
of X.
|
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
of X//G in terms of the rational cohomology ring of X//T; an `integration'
formula, which expresses cohomology pairings on X//G in terms of cohomology
pairings on X//T; and an index formula, which expresses the indices of elliptic
operators on X//G in terms of indices on X//T.
(The results of this paper are complemented by the results in a companion
paper, in which different techniques are used to derive formulae for cohomology
pairings on symplectic quotients X//T, where T is a torus, in terms of the
T-fixed points of X. That paper also gives some applications of the formulae
proved here.)
|
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 that certain 6-dimensional nilmanifolds possess families of
homogeneous symplectic forms $\omega_t$ for which numbers $h_k(M,\omega_t)$
vary with respect to t. This gives an affirmative answer to a question raised
by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of
4-dimensional nilmanifolds which do not admit such variations by a remark of
Dong Yan.
|
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 relations between such homotopy properties seem to be
not analyzed. This analysis may shed a new light on topology of symplectic
manifolds. In the paper, we summarize our knowledge in tables (different in the
simply-connected and in symplectically aspherical cases). Also, we discuss the
variation of symplectically harmonic Betti numbers on some 6-dimensional
manifolds.
|
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 (essentially) the Gassner representation.
|
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 on invariants introduced by Polterovich for loops of
Hamiltonian symplectomorphisms. One of these is the minimal Hofer length in a
given Hamiltonian isotopy class. We determine the exact values of these
invariants for loops of projective Lagrangian planes. The proof uses the Gromov
invariants of an associated symplectic fibration over the 2-disc with a
Lagrangian subbundle over the boundary.
|
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 n-dimensional torus symplectic action on a 2n-dimensional
symplectic manifold, with fixed point, is Hamiltonian. We finally prove that an
extension of Kirwan's convexity theorem (for compact group actions) is also
true.
|
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 manifolds.
|
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 main result is a geometric criterion for the neighborhood
equivalence between Poisson structures over the same leaf. This criterion gives
a Poisson analog of the relative Darboux theorem due to Weinstein. Within the
category of the algebroids, coupling tensors are introduced on the dual of the
isotropy of a transitive Lie algebroid over a symplectic base. As a basic
application of these results, we show that there is a well defined notion of a
``linearized'' Poisson structure over a symplectic leaf which gives rise to a
natural model for the linearization problem.
|
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 obtain that symplectic quotients for proper Hamiltonian actions
are topologically stratified spaces in this strong sense thereby extending and
simplifying previous work.
|
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 ${\bf R}/{\bf Z}$-valued function, and the cohomology
class $[\Omega]$ is the obstruction to the lifting of that action integral to
an ${\bf R}$-valued function. The form $\Omega$ also defines a natural grading
on $\pi_1({\cal L})$.
|
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 generating the homotopy. A key step in this work is to
calculate the mod 2 homology of the group of symplectomorphisms. Although this
homology has a finite number of generators with respect to the Pontryagin
product, it is unexpected large containing in particular a free noncommutative
ring with 3 generators.
|
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 groups changes as R increases. If the base is a
sphere, this changes precisely when R passes an integer, and for general bases
it stabilizes as R goes to infinity. Our results extend and make more precise
some of the conclusions of Abreu--McDuff concerning the rational homotopy type
of these groups for rational ruled surfaces.
|
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 reduced to the group $\Ham(M,\om)$ of Hamiltonian
symplectomorphisms. In this paper, building on prior work by Seidel and
Lalonde, McDuff and Polterovich, we show that these bundles have interesting
cohomological properties. In particular, for many bases $B$ (for example when
$B$ is a sphere, a coadjoint orbit or a product of complex projective spaces)
the rational cohomology of $P$ is the tensor product of the cohomology of $B$
with that of $M$. As a consequence the natural action of the rational homology
$H_k(\Ham(M))$ on $H_*(M)$ is trivial for all $M$ and all $k > 0$.
Added: The erratum makes a small change to Theorem 1.1 concerning the
characterization of Hamiltonian bundles.
|
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 about Hamiltonian bundles gives stability results for ruled
structures as well as obstructions to their existence.
|
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 large degree always give spheres which behave `homologically' like
rational curves; contrastingly, we give the first constructive example of a
symplectic non-holomorphic Lefschetz pencil. We also prove that only finitely
many values of signature or Euler characteristic are realised by manifolds
admitting Lefschetz pencils of genus two curves.
|
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 Lefschetz fibrations arising from pencils (i.e. with exceptional
sections) never split as non-trivial fibre sums, and that no simple closed
curve can be invariant to isotopy under the monodromy representation.
|
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 of Euclidean space. The proof for general M uses Liu-Tian's
construction of S^1-invariant virtual moduli cycles. As a corollary, we find
that any semifree action of S^1 on M gives rise to a nontrivial element in the
fundamental group of the symplectomorphism group of M. We also establish a
version of the area-capacity inequality for quasicylinders.
|
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 the group $\Diff (M)$ of diffeomorphisms of $M$ naturally acts.
Elsewhere, the author classified the $\Diff (M)$-invariant differential
operators $D: T(V_{1})\otimes T(V_{2})\longrightarrow T(V_{3})$ for irreducible
fibers with lowest weight. Here the result is generalized to bilinear operators
invariant with respect to the group $\Diff_{\omega}(M)$ of symplectomorphisms
of the symplectic manifold $(M, \omega)$. We classify all first order invariant
operators; the list of other operators is conjectural. Among the new operators
we mention a 2nd order one which determins an ``algebra'' structure on the
space of metrics (symmetric forms) on $M$.
|
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 choice of orientation at each closed Reeb orbit, that is similar to the
orientation of the unstable tangent spaces of critical points in
finite--dimensional Morse theory.
|
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 $\Ham(M, \omega)$ is sufficiently close to the identity in the
$C^{2}$-topology then it may be joined to the identity by a path whose Hofer
length is minimal among all paths, not just among paths in the same homotopy
class relative to endpoints. Thus, true geodesics always exist for the Hofer
norm. The main step in the proof is to show that a "weighted" version of the
nonsqueezing theorem holds for all fibrations over $S^2$ generated by
sufficiently short loops.Further, an example is given showing that the Hofer
norm may differ from the sum of the one sided seminorms.
|
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 classes are discussed. We show that several important classes of
groups can be realized in both classes, while some of groups can be realized in
the first class but not in the second one. Also, we notice that there are some
interesting dimensional phenomena in the realization problem.
The above results are framed by a general research of symplectically
aspherical manifolds. For example, we find some conditions which imply that the
Gompf sum of symplectically aspherical manifolds is symplectically aspherical,
or that a total space of a bundle is symplectically aspherical, etc.
|
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 $\eta$. In the second one is involved the character of an
irreducible representation of $G$.
|
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 manifolds is length minimizing in its homotopy class with
fixed ends, as long as it has a fixed maximum and a fixed minimum which are not
over-twisted and all of its contractible periodic orbits of period less than
one are sufficiently $C^1$-small. Next we give a construction of new invariant
norm of Viterbo's type on the Hamiltonian diffeomorphism group of arbitrary
compact symplectic manifolds.
|
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 integrate a Lie biagebroid to a double symplectic groupoid. When combined
with symplectic geometry, this idea leads to an infinite sequence of notions,
starting with sympectic manifolds, Poisson manifolds and Courant algebroids.
They are interrelated with higher-dimensional variational problems, and one can
use them to define higher-dimensional Hamiltonian mechanics.
|
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 particular, chapter 1 covers "symplectic Picard-Lefschetz theory" in some
detail, and chapter 2 contains a generalization of the usual TQFT setup for
Floer cohomology.
|
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 Riemann surface, twisted by a connection, and couple the curvature of
the connection with the moment map. Our main theorem asserts that the genus
zero invariants of Hamiltonian group actions defined by these equations are
related to the genus zero Gromov--Witten invariants of the symplectic quotient
(in the monotone case) via a natural ring homomorphism from the equivariant
cohomology of the ambient space to the quantum cohomology of the quotient.
|
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
manifolds, which have pieces that are four-dimensional or smaller. We give
several examples of the computations that this allows.
|
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 homology and on the notion of a relative symplectic capacity.
Applications include results about propagation properties of sequential
Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle
actions, and smooth Lagrangian skeletons in Stein manifolds.
|
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 quantization of this system reproduces formulas from the
representation theory of the unitary group.
|
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 Kirwan map, a natural map from the G-equivariant
cohomology of M to the G/T-equivariant cohomology of the symplectic reduction
of M by T. We show this map is surjective. This is an equivariant version of
the well-known result that the (nonequivariant) Kirwan map $\kappa:H_G^*(M)\to
H^*(M//G)$ is surjective. We also compute the kernel of the equivariant Kirwan
map, generalizing the result due to Tolman and Weitsman in the case T=G and
allowing us to apply their methods inductively. This result is new even in the
case that dim T = 1. We close with a worked example: the cohomology ring of the
product of two $\C P^2$s, quotiented by the diagonal 2-torus action.
|
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_i tensor K^{-1}) (with certain restrictions), there is a solution
(A, psi) of the Seiberg-Witten equations for the Spin^c-structure W_E with E=
E_0 tensor E_1, obtained by `grafting' the two solutions (A_i, psi_i).
|
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 set of non-trivial infinitesimal deformations which generate the
second Poisson cohomology and such that each of the deformations changes
exactly one of the classifying invariants. As an example, we consider Poisson
structures on the sphere which vanish linearly on a set of smooth closed
disjoint curves.
|
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 theorem for relative periodic orbits, a computation for circle
actions on a complex vector space, and a theorem about the relation between the
invariants introduced here and the Seiberg--Witten invariants of a product of a
Riemann surface with a two-sphere.
|
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
characterization of the kernel of the Kirwan map. First we compute a generating
set for the kernel of the Kirwan map for the case of product of compact
connected manifolds such that the cohomology ring of each of them is generated
by a degree two class. We assume the fixed point set is isolated; however the
circle action only needs to be ``formally Hamiltonian''. By identifying the
kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply
this result to some special cases and in particular to the case of products of
two dimensional spheres. We show that the results of Kalkman and
Hausmann-Knutson are special cases of our result.
|
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 $\phi (B)$ with the disc $D \times \{x\}$, $x \in
R^{2n-2}$, is less than $\epsilon$.
|
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 their algebraic, geometric and homotopy theoretic
properties. Lecture 3 sketches the arguments used by Gromov, Abreu and
Abreu-McDuff to figure out the rational homotopy type of these groups in the
cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve
techniques. Much of the recent progress in understanding the geometry and
topology of these groups has come from studying the properties of fibrations
with the manifold M as fiber and structural group equal either to the
symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base
is S^2 has proved particularly important. Lecture 4 describes the geometry of
Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten
invariants via the Seidel representation. It ends by sketching Entov's
explanation of the ABW inequalities for eigenvalues of products of special
unitary matrices. Finally in Lecture 6 we apply the ideas developed in the
previous two lectures to demonstrate the existence of (short) paths in
Ham(M,\om) that minimize the Hofer norm over all paths with the given
endpoints.
|
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 high-order degeneracies of local nature in projective spaces.
|
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. In a so-called tame case, the structure is induced by a
Poisson structure of the manifold. Cohomology spaces and classes relevant to
geometric quantization are also considered.
|
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 considered.
|
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 the known ones.
|
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-manifolds, including the product of a circle with a genus two
surface.
|
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 characterization of this kernel given by the residue theorem
of Jeffrey and Kirwan.
|
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 application we give examples of smooth and non-empty
Seiberg-Witten and Gromov-Witten moduli spaces whose associated invariants are
zero.
|
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 symplectic manifold with $\pi_2(M) = 0$, the canonical
embedding of Ham(M) into L, f $\mapsto$ graph(f) is not an isometric embedding,
although it preserves Hofer's length of smooth paths.
|
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 loci, and extended several theorems of symplectic geometry to
real loci. In this paper, we extend another classical result of symplectic
geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute
the kernel of the real Kirwan map. These results are direct consequences of
techniques introduced by Tolman and Weitsman. In some examples, these results
allow us to show that a symplectic reduction $M/ /T$ has the same ordinary
cohomology as its real locus $(M/ /T)^{\sigma_{red}}$, with degrees halved.
This extends Duistermaat's original result on real loci to a case in which
there is not a natural Hamiltonian torus action.
|
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 Calabi-Yau manifolds (e.g. rigid ones) than previously
thought or there exist ``symplectic Calabi-Yaus'' -- non-Kaehler symplectic
6-folds with c_1=0. The analogous surgery in four dimensions, with a
generalisation to ADE-trees of Lagrangians, implies that the canonical class of
a minimal complex surface contains symplectic forms if and only if it has
positive square.
|
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 shown to determine an isotopy class of symplectic structures on X.
This provides an inverse (at least in dimensions < 8) to Donaldson's program
for constructing linear systems on symplectic manifolds. It follows that (at
least in dimensions < 8) the set of deformation classes of hyperpencils
canonically maps onto the set of isotopy classes of rational symplectic forms
up to positive scale, topologically determining a dense subset of all
symplectic forms up to an equivalence relation on hyperpencils. Other
applications of the main techniques are presented, including the construction
of symplectic structures on domains of locally holomorphic maps, and on
high-dimensional Lefschetz pencils and other linear systems.
|
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 $C^\infty$-topology. We construct a Kuranishi structure
with corners in the sense of Fukaya and Ono. This Kuranishi structure is
orientable if $L$ is spin. In the special case where the expected dimension of
the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$
which preserves $J$ and acts freely on $L$, we define the Euler number for this
$S^1$ equivariant pair and the prescribed topological data. We conjecture that
this rational number is the one computed by localization techniques using the
given $S^1$ action.
|
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 groups including U(n) and Spin^C(n).
|
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 application, we prove that any autonomous Hamiltonian
path on arbitrary closed symplectic manifolds is length minimizing in {\it its
homotopy class} with fixed ends, when it has no contractible periodic orbits
{\it of period one}, has a maximum and a minimum point which are generically
under-twisted and all of its critical points are nondegenerate in the Floer
theoretic sense. This is a sequel to the papers math.SG/0104243 and
math.SG/0206092.
|
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 whenever the base is a sphere. Using their theorem we will
prove the c--splitting conjecture for arbitrary base $B$ and fibers $M$ which
satisfy a weakening of the Hard Lefschetz condition.
|
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 blocks associated to Kaehler surfaces are Brill-Noether
special, although the associated flat connexions may be irreducible if the
surface is simply-connected and not spin.
|
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 particular, counting the number of distinct ideals I_p as p varies over
different chambers is equivalent to counting the number of chambers.
|
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 the Lagrangian submanifold, then the Arnold
conjecture is true in the degenerated (non-transversal) case.
|
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 allow reduction procedures, and obtain in the torus case a version of
the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we
reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions
are Hamiltonian, and give a symplectic proof of the finiteness of certain
symmetry groups of compact oriented surfaces.
|
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 surfaces in S^2-bundles over S^2, of relative degree up to 7 over
the base, and of symplectic surfaces in CP^2 of degree up to 17.
|
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 symplectic Poisson homogeneous space is a torus bundle over a dressing
orbit.
|
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 one-dimensional orbits onto $M/T$ is a regular
$d$-valent graph; and Goresky, Kottwitz and MacPherson have proved that the
equivariant cohomology of $M$ can be computed from the combinatorics of this
graph. (See \cite{GKM:eqcohom}.) In this paper we define a ``GKM action with
non-isolated fixed points'' to be an action, $\tau$, of $T$ on $M$ with the
property that for every connected component, $F$ of $M^T$ and $ p \in F$ the
isotropy representation of $T$ on the normal space to $F$ at $p$ has pair-wise
linearly independent weights. For such an action, we show that all components
of $M^T$ are diffeomorphic and prove an analogue of the theorem above.
|
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 function of genus 1 Gromov invariant; furthermore, for heuristic
reasons one also expects a relation with the 1-loop generating function in the
A-model side of mirror symmetry, which counts genus 1 holomorphic curves.
The present article makes this expected relation precise in the simplest
cases, in two variants of the I_F defined in the earlier work: the lagrangian
intersection version, I_F(L, L'), and an S^1-equivariant version, I_F^{S^1}.
As a by-product, we obtain some existence results of noncontractible periodic
orbits in symplectic dynamics. For example, the results of Gatien-Lalonde are
extended to a much wider class of manifolds.
The two versions I_F(L, L') and I_F^{S^1} are only minimally developed in
this paper, leaving fuller accounts to future work. The lagrangian intersection
version, I_F(L, L'), should be viewed as a simplest example of a rigorous
definition of the higher-loop ``open Gromov-Witten invariants'' proposed by
physicists.
|
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 exists an unipotent element T in the mapping class group $\Gamma
$(N) such that its action on the second cohomology group satisfies
$(T-id)^{2}\neq0$ and $(T-id)^{3}=0.$ A Theorem of Verbitsky implies that the
symmetric power $S^{n}(T)$ acts on $H^{2n}$ and it satisfies $(S^{n}%
(T)-id)^{2n}\neq0$ and $(S^{n}(T)-id)^{2n+1}=0.$ This fact established the
existence of Large Radius Limit for Hyper-K\"{a}hler manifolds for polarized
algebraic Hyper-K\"{a}hler manifolds. Using the theory of vanishing cycles it
is proved that if a Hyper-K\"{a}hler manifold admits a Lagrangian fibration
then the rank of the second homology group is greater than or equal to five. It
is also proved that the fibre of any Lagrangian fibration of a Hyper-K\"{a}hler
manifold is homological to a vanishing invariant $2n$ cycle of a maximal
unipotent element acting on the middle homology. According to Clemens this
vanishing invariant cycle can be realized as a torus. I conjecture that the SYZ
conjecture implies finiteness of the topological types of Hyper-K\"{a}hler
manifolds of fix dimension.
|
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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