Hamiltonian systems, symplectic flows, classical integrable systems

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4
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2answers
826 views

A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...
0
votes
1answer
178 views

Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
2
votes
1answer
136 views

Symplectic (contact) structure on $M_{n}(\mathbb{R})$

Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under ...
0
votes
0answers
111 views

On Gromov's Theorem on Symplectic Homotopy

I want to understand the proof of the following theorem due to Gromov which I'll state in the context of Euclidean spaces. While I tried to read the proof from Macduff-Salamon, it turned out that my ...
3
votes
1answer
192 views

Boundary geometry of a contact manifold

Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact ...
1
vote
1answer
103 views

Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...
2
votes
1answer
194 views

An identity for Futaki-Donaldson invariant

Let $(X,L)$ be a polarized projective variety Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of : a scheme $\mathfrak X$ with a $\mathbb C^*$-action a ...
2
votes
3answers
200 views

Computing the coefficients of the polynomial $\dim H^0(X,L^k)$ in non-smooth case

Let $(X,L,\omega)$ be a projective variety with polarization $L$. then we can write $$\dim H^0(X,L^k)=a_0k^n+a_1k^{n-1}+...$$ If $X$ is smooth then $a_0=Vol(X)$ and we can compute $a_i$. If $X$ is ...
7
votes
1answer
308 views

Examples of Symplectic Questions Solved by ``Mirror Symmetry Translation'' to Complex Questions

According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them ...
7
votes
0answers
129 views

Can a symplectic manifold be recovered from its Lagrangians?

Something I have wondered idly about from time to time is: If $(M,\omega), (M',\omega')$ are symplectic manifolds, and you "know what the Lagrangians $L \subset M$ resp. $L' \subset M'$ are," can ...
14
votes
4answers
1k views

How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...
11
votes
1answer
282 views

Thom conjecture in CP3

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...
0
votes
0answers
103 views

moduli space of meromorphic $G$-Higgs bundles

I want to clarify with some topics in moduli space of semistable $G$-Higgs bundles on curve $X$ (genus $g$ is large enough) of fixing topological type $d \in \pi_1(G)$. Simpson's construction gives us ...
4
votes
0answers
273 views

A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...
15
votes
5answers
2k views

Roadmap for Mirror Symmetry

I am interested in learning Mirror Symmetry, both from the SYZ and Homological point of view. I am taking a reading course in Mirror Symmetry, which will focus on the SYZ side. I know basic Complex ...
37
votes
9answers
4k views

Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...
4
votes
1answer
85 views

Abstract stationary phase

I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what ...
0
votes
0answers
93 views

Lagrangian fibrations with isolated singular fibers

Let $(M,\omega)$ be a symplectic manifold, and assume $\dim_\mathbb{R}(M)>4$. Suppose there exists a real Lagrangian fibration $\pi:M\rightarrow Q$ so that $Q$ becomes an integral affine manifold ...
3
votes
1answer
236 views

Generalization of Giroux's Theorem for Higher Dimensions?

Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized: In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any ...
0
votes
0answers
66 views

Floer homology for manifolds with contact boundary

I am reading the paper " A survey of floer homology for manifolds with contact boundary" by A. Oancea. In theorem 2.1, he discussed the invariance of Viterbo's theory of symplectic homology with ...
1
vote
0answers
121 views

Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...
3
votes
0answers
110 views

hard Lefschetz property of symplectic manifolds

Let $(M^{2n},\omega)$ be a compact symplectic manifold. We say that $M$ has the hard Lefscthetz property iff the homomorphisms of cohomology $$L^{k}:H^{n-k}(M)\longrightarrow H^{n+k}(M)$$ defined by ...
1
vote
1answer
195 views

Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles

Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl_p(\mathbb CP^2)$, then for the Line bundle $L=-K_X$, we have for Futaki invariant $Fut_L\neq 0$, I want to see, what about ...
4
votes
2answers
657 views

Applications of Floer homology

Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture. Every answer would be appreciated.
0
votes
1answer
167 views

floer homology and viterbo's theorem

Let $M$ be a compact manifold. In their paper "On the Floer Homology of Cotangent Bundles", A. ABBONDANDOLO and M. SCHWARZ define the Floer homology of $T^*M$ by looking at 1-periodic Hamiltonian ...
0
votes
1answer
111 views

Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds

We know every contact manifold admits a symplectic submanifold, e.g., by Giroux's bijection : if $\omega$ is a contact form for a 3-manifold $M^3$ , then $d \omega$ is symplectic on the fibers of the ...
1
vote
1answer
169 views

Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much. Let $M$ be a n dimensional manifold. ...
45
votes
4answers
3k views

Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
2
votes
0answers
147 views

Fully faithful embedding of the exact Fukaya category

Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...
3
votes
1answer
479 views

a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
2
votes
1answer
387 views

Symplectic quotient of projective variety is projective?

Let $G$ be a compact connected Lie group and $\mathfrak g^*$ be dual of Lie algebra $\mathfrak g$. Let $M$ be a compact projective variety and $G$ act on $M$ freely and $M$ is $G$ equivariant, and ...
0
votes
1answer
203 views

Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...
2
votes
0answers
154 views

Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...
3
votes
1answer
126 views

What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...
0
votes
0answers
65 views

About some 'rigidity theorem' for the Kahler forms on projective bundles

Let $E\to X$ be a holomorphic vector bundle over a compact Kahler manifold $X$ with Kahler form $\omega_{X}$. For a given hermitian metric on $E$, let $\omega_{E}$ be the Chern form of the line bundle ...
3
votes
1answer
79 views

Embedded Contact Homology and Manifold Decompositions

Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them. In the symplectic ...
4
votes
0answers
96 views

Gromov width of cotangent disk bundle

Given a symplectic manifold $(M^{2n},\omega)$, the Gromov width of $M$ is defined to be $w(M)=sup\{{\pi r^2| B^{2n}(r) \rightarrow M}\}$ My question is: what is the explicit value of $w(D^*S^n)$, ...
1
vote
0answers
98 views

Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$

I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...
1
vote
0answers
74 views

Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold. Suppose $Y$ is a conifold transition on a single nullhomologous Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to ...
0
votes
0answers
68 views

computation of floer homology of cotangent bundle of spheres

I am wondering the computation of the floer homology of cotangent bundle of spheres. By a theorem of Viterbo, it is isomorphic to the homology of free loop space of sphere. However, I am wondering ...
4
votes
1answer
238 views

Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...
4
votes
0answers
98 views

Existence of a torus fibration with given vanishing cycles

Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be ...
0
votes
1answer
98 views

symplectic homotopy equivalence

Let $(M,\omega)$ and $(M^{\prime},\omega^{\prime})$ are two symplectic manifolds. Then we may define a natural homotopy equivalence as follows. We say that the smooth map $f:M\longrightarrow ...
1
vote
0answers
133 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: ...
0
votes
0answers
15 views

Condition for local Lipschitzness of pullback map for exterior forms

Given $w\in\Lambda^k(\mathbb{R}^n)$, determine the condition under which the map $T\rightarrow T^*(w)$ is locally Lipschitz, where $T\in GL_n(\mathbb{R})$ and $T^*(w)$ denotes the Pullback of $w$ by ...
1
vote
1answer
197 views

Symplectic and Holomorphic Vector Bundles

As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the ...
2
votes
0answers
82 views

Nature of separatrix in Fokker--Planck Hamiltonian with two degrees of freedom

Background The semiclassical (weak noise, small $D$) limit of the Fokker--Planck equation $$\frac{\partial P}{\partial t}=D\frac{\partial^2 P}{\partial x^2}-\frac{\partial}{\partial x}(v(x) P)$$ can ...
2
votes
1answer
317 views

Kodaira dimension of co-adjoint orbit

Let $G$ be a compact Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also ...
1
vote
0answers
162 views

Dropping the closed requirement from the symplectic manifold definition?

A symplectic manifold is a pair $(M,\omega)$, where $\omega$ is a non-degenerate closed two-form. When $M$ is compact, Hodge decomposition implies that such manifolds have non-zero second ...
0
votes
0answers
85 views

Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds. My question is: Is the group of symplectic automorphisms of $S^2$ with respect to this ...