1
vote
0answers
111 views

Lagrangian complement in a symplectic vector bundle

A standard, folk result in symplectic geometry states that: in a symplectic vector bundle $(E,\pi,B,\omega)$, any lagrangian subbundle $L$ admits a lagrangian complement $L'$. Having to use this ...
3
votes
1answer
153 views

Special Lagrangians and fat

I am unable to find the MO comments about the first use of the phrase "fat slags" in an article. On page 26 of this we find "these correspond to thickenings of the corresponding special Lagrangian ...
2
votes
0answers
123 views

Geometric meaning of a certain form in almost-Kähler geometry

I have difficulties finding an appropriate reference for the following question: Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of ...
3
votes
1answer
123 views

Level sets of Hamiltonians of S^1 actions

Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of ...
0
votes
1answer
135 views

Reference on elements of finite order in principal congruence subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
1
vote
3answers
357 views

Reference on generators of subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...
1
vote
0answers
149 views

Question in the paper of Robert Bryant “Calibrated embeddings in the special Lagrangian and coassociative cases”

Hallo, I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...
8
votes
1answer
454 views

Known size invariant for Riemannian manifolds?

Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under ...
5
votes
0answers
249 views

Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal ...
1
vote
2answers
262 views

Analytic Lagrangian Submanifolds

Hallo, I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
5
votes
1answer
284 views

The Fukaya category of a simple singularity (reference request)

I have heard that for an ADE singularity $f$, $ D^b\mathrm{Fuk}(f) \simeq D^b(\mathrm{Rep}\\ Q)$ where $Q$ is the corresponding Dynkin quiver. (As one would hope, if $\mathrm{Fuk}$ is some kind of ...
31
votes
2answers
2k views

About a letter by Richard Palais of 1965.

In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of ...
2
votes
0answers
99 views

Concerning the classification of transversally integral affine structures on symplectic foliations $F$

Recently, in http://arxiv.org/pdf/1207.3655.pdf, the authors have determined that an element $c$ in $H^2(P, P_v)$ is the Chern class of a twisted isotropic realization $p$: $(M, ...
1
vote
0answers
245 views

Transversality and isolated degenerate critical points

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ ...
5
votes
3answers
313 views

Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator

Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
1
vote
0answers
176 views

Co-normal bundle of orthogonal compliment

Is the following fact well known? Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X ...
2
votes
1answer
288 views

Non-Abelian Duistermaat-Heckman Measure (not just a reference request)

Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of ...
17
votes
3answers
670 views

Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf I would like to find a reference for a beautiful ...
2
votes
0answers
250 views

$A_{infty}$-categories and mirror symmetry.

I have been looking for an electronic version of Kontsevchi's talk in the Arbeitstagung at 1993. In Kontsevich's publications webpage, I found this reference preprint MPI/93-57, but there is no link ...
2
votes
1answer
247 views

On the Complete integrability of a tangent distribution

Reading about the geometrical theory of systems of first order pdes, I have met a result from symplectic geometry, that is easy to prove, but I am unable to give a reference for it. So my question ...
9
votes
1answer
686 views

What is the “symplectic duality” between holomorphic symplectic manifolds? Where can I read more about it?

I'm recently working on something called 3d mirror symmetry in QFT literature, which involves two hyperkähler manifolds. There seems to be a corresponding(?) mathematical theory called symplectic ...
6
votes
0answers
384 views

How to prove that a certain action is hamiltonian?

I posted this question here on math.stackexchange.com. I have not had an answer, and I thought it could be more appropriate here. (Please, If you judge this my opinion is wrong, then I will delete ...
9
votes
0answers
625 views

Kontsevich's lecture at Rutgers (1996)

Reading some papers about (homological) mirror symmetry I have found the reference to the unpublished Kontsevich's lecture at Rutgers University (Nov 11, 1996). I would like to know if someone has ...
13
votes
3answers
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 ...
11
votes
0answers
596 views

Compact Symplectic Fano (strongly monotone) manfiolds

What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry? We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that ...