3
votes
0answers
76 views

Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle. I ...
1
vote
0answers
110 views

Simply connected pencils

Is it true/false that for every complex projective variety $V$, there is a smooth simply-connected one parameter family $\pi:X\to \mathbb{P}^1$, one of whose smooth fibers is $V$? (i.e. we want ...
2
votes
0answers
105 views

Symplectic form on moduli space of connections

Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a ...
1
vote
1answer
172 views

Hamiltonian Isotopy class of Lagrangian Submanifold

Let $(X,\omega)$ be a symplectic manifold, $L\subset X$ be a Lagrangian submanifold, $[L]$ denotes the Hamiltonian isotopy class. How to represent $L'\in[L]$ via $L$ (for example, a graph over $L$)? ...
13
votes
0answers
287 views

Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...
11
votes
0answers
336 views

Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
6
votes
1answer
254 views

Lifting a Diffeomorphism to the Cotangent Bundle

Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving. ...
2
votes
1answer
162 views

How to find Darboux coordinates?

I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be ...
1
vote
1answer
198 views

Computation with the Legendre Transform

Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x_1,\dots x_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates ...
0
votes
0answers
111 views

Gromov-Floer compactness for C^0 convergence of complex structure/ C^1 convergence of Hamiltonian

Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of ...
2
votes
3answers
350 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
7
votes
1answer
447 views

Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know Let $(M,\omega)$ be a compact ...
1
vote
1answer
157 views

When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and $\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...
1
vote
0answers
174 views

Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$ I am looking for a proof for ...
1
vote
1answer
302 views

A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding $$\phi: (M,\omega)\to  (\mathbb CP^N, ...
3
votes
1answer
173 views

Displaceability of submanifolds

My question is motivated by the following question. How transitive are the actions of symplectomorphism groups ? A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there ...
0
votes
1answer
195 views

A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$ is integral, for some almost complex structure ...
3
votes
1answer
135 views

All symplectic manifolds have $Mp^c$-structures?

Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times_{\mathbb Z_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a ...
2
votes
0answers
183 views

Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact $$ ...
1
vote
1answer
148 views

Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...
4
votes
2answers
475 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 ...
4
votes
0answers
97 views

Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...
3
votes
0answers
76 views

DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...
3
votes
0answers
112 views

Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...
2
votes
1answer
250 views

Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits. A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold: $X$ is holomorphically convex, ...
7
votes
1answer
305 views

When a symplectic manifold is formal?

Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, then we have the symplectic Hodge operator $$*:\Omega^{k}(M)\rightarrow\Omega^{2n-k}(M)$$ Furthermore, we can define a differential ...
0
votes
0answers
158 views

Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
2
votes
1answer
173 views

Extending Reeb field from contact submanifold to ambient contact manifold

Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb ...
1
vote
1answer
268 views

A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P ...
2
votes
1answer
183 views

almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
1
vote
1answer
155 views

The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...
2
votes
2answers
174 views

Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration?

Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold ...
0
votes
1answer
136 views

model compact coisotropic submanifold

$\{x_1,\ldots,x_n,\xi_{m+1},\ldots,\xi_n\in\mathbb{R},\xi_1=\ldots=\xi_m=0\}$ is a "model" codimension $m$ coisotropic submanifold of $T^*\mathbb{R}^n$, and is of course noncompact. Is there such a ...
2
votes
2answers
651 views

Kenji Fukaya's Lecture series at Simons center

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry. ...
0
votes
0answers
78 views

about energy bound in Fukaya category

In Fukaya category, moduli spaces is defined, which are solutions of certain $C$-$R$ equations, which involve strip ends in boundary condition. When the number of strip ends $>2$, a curvature term ...
1
vote
2answers
192 views

coisotropic submanifolds

I'm thinking about coisotropic/involutive submanifolds of the symplectic phase space $T^*\mathbb{R}^n$ (with coordinates $x_1,\ldots,x_n,\xi_1,\ldots,\xi_n$). As I understand, the smallest ...
1
vote
0answers
188 views

quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement. Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...
1
vote
1answer
144 views

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of ...
3
votes
0answers
116 views

Classification of Compact Symplectic Homogeneous Spaces

Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
0
votes
0answers
111 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
3
votes
1answer
222 views

Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
1
vote
0answers
130 views

Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...
0
votes
0answers
142 views

Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...
5
votes
1answer
566 views

What is the geometric interpretation of this quantity?

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic structure on the tangent bundle, $(TM, ...
0
votes
1answer
159 views

An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
0
votes
0answers
94 views

Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...
1
vote
1answer
275 views

The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation. Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...
2
votes
1answer
83 views

An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar ...
-1
votes
1answer
142 views

$S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
5
votes
0answers
271 views

Laplacians associated to symplectic cohomologies

I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...