Hamiltonian systems, symplectic flows, classical integrable systems

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0
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1answer
40 views

Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$. Assuming I can find ...
2
votes
1answer
73 views

First Chern class and second Chern class in Quantizable Kaehler manifolds

Assume that $(X,\omega)$ is a K\"ahler manifold and $L\to X$ be a pre-quantum line bundle, then is there any relation between first Chern class and second chern class?
4
votes
1answer
146 views

A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following: Let $G=K_{\Bbb C}$ be a ...
1
vote
0answers
89 views

Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth ...
8
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0answers
254 views

Why do we study symplectic geometry? [on hold]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form? The subject certainly originated from physics, but is there a deeper reason for why it is still an ...
1
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0answers
90 views

A question about canonical bundle of moduli space of Kahler Einstein metrics

Let $\mathcal M$ be a moduli space of Kahler-Einstein metrics with negative Ricci curvatures on pairs $(X,D)$. Is the canonical bundle of $\mathcal M$ nef? Motivation: If we know the nefness ...
3
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0answers
189 views

Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact ...
5
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0answers
125 views

Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$. My ...
15
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1answer
814 views
4
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1answer
151 views

How to understand Taubes' moduli space of holomorphic curves?

Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is ...
2
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0answers
47 views

Order of metaplectic operator

I have a weak background on this subject. Suppose $S$ be a $2m \times 2m$ symplectic matrix of order $n$. Suppose $W_S$ be the corresponding metaplectic operator on $\mathcal{S}(\mathbb{R}^m)$, the ...
2
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0answers
59 views

Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...
2
votes
1answer
75 views

multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$. That means: 1) There exists an ...
3
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0answers
54 views

Principal orbits for hamiltonian actions

Let $G$ be a compact Lie group which acts by symplectomorphisms on a symplectic manifold $(M,\omega)$. Futhermore let $\mu \colon M \to \mathfrak g$ be a moment map for this action. Denote by $\Omega ...
6
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0answers
399 views

When a Spherical variety is $K$-stable

Let $X=G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then we call $X$ spherical. My question is When a Spherical variety is $K$-stable? Is ...
7
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3answers
2k views

Almost Complex Structures: 'Tame' versus 'Compatible'

Consider a symplectic manifold $(M,\omega)$ and the space of almost complex structures. These are $J:TM\to TM$ with $J^2=-\text{id}$. A given $J$ is $\omega$-tame when $\omega(v,Jv)>0$, and $J$ is ...
5
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0answers
76 views

Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...
3
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0answers
251 views

Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that $$\frac{\partial \omega(t)}{\partial ...
4
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1answer
266 views

AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?
4
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0answers
166 views

Kähler quotients of affine varieties and GIT

Let $X\subseteq \Bbb C^n$ be a smooth affine variety and $G=K_{\Bbb C}$ a complex reductive group acting linearly on $\Bbb C^n$ preserving $X$ (where $K$ is a maximal compact subgroup of $G$). Suppose ...
4
votes
1answer
139 views

Deformation long exact sequence of GW theory in the analytical setting

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume ...
2
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1answer
144 views

Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow $$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$ starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...
5
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1answer
68 views

Is the Hofer topology second countable?

Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}^c(M,\omega)$ denote the group of compactly supported Hamiltonian diffeomorphisms of $(M,\omega)$. Is the Hofer topology on ...
3
votes
1answer
82 views

coisotropic action on $TS^{2n+1}$

Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of ...
6
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0answers
79 views

Fubini-Study form on weighted projective spaces

As it is known, $\mathbb CP^n$ with Fubini-Study symplectic form can be get by the symplectic reduction of $\mathbb C^{n+1}$ with a symplectic form $\sum_{i=0}^n dz_i\wedge d\bar z_i$ by a hamiltonian ...
3
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0answers
63 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
5
votes
0answers
88 views

$H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the ...
2
votes
1answer
303 views

Show that the symplectic action 1-form on loop space is closed

I am struggling to see how the symplectic action 1-form $\alpha_H$ on the loop space of a symplectic manifold $(M,\omega)$ is closed. It is defined by $\alpha_H(Y) = \int_0^1 ...
1
vote
1answer
67 views

Lagrangian foliation

Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f_1$, so we find $n = \frac{1}{2}\dim M$ functions ...
7
votes
1answer
294 views

Different complexifications of a real analytic Riemannian manifold

I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero ...
1
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0answers
66 views

Vanishing first Chern class on fibers and Symplectic reduction

Suppose that a Lie group $G$ acts on compact Kahler manifolds $M$ and $N$ via symplectomorphisms take $\eta\in \mathfrak g^*$. and let $M_\eta$ and $N_\eta$ are symplectic quotients of $M$ and $N$ ...
5
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0answers
84 views

The autonomous diameter of the group of Hamiltonian diffeomorphisms of the standard symplectic space

The autonomous norm of a Hamiltonian diffeomorphism $h$ of a symplectic manifold $(M,\omega)$ is the smallest number $n\in \mathbf N$ such that $h=a_1\dots a_n$, where $a_i$ are autonomous ...
1
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0answers
109 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
4
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1answer
197 views

Understanding “Decategorified” symplectic Khovanov homology

In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
1
vote
1answer
189 views

What are the finite subgroups of $\operatorname{Sp}_{2n}(\mathbb{Z})$?

I've read the following question: Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request) and it made me wonder. It's easy to see that ...
3
votes
0answers
91 views

Moduli space of null Sasaki $η$-Einstein structures for higher dimensions(Calabi-Yau structures in Sasakian setting)

The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric $$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such ...
1
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0answers
49 views

Momentum Map on cotangentbundle as submersion

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
8
votes
1answer
120 views

Example where Calabi invariant is nontrivial?

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of ...
50
votes
4answers
4k 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 ...
3
votes
0answers
56 views

What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic ...
2
votes
1answer
292 views

A conjecture from Jean Varouchas on Kahler varieties

Conjecture: Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces. If $X$ is Kahler, is $X'$ Kahler? This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from ...
4
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0answers
47 views

Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a ...
13
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4answers
2k views

What is the role of contact geometry in the hamiltonian mechanics?

Let us assume someone is interested in the study of Hamiltonian mechanics. What are good examples to illustrate him of the usefulness of contact geometry in this context? On one hand the Hamiltonian ...
1
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0answers
26 views

Translating between complex blow-up and symplectic cut

Symplectic cutting is a way of describing a symplectic form on the blow-up $\tilde M$ of a complex manifold $M$, if we have a symplectic form on $M$ (that is compatible with the complex structure). ...
3
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0answers
54 views

Interpolating from a Hard Lefschetz class to a Kaehler class

Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures. There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...
2
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0answers
117 views

Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$? Think of $\Bbb R ...
0
votes
1answer
38 views

Closing the commutative diagram for symplectic realizations

Let $f: (M_1, P_1) \to (M_2, P_2)$ be a Poisson map between Poisson manifolds. Let $\pi_i : (S_i, \omega_i) \to (M_i, P_i), \ i=1,2$ be symplectic realizations. Putting these objects in a rectangular ...
2
votes
0answers
154 views

Ricci curvature in resolution of singularities

Let $X$ and $X'$ are Kahler variety and $f: (X',\omega')\to (X,\omega)$ be the resolution of singularities of $X$ then from $K_X=f^*K_X'+E$ how can we find the relation between $Ric(\omega)$ and ...
2
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0answers
128 views

Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...
3
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0answers
62 views

Using and understanding the Atiyah-Bott localization theorem/integration formula

I posted this on r/math, but was told I might have better success here given the level of the question. Basically, I need to learn how to use the localization theorem to compute integrals on ...