Hamiltonian systems, symplectic flows, classical integrable systems

**4**

votes

**0**answers

43 views

### Maslov class of a diagonal

Let $(M,\omega)$ be a symplectic manifold. Which condition on $M$ guarantees that the diagonal of $(M \times M, (\omega,-\omega))$ has a vanishing Maslov class? $H^1(M,\mathbb{Z})=0$ is enough, but I ...

**1**

vote

**1**answer

79 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 ...

**1**

vote

**0**answers

101 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 ...

**4**

votes

**1**answer

248 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 ...

**3**

votes

**0**answers

86 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?

**9**

votes

**0**answers

271 views

### Why do we study symplectic geometry? [closed]

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 ...

**5**

votes

**0**answers

129 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 ...

**4**

votes

**1**answer

152 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**

votes

**0**answers

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**

votes

**0**answers

72 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 ...

**3**

votes

**0**answers

56 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 ...

**9**

votes

**1**answer

123 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 ...

**2**

votes

**1**answer

76 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 ...

**1**

vote

**0**answers

91 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 ...

**4**

votes

**0**answers

171 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 ...

**5**

votes

**1**answer

69 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

**1**answer

84 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**

votes

**0**answers

80 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**

votes

**0**answers

190 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 ...

**3**

votes

**0**answers

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

**0**answers

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 ...

**1**

vote

**1**answer

70 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 ...

**5**

votes

**0**answers

86 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**

vote

**0**answers

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 $N$ ...

**4**

votes

**1**answer

143 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 ...

**1**

vote

**0**answers

110 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**

votes

**1**answer

203 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

**1**answer

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

**0**answers

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**

vote

**0**answers

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 $l\colon G \times G/K \to G/K$ ...

**2**

votes

**1**answer

145 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 $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...

**8**

votes

**1**answer

122 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 ...

**3**

votes

**0**answers

57 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 ...

**4**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

27 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**

votes

**0**answers

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**

votes

**0**answers

118 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 ...

**2**

votes

**0**answers

155 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 ...

**3**

votes

**0**answers

63 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 ...

**4**

votes

**1**answer

136 views

### Smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$ without a point admits symplectic structure?

See my previous question here.
Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?
This was answered in the ...

**0**

votes

**1**answer

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 ...

**3**

votes

**0**answers

87 views

### Symplectic Hodge Maps and Mirror Symmetry

The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau:
http://arxiv.org/abs/1011.1250
...

**3**

votes

**1**answer

88 views

### Characterization of Lagrangian planes in symplectic vector spaces over finite fields [closed]

EDIT: As L Spice pointed out, there is an error in the observation. The question is void therefore
Let $p$ be a prime and $q=p^r$. Let $V$ be a $\mathbb F_q$-vector space of dimension four, with a ...

**1**

vote

**0**answers

80 views

### Ohsawa-Takegoshi extension theorem along snc divisor

Let $f:X\setminus D\to Y$ be a holomorphic fibre space and $X,Y$ are Kahler manifolds, where $D$ is a snc divisor on $X$. Can Ohsawa-Takegoshi extension theorem, imply that there exists a holomorphic ...

**1**

vote

**0**answers

106 views

### horizontal lift along fibres

Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local
coordinate $(s_1,...,s_d)$ of
$Y$
and a local coordinate $(z_1,...,z_n)$ of ...

**0**

votes

**1**answer

64 views

### Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of ...

**2**

votes

**0**answers

84 views

### Pulled back foliation is completely integrable

There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm".
Assume $M$ is a symplectic ...

**15**

votes

**1**answer

819 views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?