# Tagged Questions

Hamiltonian systems, symplectic flows, classical integrable systems

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votes

**0**answers

417 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics B,...

**0**

votes

**0**answers

134 views

### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

**6**

votes

**1**answer

544 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

**2**

votes

**1**answer

214 views

### Boundary conditions and the relationship between Hamiltonian and Lagrangian Floer theories

I am looking for a full account of the relationship between the various versions of Floer theory on a symplectic manifold $M$. If we take the usual Floer equation (Hamiltonian version)
\begin{equation*...

**0**

votes

**1**answer

192 views

### Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...

**20**

votes

**2**answers

775 views

### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...

**6**

votes

**1**answer

307 views

### Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H_2(X_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**3**

votes

**1**answer

236 views

### What are the indecomposable classes on a del-Pezzo surface?

Let $X_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to $8$).
Let $\beta \in H_2(X_k, \mathbb{Z}) $ be a homology class given by
$$ \beta := n L + m_1 E_1 + \ldots + m_k E_k $$
...

**4**

votes

**2**answers

217 views

### actual dimension of concrete moduli space of holomorphic curves vs its virtual dimension

I am looking at exercise 6.3.3 in Mcduff's and Salamon's book J-holomorphic curves and Symplectic topology, which basically gives an example of a moduli space whose actually dimension is greater than ...

**3**

votes

**1**answer

128 views

### neighborhood of symplectic surfaces

I want to know if there is a uniqueness (in any sense) theorem for the symplectic structure in a neighborhood of a symplectic surface in a four dimensional symplectic manifold. Or more generally for ...

**5**

votes

**1**answer

262 views

### Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-...

**4**

votes

**1**answer

332 views

### Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer theory",...

**2**

votes

**2**answers

199 views

### Hamiltonian group actions in the context of holomorphic line bundles

When studying Hamiltonian group actions, a very nice set up might be to take the following:
Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian $G$...

**10**

votes

**0**answers

138 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...

**0**

votes

**1**answer

115 views

### Lagrangian flow preserves symplectic form

Let $X$ be a configuration space and $L: TX \rightarrow \mathbb{R}$ a Lagrangian. Then I want to show that the Lagrangian flow $F^t(x(0),x'(0)) = (x(t),x'(t))$ preserves the symplectic form just like ...

**4**

votes

**1**answer

316 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?

**1**

vote

**1**answer

265 views

### symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?

**12**

votes

**2**answers

427 views

### Is there a relationship between Fourier transformations and cotangent spaces?

Maybe a trivial question but I can't seem to find it treated anywhere.
Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function $F:Q\to\mathbb{...

**0**

votes

**0**answers

78 views

### Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?

Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...

**4**

votes

**1**answer

124 views

### Singularities of the moduli stack of polarized hyperkahler varieties

Inspired by the recent question on singularities of the moduli stack of Calabi-Yau threefolds (Singularities of the moduli stack of Calabi-Yau threefolds) I'd like to ask the following question.
Is ...

**3**

votes

**0**answers

195 views

### Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family
of degree $2$ maps defined (for $t$ small and non zero) by
$$u_t([X,Y]) := [X^2, t Y^2, XY].$$
Note that as $t$ goes to zero, $u_t$...

**4**

votes

**0**answers

109 views

### Generation of compact Lagrangians over fields with characteristic 2

Let $\pi:X\rightarrow\mathbb{C}$ be a Lefschetz fibration, and assume that the fibers are exact or monotone. A classical result of Seidel says that all the closed weakly unobstructed Lagrangian ...

**2**

votes

**0**answers

107 views

### Is there an action functional for the s-dependent Floer equation?

The usual Floer equation (in local coordinates)
\begin{equation*}
\partial_su+J(t,u)(\partial_tu-X_{H_t}(u))=0
\end{equation*}
is derived as the gradient flow of the symplectic action functional $\...

**4**

votes

**2**answers

238 views

### Momentum a cotangent vector

Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources ...

**3**

votes

**1**answer

122 views

### Reference of $\hbar$-differential operator from symplectic geometry perspective

I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian ...

**3**

votes

**1**answer

352 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 \omega(\dot{x}(t)-X_H,Y)...

**2**

votes

**0**answers

201 views

### Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p,q+...

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votes

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32 views

### When does a symmetric Poisson manifold decompose into homogeneous pieces?

When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct ...

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votes

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272 views

### Schoenflies and symplectic topology

The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ...

**6**

votes

**1**answer

307 views

### Symplectic form/Kahler metric on a toric manifold

I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form $...

**1**

vote

**1**answer

73 views

### Principal bundle associated to a Courant algebroid

A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition ...

**1**

vote

**1**answer

175 views

### Contact structures on circle cross plane

Can anyone provide an explicit contactomorphism between the following two contact structures on the circle cross the plane?
1) The standard contact structure on threespace, but with the line that ...

**1**

vote

**1**answer

156 views

### Symplectic reduction: from indefinite signature to Riemannian signature

Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...

**3**

votes

**0**answers

238 views

### Complex symplectic reduction

Oddly I find about zero resources talking about "complex symplectic reduction" upon a web search. Is there anything wrong with it?
I guess maybe there are two competing settings a priori: a complex ...

**0**

votes

**1**answer

95 views

### Is the group $Ham(M,\omega)\cap Iso_{0}(M,g)$ compact?

let $(M,J,g,\omega)$ be a compact K\"ahler manifold of complex dimension at least $2$. As usual $J$ is the complex structure, $\omega$ is the symplectic form, $g$ is the Riemannian metric and
$$\...

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vote

**2**answers

451 views

### Semistability in GIT

If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose ...

**2**

votes

**1**answer

172 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

**1**answer

255 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

**2**answers

369 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 ...

**2**

votes

**1**answer

362 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 ...

**3**

votes

**3**answers

243 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 ...

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votes

**0**answers

408 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**

votes

**1**answer

509 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 ...

**9**

votes

**1**answer

408 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 ...

**11**

votes

**1**answer

334 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 ...

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vote

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213 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 ...

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votes

**1**answer

116 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 $a_i$...

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votes

**0**answers

292 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 \chi^{\infty}(M)\...

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vote

**0**answers

128 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 ...

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vote

**1**answer

233 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 $\mathcal{G}$ ...