# Questions tagged [lagrangian-submanifolds]

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53
questions

**4**

votes

**1**answer

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### Why the Euler characteristics of a compact connected lagrangian submanifold of $\mathbb{R}^4$ is zero?

Let's consider space $\mathbb{R}^4$ with the standard symplectic structure and let $L\subseteq \mathbb{R}^4$ be a compact connected embedded submanifold. There is a fact that if $L$ is lagrangian ...

**7**

votes

**2**answers

219 views

### How to find equations of a sub-Riemannian problem

I am working on sub-Riemannian geometry and try to understand what are the tools to find the equations of a sub-Riemannian problem. Here is an example:
Let us consider the system defined by a ...

**1**

vote

**0**answers

117 views

### Books and References on Geometry of Submanifold [closed]

In this semester I want to study Geometry of Submanifolds. I know Chen Bang Yen's book: Geometry of submanifolds, but it is too hard to read since its strange print. Can people recommend textbooks and/...

**3**

votes

**1**answer

138 views

### Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds

I started reading the book "Lagrangian intersection Floer theory anomaly and obstruction", and there are a couple of details and assumptions in the definition of the Novikov covering that I ...

**1**

vote

**0**answers

58 views

### Action functional for the definition of Lagrangian Floer homology

I have been starting to learn about Lagrangian Floer homology using notes by A. Pedroza (arXiv link).
Consider $(M,\omega)$ a symplectic manifold that is symplectic aspherical and $L_0,L_1$ two ...

**3**

votes

**0**answers

106 views

### What is the significance of a Lagrangian Submanifold and what are the implications of the symplectic form being zero?

I'd like to understand better the relevance of Lagrangian submanifolds in Hamiltonian Mechanics. A Lagrangian Manifold is defined as a submanifold of a symplectic manifold upon which the restriction ...

**2**

votes

**0**answers

45 views

### Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...

**2**

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**0**answers

147 views

### Two possible meanings of “totally real” submanifold

It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent ...

**3**

votes

**1**answer

272 views

### Viterbo restriction map surjective on Weinstein neighbourhood

In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^*(i): SH^*(M)\rightarrow SH^*(N).$$
In ...

**7**

votes

**1**answer

301 views

### Lagrangian intersection Floer homology: understanding some assumptions

Let $(X,\omega)$ be a symplectic manifold and $L\subset X$ be a Lagrangian subspace.
Let $\mu_L:H_2(X,L;\mathbb{Z})\to \mathbb{Z}$ be the Maslov index
homomorphism.
Usual hypothesis
Recall that $L$...

**10**

votes

**3**answers

1k views

### Why are Lagrangian subspaces in a symplectic vector space interesting?

A subspace in a symplectic vector space could be one of two extremes: either symplectic (meaning the form is nondegenerate there) or Lagrangian. Or it could be something between the two, meaning a ...

**6**

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119 views

### A clarification in the definition of Seidel's absolute Maslov index for a pair of transverse Lagrangians

I'm reading Seidel's paper Graded Lagrangian submanifolds where he introduces the absolute Maslov index of a pair of graded lagrangians as follows:
Let $\mathcal{L}(V,\beta)$ be the Lagrangian ...

**4**

votes

**0**answers

139 views

### Lagrangian subgroup of a nonabelian Lie group

My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...

**8**

votes

**0**answers

219 views

### Lagrangian subgroups/submanifolds, 2d topological boundary and 3d “non-abelian” Chern–Simons theory

This post is meant to ask for proper references to fill a gap in the literature.
My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...

**3**

votes

**0**answers

57 views

### Reference Request: Central Curvature “Fix”

Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...

**5**

votes

**0**answers

87 views

### GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question
GSO (...

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184 views

### GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...

**1**

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**0**answers

127 views

### how to understand the manifold with boundary jet bundle and cotangent bundle with boundary

Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type ...

**4**

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110 views

### Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?
The main example for this question ...

**3**

votes

**0**answers

90 views

### Symplectic displacement energy for several intersection points?

Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as
$$
e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap ...

**13**

votes

**2**answers

370 views

### Symplectic mapping class group and the “Lagrangian sphere complex”

For a genus $g$ surface $\Sigma_g$, the mapping class group $\mathrm{Mod}(\Sigma_g)$ acts on the curve complex $\mathcal C(\Sigma_g)$: vertices being isotopy classes of essential, nonseparating, ...

**4**

votes

**0**answers

88 views

### Lagrangian embeddings in prequantizable symplectic manifolds

I'm looking for a reference for a special type of Lagrangian embedding in a prequantizable symplectic manifold.
The setting is a symplectic manifold $(M, \omega)$, whose symplectic form is the ...

**4**

votes

**0**answers

139 views

### Lagrangian homology classes in compact symplectic manifolds?

Let $X$ be a compact symplectic $2n$-fold. Which classes in $H_{n}(X, \mathbb{Z})$ can be realized by embedded (or immersed, if that matters) Lagrangian submanifolds?
My question is motivated by ...

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votes

**0**answers

147 views

### Is every monotone Lagrangian Hamiltonian isotopic to minimal Lagrangian?

Assume we have a closed Lagrangian submanifold $L$ in Kaehler-Einstein manifold of positive scalar curvature (for instance, complex projective space). Dazord has proved that 1-form $\alpha=\omega(\...

**3**

votes

**0**answers

110 views

### Excessive Lagrangian intersection?

Assume we have a monotone Lagrangian submanifold $L$ in a 'good' symplectic manifold $X$ so that Floer homology can be defined (I am interested in $\mathbb{C}P^n$). Then $L$ and its image under ...

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**0**answers

138 views

### McLean theorem for Fano varieties?

Well-known McLean theorem states that deformations of special Lagrangian $L$ submanifolds in Calabi-Yau manifold are unobstructed and in bijection with harmonic 1-forms on $L$. The proof relies on the ...

**3**

votes

**2**answers

207 views

### When is mean curvature flow a Hamiltonian isotopy?

Assume we have a compact immersed Lagrangian $L$ in a Kaehler manifold $X$. Recall that a normal vector field $v \in \Gamma(L, N)$ is called Hamiltonian iff $\omega(v, \bullet)$ is an exact 1-form. My ...

**4**

votes

**1**answer

255 views

### How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?

When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...

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167 views

### Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?

In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...

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votes

**1**answer

563 views

### Why is every Hamiltonian system locally integrable?

It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional ...

**2**

votes

**1**answer

197 views

### Irreducibility of holomorphic symplectic quotients

Let a connected algebraic group $G$ (over $\mathbb C$, say) act Hamiltonianly on an algebraic symplectic variety $M$, with moment map $\Phi: M\to \mathfrak g^*$. In the example I care about, vaguely ...

**22**

votes

**1**answer

1k views

### Why are Lagrangian submanifolds called Lagrangian?

Much of the terminology in symplectic geometry comes from classical mechanics: the symplectic manifold is modeled on a cotangent bundle $T^*N$ of some configuration space $N$ with local position ...

**14**

votes

**1**answer

573 views

### What is the motivation behind the characteristic variety of a D-module and what does it's geometry tell me about the D-module?

Given a smooth algebraic variety $X$, and an $\mathcal{M}\in \text{Mod}(D_X)$, there is the characteristic variety of $\mathcal{M}$ defined as
$$
\text{Char}(\mathcal{M}):= V\left(\sqrt{Ann(\mathcal{M}...

**5**

votes

**1**answer

248 views

### In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context.
Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...

**4**

votes

**0**answers

105 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

245 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 $...

**2**

votes

**1**answer

403 views

### Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...

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118 views

### Shape of the bubbling limit of holomorphic discs

I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...

**3**

votes

**1**answer

233 views

### What's the geometric statement of this fibrewise integration on a symplectic manifold with Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic ...

**4**

votes

**1**answer

455 views

### Lagrangian fibration on Schoen's Calabi-Yau 3-fold

Schoen's Calabi-Yau 3-fold is the fiber product $X=Y_1\times_{\mathbb{P}^1}Y_2$ of two rational elliptic surfaces $Y_1\rightarrow\mathbb{P}^1$ and $Y_2\rightarrow\mathbb{P}^1$ with $\chi(X)=0$ and $h^{...

**4**

votes

**1**answer

568 views

### What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...

**2**

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**1**answer

308 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 (...

**8**

votes

**2**answers

676 views

### How many Lagrangian submanifolds?

An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are.
For each ...

**0**

votes

**2**answers

547 views

### about decomposition of three forms

Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...

**8**

votes

**1**answer

783 views

### For which Calabi-Yau threefolds is SYZ conjecture known to hold?

I would like to know for which Calabi-Yau threefolds SYZ conjecture is known to hold. I am aware of works by Gross-Wilson (Borcea-Voisin CY3s) and Ruan (quintic CY3), but they are quite classical ...

**2**

votes

**1**answer

217 views

### Image of an isotropic manifold under lagrangian correspondence is isotropic?

Is the following statement well known?
Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth)
Lagrangian correspondence. For a subset $X \subset M$ we denote $L(X):=(...

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votes

**4**answers

588 views

### Lagrangian Kleinian bottles

I remember some talks some time ago about proofs of nonexistence of Lagrangian Kleinian bottles in C^2 for the standard symplectic structure, mentioning that this were the only compact surface for ...

**1**

vote

**1**answer

368 views

### holomorphic sections on elliptic K3 surface

Hi all,
I want to ask something about the holomorphic sections on elliptic K3:
Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some ...

**1**

vote

**0**answers

209 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 \...

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votes

**1**answer

1k views

### Cohomology theory for symplectic manifolds

Suppose I have a symplectic manifold $(M,\omega)$ and a line bundle $\mathcal L$ with a connection with curvature $\omega$ (or perhaps it's more standard to say $\frac i{2\pi}\omega$; anyway, the ...