# Questions tagged [lagrangian-submanifolds]

The lagrangian-submanifolds tag has no usage guidance.

58
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### Symplectomorphism and Hamiltonian isotopy

I would like to ask whether a symplectomorphism of a given symplectic manifold respects Hamiltonian isotopy classes of Lagrangian submanifolds. In other words, given two Hamiltonian isotopic ...

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### Understanding the Lagrangian Lift of a given isotopy

I was looking at the paper "Sheaf quantization of Hamiltonian isotopies
and applications to non-displaceability
problems" by Guillermou-Kashiwara-Schapira(GKS). So firstly, GKS doesn't ...

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155
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### Neighborhood theorem for conical Lagrangian

Let $(M,\omega)$ be a compact $2n$ dimensional symplectic manifold and $T$ be a compact smooth $(n-1)$ dimensional manifold.
Let $CT$ be the cone over $T$, i.e. $CT = T\times [0,1] / \sim $ where $\...

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### Are all exact Lagrangian spheres, vanishing cycles?

Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F_\theta = \pi^{-1}(\theta)$ over that ...

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### Can Lagrangian fibrations have multiple fibres in codimension $1$?

I know that if $\pi: S \to \mathbb P^1$ is an elliptic fibration of a K3-surface $S$, then $\pi$ does not have multiple fibers. A proof of this can be found in Huybrechts' Lectures on K3 surfaces, ...

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### Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$

Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...

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### Composition of coproduct and product in Lagrangian Floer (co)homology

Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...

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### 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 ...

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### 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/...

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### 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 ...

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### 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 ...

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### 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, ...

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345
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### 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 ...

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342
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### 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 ...

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543
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### 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 $L$...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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{...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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528
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### 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, ...

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### 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 ...

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153
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### 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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### 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(\...

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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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 ...

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### 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 $...

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### 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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### 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 ...

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### 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 ...

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### 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^{...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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), $W^\ast = Ann(X)$ ...

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### 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 ...