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2
votes
1answer
60 views

Hofer-Zehnder capacity of toric varieties

Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset \...
2
votes
0answers
61 views

Grading in Lagrangian Floer homology

What are the conditions on a symplectic manifold (M,w) and on a Lagrangian submanifold L so that Lagrangian Floer complex CF(L, f(L)) is Z-graded? Here f is a compactly supported Hamiltonian isotopy. ...
4
votes
1answer
163 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 ...
0
votes
0answers
20 views

Extending a symplectomorphism of a surface with boundary, properly embedded in a 4-disk, to the ambient space

Let $S$ be a surface with boundary that is properly embedded, via a map $f$ say, inside a $4$-disk. When is the embedding isotopic to the identity? -- All embeddings and isotopes are in the symplectic ...
3
votes
0answers
198 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 K\...
3
votes
0answers
71 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 ...
1
vote
0answers
73 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 $N$ ...
8
votes
1answer
124 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 $...
0
votes
0answers
101 views

Triangulation of S^2xS^2

Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ? Thanks,
0
votes
0answers
64 views

Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants before....
1
vote
0answers
46 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 ...
2
votes
0answers
105 views

holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks. We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...
6
votes
0answers
164 views

Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...
3
votes
0answers
179 views

What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion). Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...
4
votes
1answer
179 views

Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
4
votes
0answers
141 views

Maslov class as a relative cohomology class in H^2(M, L)

Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to ...
0
votes
1answer
259 views

Steps in paper on sympl. geometry unclear

I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé. It deals with the question how the stability properties of a ...
3
votes
1answer
176 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
3
votes
2answers
243 views

Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...
6
votes
1answer
174 views

Homology class of a Lagrangian Klein bottle

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...
3
votes
0answers
59 views

Flux group of surfaces with genus $g\ge2$

I have read in this paper by C. Campos-Apanco and Pedroza (at the end of page 2) that: When $(M,\omega)$ is a closed surface of genus greater than one, the flux group is trivial. Does anyone ...
6
votes
1answer
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 ...
1
vote
0answers
55 views

Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper http://de.arxiv.org/abs/1310.5080 by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...
0
votes
1answer
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 ...
0
votes
0answers
77 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
0answers
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
0answers
144 views

McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...
2
votes
0answers
106 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 $\...
3
votes
1answer
330 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)...
0
votes
1answer
253 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 ...
1
vote
0answers
136 views

Complex but not Symplectic

For every $n$ there exist a smooth manifold $M$ of $dim M = n$ that admits a complex structure but not a symplectic one?
1
vote
0answers
70 views

why is there such a 1-form on a planar open book?

Suppose $B$ is the binding (with more than one component) of a planar open book on a 3-manifold $Y$ and let $L\subset B$ be the complement of a single component of the binding. Now we perform page-...
1
vote
0answers
127 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 ...
2
votes
0answers
173 views

Fully faithful embedding of the exact Fukaya category

Let $\mathscr{F}(X)$ be the exact Fukaya category of an exact symplectic manifold $(X^{2n},\omega)$, i.e. the objects in $\mathscr{F}(X)$ are all closed exact Lagrangian submanifolds with Maslov index ...
2
votes
0answers
118 views

Examples of symplectic manifolds which are twisted $T^n$ bundles over $T^n$

I'm looking for certain (higher-dimensional) analogues of the Kodaira-Thurston manifolds, i.e. I want to know whether in $\dim_\mathbb{R}X\geq6$ we have examples of symplectic manifolds satisfying the ...
1
vote
0answers
90 views

Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold. Suppose $Y$ is a conifold transition on a single nullhomologous Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to ...
0
votes
2answers
389 views

When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?

My goal is to better understand the Maslov-index of pseudoholomorphic disks. For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic ...
0
votes
0answers
108 views

Connectedness of the symplectic automorphism of the 2-sphere $S^2$

The 2-sphere, endowed with the round Riemann metric with constant curvature 1, is a symplectic manifolds. My question is: Is the group of symplectic automorphisms of $S^2$ with respect to this ...
10
votes
6answers
943 views

When do you go hunting for Lagrangian submanifolds?

Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...
2
votes
0answers
146 views

Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres

Given a symplectic 6-manifold $(M,\omega)$ and an embedded symplectic 2-sphere $C\subset M$ whose normal bundle has the first Chern class -2. How to find on $M$ another closed 2-form $\eta$ which only ...
1
vote
0answers
75 views

Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$ acts holomorphically such that the group $K$ preserves the Kähler ...
1
vote
0answers
90 views

Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...
3
votes
0answers
115 views

The relative h-principle and extension problems

As a beginner for h-principles, I want to know why the relative h-principle cannot imply a positive solution to the problems for extending symplectic structures. The following is a relative h-...
3
votes
2answers
387 views

Reference Request: “Neck Stretching Procedure” (In Symplectic Field Theory)

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/...
4
votes
0answers
161 views

Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...
2
votes
1answer
197 views

On Lerman's description of symplectic cut

Assume $(X,\omega)$ is a compact real $2n$-dimensional symplectic manifold with a Hamiltonian torus action corresponding to the moment map $\mu:X\to \mathfrak{t}^*\cong \mathbb{R}^k$. In this ...
9
votes
2answers
312 views

The importance of differentiable dynamics from outside dynamics? (mainly topology)

I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics. This could potentially include ...
4
votes
0answers
167 views

What can be said about compact embedded exact Lagrangians in the generalized pair of pants?

What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation: $$ 1+\Sigma_i z_i = ...
2
votes
0answers
249 views

$C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...
1
vote
1answer
240 views

A question on existence of $Spin^c$-structure $P\to M$

Let $(M,\omega)$ be a compact symplectic manifold and the cohomology class $$[\omega]+\frac{1} {2}c_1(\wedge_{\mathbb C}^{0,n}(TM, J))\in H_{dR}^2(M)$$ is integral, for some almost complex structure $...