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The reason is related to orientations, and more specifically to bad orbits (see the end of paper :Coherent orientation in SFT). Assume that a holomorphic curve is asymptotic to a bad orbit ( means $ …

In Symplectic field theory ( Hofer-Eliashberg....) and considering moduli of J-holomorphic curves asymptotic to Reeb orbits at punctures (J-holomorphic curve into a symplectic cobordism), The authors …

Let me explain my reason: Let's $w$ be the symplectic structure on D and $\tilde{w}$ be the one on total space of line bundle . Then we have Gysin exact sequence above which gives $H^2(V)=H^2(D)/ \le …

Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi …

Consider a Symplectic manifold D (with $H^1(D)=0$) with symplectic form $w$. Let V be the total space of a circle bundle over D with non-trivial Euler class $e\in H^2(D)$. You may think of V as the se …

Also there are C.Y 3-folds with this property constructed via toric geometry (I think due to Batyrev)

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 situati …

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 origin …

Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the stand …

Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then $$ \psi_1 = \psi_{\frac …

I got this answer from Dusa McDuff (and she got it from some body else): Suppose given $f:[0,1]\to [0,1]$ such thqt 0 is repelling fixed point and 1 is attracting fixed point and there are no others. …

I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$). Of course, the result/conjecture is that the …

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$ Question: For which $g$, there is such …

I found the following paragraph in the paper " Intro to symplectic field theory " which I don't understand what does it mean precisely? Suppose W is a symplectic (or Kahler) manifold. D, smooth divis …

A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$. We usually like these curves because of their rigidity. But, Is there any t …