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I think a mainstream answer would be that symplectic geometry has two (seemingly opposiate, but actually related) aspects: rigidity and flexibility, it is the rigidity aspect that makes symplectic geo …

Lagrangian submanifolds (not necessarily closed) always exist locally in $(M,\omega)$, as you can easily see from Darboux coordinate theorem, so its existence in general is quite trivial. The non-triv …

A perfect example is Abouzaid-Smith's classification of genus 2 Lagrangian surfaces in $(T^4,\omega_{std})$. In this paper (http://arxiv.org/pdf/0903.3065v2.pdf), they proved that any Lagrangian genus …

Let's say $X$ is an exact symplectic manifold, and $\pi:X\rightarrow\mathbb{C}$ is a Lefschetz fibration whose critical values are $z_1,\cdot\cdot\cdot,z_k\in\mathbb{C}$. Let $\gamma_1,\cdot\cdot\cdot …

A partial answer is as follows. In order to make use of the theory of holomorphic curves (with boundaries or asymptotics, whatever), we should restrict ourselves to symplectic manifolds with convex b …

For compact toric manifolds, Floer cohomology of non-displaceable Lagrangians can be detected by their superpotentials. This is in some sense the $\mathfrak{m}_0$ term in the $A_\infty$ structure whic …

I'd like to mention the work of Castaño-Bernard-Matessi-Solomon, who proved the existence of an anti-symplectic involution for symplectic manifolds carrying a Lagrangian torus fibration of a certain c …

If you take away $D$, then $X\setminus D$ is a non-compact complex manifold, so $\partial\bar{\partial}$-lemma in general does not hold in this case. However, by the work of Bott-Chern in 1965, for an …

You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: https://link.springer.com/article/10.1007/s10240-016-0081-9. Basically, they studied the case when $X$ is $(\mathbb{C}^\ …

The geometric interpretation is quite simple: there is a unique torus fiber $L\subset M$ of the moment map $\mu:M\rightarrow\Delta_M$ which is monotone, and this fiber lies over the unique integral po …

In general, if $X$ is a compact smooth $n$-dimensional Calabi-Yau manifold, and $D\subset X$ is an ample (or numerically effective) divisor, then the mirror of $X$ is usually a degeneration of the mir …

It depends on the positions of the points that you blow up. If you blow up respectively $p,q$ and $r$ points on the 3 irreducible components of the toric divisor $D\subset\mathbb{CP}^2$, with $p,q,r\g …

These are all standard TQFT structures which reflect the combinatorics of Riemann surfaces. In general, one starts with a compact Riemann surface (possibly with boundary $\bar{S}$ and a finite set of …

There is a Morse-Bott spectral sequence computing the symplectic cohomology of affine varieties which are complements of normal crossing divisors in smooth projective varieties. For simplicity, let's …

First, the closed-open string map is expected to be an isomorphism between BV algebras. In the case when $X$ is a Weinstein manifold, it is known to be an isomorphism of BV algebras, just make sure th …