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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 …

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 …

This is almost never true in general, although it's obviously true for boundary connected sums. For example, it is usually the case that Weinstein handle attachment will kill (non-trivial) invertible …

Yes. Recently Auroux (jointly with Efimov and Katzarkov) has proposed a definition of the Fukaya category for trivalent configurations of rational curves. If $\Sigma_g$ is a genus $g$ Riemann surface …

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 …

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 …

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 …

Yes, it's possible for the specific case that you are looking at. For $k=1$, this is obvious. For $k\geq2$, such a Lefschetz fibration can be constructed by applying a standard construction to the sta …

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}^\ …

Let's assume that $M$ is an $n$-dimensional exact symplectic manifold and we are only interested in Fukaya categories of closed exact Lagrangian submanifolds. Then for any subcritical Weinstein manifo …

$S^3$ as a boundary cannot be both convex and concave. This is proved in the famous paper of Eliashberg-Gromov.

In the case when you are blowing up a smooth submanifold this actually coincides with the symplectic blow-up (with the symplectic form forgotten). For simplicity, let's take the submanifold to be a po …

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 …

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 …

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 …