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For the mirror of $\mathbb{CP}^2$, a smooth fiber of $W$ is a 3-punctured torus, and you can compactify the fibers by filling in 1, 2 or 3 punctures. This corresponds, in $\mathbb{CP}^2$, to different …

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 …

Let me summarize and supplement my remarks above. A quaternion-Kahler manifold $X$ is a Riemannian manifold with holonomy $Sp(n)Sp(1)$, its definition of course includes hyperkahler manifolds as a spe …

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 …

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 …

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. One way outlined in the work of Abouzaid-Auroux-Katzarkov's work (http://arxiv.org/pdf/1205.0053.pdf) is to look at the space $\mathbb{P}^n\times\mathbb{C}^r$ blown up along the codimension 2 sub …

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 …

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 …

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}^\ …