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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 …

You must restrict to the case of blowing up at a fixed point of the torus action, otherwise the manifold is no longer toric, and the remainings are nonsense. Naively one just replaces the corner with …

The answer is trivially no, even in the topological sense. For example, you consider $\mathbb{CP}^2$, the toric fibration $f_1:\mathbb{CP}^2\rightarrow B_1$ is of course almost-toric, since only ellip …

I'd like to add two more examples. The first one is in the negative direction. Let $\pi:E^{2n}\rightarrow\mathbb{C}$ be an exact Lefschetz fibration, by the work of Seidel one can associate its Fukay …

If the complex structure $J$ compatible with $\omega$ is integrable for $X$, then of course you have $\partial\bar{\partial}$ lemma. A natural generalization of the $\partial\bar{\partial}$ lemma in s …

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

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …