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When does a holomorphic symplectic manifold compactify to a Poisson manifold?

No, not even under the nicest possible algebraicity assumptions like quasiprojective. Let $Z$ be the product of two curves of genus $\geq 2$. Choose a nonzero $2$-form $\omega$ on $Z$, the wedge of a ...
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Influence of symplectic invariants of the complement on being superheavy

Two things I am aware of: If $K^{c}$ is a finite disjoint union of stably-displaceable sets. Then $K$ is a stable stem. (This is explained in 1.2 in the paper you cited Entov and Polterovich - Rigid ...
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Generic choice of non-degenerate Hamiltonians $H$ in Floer theory

You can find a statement (and proof) of such a theorem in Hofer-Salamon's Floer homology and Novikov rings, where it appears as Theorem $3.1$. They require also that no holomorphic spheres with first ...
For a lot of things, you can work with a generic time-independent $J_0$ as you suggest; for instance, Audien & Damien work in this context in their book (so for most of the "fundamental" ...