7 votes

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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5 votes
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Choice of a family of almost complex structures when defining Floer Homology

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" ...
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3 votes
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Cotangent bundles to Riemannian manifolds and submanifolds

May I please call the submanifold $M$ instead of $\tilde N$? For a smooth manifold $N$, $T^\ast N$ does not just have a canonical symplectic form $\omega$. It also has a canonical $1$-form $\alpha$ ...
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3 votes
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Holomorphic/Symplectic embedding of Riemann surfaces

I am just posting my comment as one answer. Lemma. For integers $\ell < g$, for every continuous map from $\Sigma_\ell$ to $\Sigma_g$, the pullback map on $H^2$ is the zero map. Proof. The ...
1 vote
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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 ...
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