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

Let $$X$$ be a complex manifold endowed with a holomorphic closed 2-form $$\omega$$ whose associated map $$\omega : TX \to T^*X$$ is invertible. Can we always embed $$X$$ as an open subset of a compact complex manifold $$Y$$ endowed with a holomorphic Poisson structure $$\pi : T^*Y \to TY$$ such that $$\pi|_X = \omega^{-1}$$? We may assume that $$X$$ is quasi-projective, or other reasonable assumptions, if that helps.

Note that this holds, for example, if $$X$$ is the cotangent bundle of a compact manifold.

Let $$Z$$ be the product of two curves of genus $$\geq 2$$. Choose a nonzero $$2$$-form $$\omega$$ on $$Z$$, the wedge of a nonzero one-form on each of the two curves. Let $$X$$ be obtained from $$Z$$ by removing the locus where $$\omega$$ vanishes. Then $$X$$ clearly has a nowhere vanishing (and thus invertible as $$\dim X=2$$) two-form.
Now any compactification $$Y$$ of $$X$$ is a compact complex manifold birational to the algebraic variety $$Z$$, thus is Moishezon, and because it's a surface, must be projective algebraic. Since $$Z$$ is a minimal surface, $$Y$$ is the blow-up of $$Z$$ at finitely many points. So if the Poisson structure on $$X$$ extends to $$Y$$, then it extends to $$Z$$ less finitely many points, hence to $$Z$$.
But $$Z$$ does not have a nontrivial Poisson structure since $$T Z \otimes TZ$$ is a sum of line bundles of negative degree and thus has no global sections.