When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Question

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset of these is also compatible with $\omega$, in that $\omega(\cdot, J\cdot \cdot)$ defines a Riemannian metric on the manifold. How do know, other than things like odd Betti numbers being even, if $\omega$ has an integrable member $J_{\omega, int}$ of $\mathcal{J}_{\omega}$, so that $(X,\omega, J_{\omega, int}, \omega(\cdot, J_{\omega, int}\cdot \cdot))$ is a Kaehler manifold?

Answer 1

There are LOTS of additional obstructions (for example, not every finitely presented group whose abelianization has even rank is a fundamental group of a Kaehler manifold). Most of the known obstructions are listed, for example, here.