Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My question is: is it true that Hodge numbers are symplectic invariants, i.e. depend only on $(X,\omega)$ and not on the extra choice of a compatible complex structure $J$?
Remarks:
1) I assume that there exists $\omega$-compatible complex structures for $(X,\omega)$. My question makes sense only for these very special symplectic manifolds.
2) it is known that Hodge numbers are not topological or smooth invariants, see for example Diffeomorphic Kähler manifolds with different Hodge numbers
3) obviously, Hodge numbers are complex invariants: the Hodge numbers of the Kähler manifold $(X,J,\omega)$ only depend on the complex manifold $(X,J)$. My question is in the "orthogonal" direction, fixing $\omega$ and varying $J$.
4) Hodge numbers are constant in a smooth family of compact Kähler manifolds. So a first step could be:
Is there a symplectic manifold $(X,\omega)$ whose space of $\omega$-compatible complex structures is not connected?
(I vaguely remember having seen that the answer is yes but I am unable to find where). Of course, the space of almost complex structures compatible with a given symplectic structure is known to be connected (in fact contractible) but the difficulty comes from the integrability condition necessary to have complex structures.
