I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two points $c$ and $d$, which are nodes for the (reducible) union of the two curves. One blows up $C$ at point $c$, and $D$ at point $d$. The resultant manifold is a compact 3-manifold that that contains two smooth rational curves $L$,$M$ lying over $c$ and $d$ such that $L+M$ is algebraically equivalent to $0$, and hence, the Hironaka example is not Kaehler.
I would like to know if the Hironaka example admits non-Hard Lefschetz symplectic structures. This seems quite likely, but my web searches so far have turned up no confirmation. Do any of you know of a specific paper providing examples of such symplectic structures?
I am a novice in algebraic geometry (I'm coming from a symplectic geometry background), so do you any of you know the best source to learn how to compute real de Rham cohomology of manifolds like Hironaka's example?