For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index theorem, from Voisin's book on Hodge theory and complex geometry).
I am still studying Voisin's proof, but it seems that the only aspect of Kaehler-ness that she uses is the Lefschetz decomposition at the level of cohomology. This begs the question: does there exist another useful formula for the signature of the intersection form on middle cohomology, for compact symplectic manifolds that are non-Kaehler, but satisfy the Lefschetz decomposition on cohomology?
Do any of you know of such a result? In seeking it out, where might I have the most luck in finding it (if it exists)? The arXiv has not been helpful.