A symplectic manifold is a pair $(M,\omega)$, where $\omega$ is a non-degenerate closed two-form. When $M$ is compact, Hodge decomposition implies that such manifolds have non-zero second betti-number, and one sometimes analogues of the Hard Lefschetz Theorem.
What happens if one drops the assumption that $\omega$ is closed. For sure $\omega$ will not give a class, but maybe something interesting will happen. What are examples of such spaces.