  Let $(M,J)$ be a $n$-dimensional complex manifold: a $p$-K\"ahler structure
(resp. $p$-symplectic structure) on $M$ is a real, closed $(p,p)$-transverse
form $\Omega$ (resp. real, closed $2p$-form whose $(p,p)$-component is
transverse). We give obstructions to the existence of such structures on
compact complex manifolds. We provide several families of compact complex
manifolds which admit both $(n-1)$-symplectic structures and special Hermitian
metrics.
