In this paper, we introduce the notions of $p$-Hermitian-symplectic and $p$-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer $p$ not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case $p=1$. We then notice that these two properties are equivalent on $\partial\bar{\partial}$-manifolds and go on to prove that in (smooth) complex analytic families of $\partial\bar{\partial}$-manifolds, they are deformation open. Concerning closedness results, we prove that the cones $\mathcal{A}_p$, resp. $\mathcal{C}_p$, of Aeppli cohomology classes of strictly weakly positive $(p,p)$-forms $��$ that are $p$-pluriclosed, resp. $p$-Hermitian-symplectic, must be equal on the limit fibre if they are equal on the other fibres and if some rather weak $\partial\bar{\partial}$-type assumptions are made on the other fibres.

16 pages