Deformation stability of p-SKT and p-HS manifolds

Abstract

We study the notions of p-Hermitian-symplectic and p-pluriclosed compact complex manifolds, which are defined 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 notions are equivalent on $$\partial \overline{\partial }$$ -manifolds and go on to prove that in (smooth) complex analytic families of $$\partial \overline{\partial }$$ -manifolds, the properties of being p-Hermitian-symplectic and p-pluriclosed are deformation-open. Concerning closedness results, we prove that the cones , resp. , of Aeppli cohomology classes of strictly weakly positive (p, p)-forms $$\Omega $$ 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 $$\partial \overline{\partial }$$ -type assumptions are made on the other fibres.