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 ∂∂-manifolds and go on to prove that in (smooth) complex analytic families of ∂∂-manifolds, they are deformation open. Concerning closedness results, we prove that the cones A p , resp. C p , of Aeppli cohomology classes of strictly weakly positive (p, p)-forms Ω 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 ∂∂-type assumptions are made on the other fibres.
The main result of this paper is to study the local deformations of Calabi-Yau ∂ ∂-manifold that are co-polarised by the Gauduchon metric by considering the subfamily of co-polarised fibres by the class of Aeppli/De Rham-Gauduchon cohomology of Gauduchon metric given at the beginning on the central fibre. In the latter part, we prove that the p-SKT h-∂ ∂-property is deformation open by constructing and studying a new notion called hp-Hermitian symplectic (hp-HS) form.
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