Co-polarised Deformations of Gauduchon Calabi-Yau $\mathrm{\partial }\bar{\mathrm{\partial }}$-Manifolds and Deformation of $p$-SKT $h$-$\mathrm{\partial }\bar{\mathrm{\partial }}$-Manifolds

The main result of this paper is to study the local deformations of Calabi-Yau $\mathrm{\partial }\bar{\mathrm{\partial }}$-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 $h$-$\mathrm{\partial }\bar{\mathrm{\partial }}$-property is deformation open by constructing and studying a new notion called $hp$-Hermitian symplectic ($hp$-HS) form.