  Given a $\partial\bar\partial$-manifold $X$ with trivial canonical bundle and
carrying a metric $\omega$ such that $\partial\bar\partial\omega=0$, we
introduce the concept of small deformations of $X$ polarised by the Aeppli
cohomology class $[\omega]_A$ of an SKT metric $\omega$. There is a
correspondence between the manifolds polarised by $[\omega]_A$ in the Kuranishi
family of $X$ and the Bott-Chern classes that are primitive in a sense that we
define. We also investigate the existence of a primitive element in an
arbitrary Bott-Chern primitive class and compare the metrics on the base space
of the subfamily of manifolds polarised by $[\omega]_A$ within the Kuranishi
family.
