Deformed Aeppli cohomology: canonical deformations and jumping formulas

Given a complex analytic family of complex manifolds, we consider canonical Aeppli deformations of $(p,q)$-forms and study its relations to the varying of dimension of the deformed Aeppli cohomology $\dim H^{\bullet,\bullet}_{A\phi(t)}(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H^{\bullet,\bullet}_{A\phi(t)}(X)$. As a direct consequence, $\dim H^{p,q}_{A\phi(t)}(X)$ remains constant iff the Bott-Chern deformations of $(n-p,n-q)$-forms and the Aeppli deformations of $(n-p-1,n-q-1)$-forms are canonically unobstructed. Furthermore, the Bott-Chern/Aeppli deformations are shown to be unobstructed if some weak forms of $\partial\bar{\partial}$-lemma is satisfied.