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 $\dim H_{A\phi (t)}^{\bullet ,\bullet }(X)$. In particular, we prove the jumping formula for the deformed Aeppli cohomology $H_{A\phi (t)}^{\bullet ,\bullet }(X)$. As a direct consequence, $\dim H_{A\phi (t)}^{p,q}(X)$ 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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma is satisfied.