Deformed Aeppli cohomology: canonical deformations and jumping formulas

Abstract 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.