On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-K\"ahler structures

Inspired by a recent work of D. Wei–S. Zhu on the extension of closed complex differential forms and Voisin’s usage of the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-K"ahler structures with the $\mathrm{\partial }\bar{\mathrm{\partial }}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{{\iota }_{\phi }}$. Furthermore, we prove the local stabilities of transversely $p$-K"ahler structures with mild $\mathrm{\partial }\bar{\mathrm{\partial }}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui–B. Gmira and P. Ra'zny on that of the transversely K"ahler foliations with homologically orientability. We observe that a transversely K"ahler foliation, even without homologically orientability, also satisfies the $\mathrm{\partial }\bar{\mathrm{\partial }}$-property. So even when $p=1$ (transversely K"ahler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott–Chern numbers with mild $\mathrm{\partial }\bar{\mathrm{\partial }}$-properties are also presented.