On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-Kähler structures

Abstract

Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial \bar{\partial }$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-Kähler structures with the $\partial \bar{\partial }$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{{\hat{I}}_{\ddot{I}}^1†}$. Furthermore, we prove the local stabilities of transversely $p$-Kähler structures with mild $\partial \bar{\partial }$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Raźny on that of the transversely Kähler foliations with homologically orientability. We observe that a transversely Kähler foliation, even without homologically orientability, also satisfies the $\partial \bar{\partial }$-property. So even when $p=1$ (transversely Kähler), 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 $\partial \bar{\partial }$-properties are also presented.

V3: Minor revison. Final Version, to appear in Annali di Matematica Pura ed Applicata (1923 -). Note that the final title is "...(basic) Hodge numbers...". V2: New Version. 50 pages. Particularly, Subsection 6.4 and Example 6.12 are new added. All comments are still welcome!