I would like to know if there are any known examples of families of complex manifolds for which the Frölicher spectral sequence of one fibre degenerates on the $E\_m$ page and the spectral sequence of a different fibre degenerates on the $E\_n$ page for $m \neq n$.


Since the spectral sequence degenerates on the $E\_1$ page for compact Kähler manifolds and since small deformations of compact Kähler manifolds are compact and Kähler, one would have to look at either non-compact manifolds or non-Kähler manifolds (or "large" deformations).



Does the degeneracy of the Frölicher spectral sequence vary in families?
For example, in Corollary 4.7 in "[Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics](http://arxiv.org/pdf/1111.5873.pdf)" by Manuel Ceballos, Antonio Otal, Luis Ugarte, Raquel Villacampa (JGA), a family of complex non-Kähler structures such that the Frölicher spectral sequence degenerates for any fibre except $X\_0$ is provided on the Iwasawa manifold. For other examples, see the references therein and other works by the authors.


