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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).

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For example, in Corollary 4.7 in "Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics" 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.

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