This is partly inspired by answers to the question:
Question about Hodge number .
Is there a family of compact complex manifolds, where the general fibres are
Kähler, but for which $E_1$ degeneration of the Hodge to de Rham spectral sequence fails
at the special fibre? Or, even better, such that the special fibre has nonclosed
holomorphic forms?
I feel like I should know the answer, but somehow I don't. All
the examples I know where the spectral sequence doesn't degenerate are nilmanifolds*,
so they aren't even homotopic to Kähler manifolds by standard rational homotopy theoretic obstructions (e.g. they aren't formal).
Also the famous Hironaka example [Ann. Math 1962] won't work either, because
the special fibre is an algebraic variety, so the spectral sequence will degenerate
(by an argument that can found in Deligne [Théorème de Lefschetz…]).
Obviously, I haven't thought about this deeply enough, but perhaps someone else has**.
Footnotes
*I was bit sloppy yesterday, since the examples I have in mind include
solvmanifolds. However, there are still topological obstructions to these being Kähler
due to Nori and myself.
** From the answers, I gather that the work of Popovici suggests that
there may be no counterexample.
Best Answer
This is known, for projective (even Moishezon) manifolds as shown by Dan Popovici in his paper http://arxiv.org/abs/1003.3605 For general Kaehler manifold, this is conjectured. Popovici has proved that a property of "strong Gauduchon" is preserved in limits http://arxiv.org/abs/1009.5408 and (I think) there are no example of strong Gauduchon manifold without Hodge decomposition.