Deformations of Kähler manifolds where Hodge decomposition fails?

Question

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.

Answer 1

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.

Answer 2

If any example exists, then the general fibre of the family cannot be projective.

In fact, Dan Popovici ["Limits of projective manifolds under holomorphic deformations", arXiv.09102032] recently proved the following

Theorem. Let $\pi \colon \mathcal{X} \to \Delta$ be a complex analytic family of compact complex manifolds such that the fibre $X_t:=\pi^{-1}(t)$ is projective for all $t \neq 0$. Then $X_0:=\pi^{-1}(0)$ is Moishezon.

Since Moishezon manifolds admit a projective algebraic modification, it follows that their Hodge-Frolicher spectral sequence degenerates at $E_1$. In particular, Hodge decomposition holds for $X_0$. Notice that in this case $X_0$ is Kähler if and only if it is projective.

Answer 3

Just as a comment.

In general, given a family of sG (i.e., strongly-Gauduchon) manifolds, the limit need not to be still sG: an example was provided by Ceballos, Otal, Ugarte, and Villacampa, http://arxiv.org/abs/1111.5873 (see also http://arxiv.org/abs/1210.0406). It may be possible that the sG property is preserved to the limit assuming stronger conditions on the fibres: however, the published version of the paper by Popovici does not prove this fact, as noticed also by YangMills.

On the other hand, an example of a family of (non-Kähler) manifolds satisfying the Hodge decomposition (namely, the $\partial\overline{\partial}$-Lemma) and whose limit does not, is studied in http://arxiv.org/abs/1305.6709.

Note that there are several examples of sG manifolds non-satisfying the Hodge decomposition: for example, every nilmanifold admitting a balanced or sG metric.