When we talk about the theory of variation of Hodge structures, we always assume that the central fiber is a Kähler manifold $X$, then consider a family of deformations $\pi:\mathcal X\to B$ and the period map $\mathcal P:B\to Grass(b^{p,k},H^k(X,\mathbb C))$, $b^{p,k}=dim F^pH^k(X,\mathbb C)$.


What if we replace the Kähler manifold by a $\partial\bar\partial$-manifold and consider a holomorphic family of $\partial\bar\partial$-manifolds $\pi:\mathcal X\to B$?


Recall a $\partial\bar\partial$-manifold is a compact complex manifold which satisfies for any $\partial$, $\bar\partial$-closed $d$ exact $(p,q)$ form $\alpha$, there this a $(p-1,q-1)$ form $\beta$ such that $\alpha=\partial\bar\partial \beta$.


Is the theory of variation of Hodge structures remains the same? And is there also a period map $\mathcal P:B\to Grass(b^{p,k},H^k(X,\mathbb C))$?
If so, is the period map holomorphic and the Griffiths transversality still holds?


In my opinion, first we should define what $F^pH^k(X,\mathbb C)$ means for a non-Kähler manifold, since for a Kähler manifold, we have the Kähler identity $\Delta=2\Delta\_{\bar\partial}=2\Delta\_{\partial}$ from which we deduce the Hodge decomposition: $H^k(X,\mathbb C)=\oplus\_{p+q=k}H^{p,q}(X)$, so we can define
$F^pH^k(X,\mathbb C)=H^{p,k-p}(X)\oplus H^{p+1,k-p-1}(X)\oplus...\oplus H^{k,0}(X)$ which is obviously a subspace of $H^k(X,\mathbb C)$,
but for a non-Kähler manifold, there is no Kähler identities. But from [AT13](https://link.springer.com/article/10.1007/s00222-012-0406-3), we know that the Bott-Chern cohomology $H^{p,q}\_{BC}(X,\mathbb C):=\frac{ker\partial\cap ker\bar\partial}{im\partial\bar\partial}$ is isomorphic to Dolbeault cohomology $H\_{\bar\partial}^{p,q}(X)$ and $H\_{\partial}^{p,q}(X)$, so if we define $F^pH^k(X,\mathbb C)=H^{p,k-p}\_{BC}(X,\mathbb C)\oplus H^{p+1,k-p-1}\_{BC}(X,\mathbb C)\oplus...\oplus H^{k,0}\_{BC}(X,\mathbb C)$, is it reasonable to treat it as a subspace of $H^k(X,\mathbb C)$ and define a period map as in the Kähler case?


Has anyone thought it before? or any reference about this issue?



Period map for $\partial\bar\partial$-manifolds
Let me start with a disclaimer that I think the following facts are true, but I'm doing this over coffee and I haven't checked the details carefully. First, I'll redefine $F^pH^k(X,\mathbb{C})$ to be the space of de Rham classes represented by the sum of $(p', p'-k)$ forms with $p'\ge p$, or equivalently as
$$F^pH^k(X,\mathbb{C})= im(H^k(\Omega\_X^{\ge p})\xrightarrow{\iota} H^k(X,\Omega\_X^\bullet))$$
Then the $\partial\bar\partial$-lemma is sufficient to guarantee the filtration is *strict* in the sense that the maps above are injective (or equivalently that the Hodge to de Rham spectral sequence degenerates); **edit** see remark 5.21 of Deligne, Griffiths, Morgan, Sullivan, *Real homotopy theory of Kähler manifolds*. Then, if I understand the notation of the paper you linked, this should probably give decomposition in terms of BC cohomology as you wrote. The other thing I want to remark is that if $\mathcal{X}\to B$ is a smooth proper family such that the fibres satisfy $\partial\bar\partial$-lemma, then usual arguments should imply that
$$F^p R^kf\_\*\Omega\_{\mathcal{X}/B}^\bullet= im(R^kf\_\*\Omega\_{\mathcal{X}/B}^{\ge p}\xrightarrow{\iota}R^kf\_\*\Omega\_{\mathcal{X}/B}^\bullet) $$
satisfies Griffiths transversality etc. So in this sense, things work. However, you would be missing the polarization, which need for most of the deeper results about the Griffiths period map.


