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, 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 is 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?