It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H\_{BC}^{\bullet,\bullet}:=\frac{\ker \partial\cap\ker\bar\partial}{\text{im }\partial\bar\partial}$ is isomorphic to the Dolbeault cohomology $H\_{\bar\partial}^{\bullet,\bullet}(X)=\frac{\ker\bar\partial}{\text{im }\bar\partial}$, and the Frölicher spectral sequence of $X$ degenerates at $E\_1$, then
$$
H^k(X,\mathbb C)=\bigoplus\_{p+q=k}H^{p,q}\_{\bar\partial}(X)=\bigoplus\_{p+q=k}H^{p,q}\_{BC}(X),
$$
so $H^{p,q}\_{\bar\partial}(X)$ can be treated as a subspace of $H^k(X,\mathbb C)$ through the isomorphism between Bott-Chern and Dolbeault cohomology.


If we only assume that the Frölicher spectral sequence of $X$ degenerates at $E\_1$, is there a natural way to treat $H^{p,q}\_{\bar\partial}(X)$ as a subspace of $H^k(X,\mathbb C)$? or is there a natural map $f:H^{p,q}\_{\bar\partial}(X)\to H^k(X,\mathbb C)$ which is injective?



When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?
I'm not sure that the following answers the question, but at least it shows that it is not always possible to construct a map $H\_{\bar\partial}\to H\_{dR}$ by assigning a $d$-closed representative to each class in $H\_{\bar\partial}$.


Suppose $\alpha$ is a $\bar\partial$-closed form, and we want to find $\beta$ such that $d(\alpha-\bar\partial\beta)=0$. Each summand inside the parentheses is $\bar\partial$-closed, so the equality is equivalent to $\partial\alpha=\partial\bar\partial\beta$. We learn that each class in $H\_{\bar\partial}$ admits a $d$-closed representative if and only if $\partial\ker\bar\partial\subset\operatorname{im}\partial\bar\partial$ — in a more fancy language, the map $H\_{\bar\partial}\to H\_{BC}$ induced by $\partial$ is zero.


As explained in [Ornea, Verbitsky, Vuletescu, Classification of non-Kähler surfaces and locally conformally Kähler geometry](https://arxiv.org/abs/1810.05768), § 2.2, for any non-Kähler compact complex surface, the image of $H^{0,1}\_{\bar\partial}$ by this map is 1-dimensional. So there is a class in $H^{0,1}\_{\bar\partial}$ with no $d$-closed representative, notwithstanding that the Frölicher spectral sequence of any compact complex surface degenerates at the first page.


