Let $X$ be a compact complex manifold with a Hermitian metric $h$, then we can define $\bar\partial^*$ as the adjoint of $\bar\partial$, define $\Delta_{\bar\partial}:=\bar\partial\bar\partial^*+\bar\partial^*\bar\partial$, $\mathbb H_{\bar\partial}:=\ker\Delta_{\bar\partial}$, and $\mathcal H_{\bar\partial}$ denotes the projection map to the harmonic space $\mathbb H_{\bar\partial}$. By Hodge theory, we have $I=\mathcal H_{\bar\partial}+G_{\bar\partial}\Delta_{\bar\partial}$ where $G_{\bar\partial}$ denotes the Green operator.
Let $\alpha$ be a $d$-closed $(p,q)$ form on $X$, then it is both $\partial$- and $\bar\partial$-closed. Here is the question:
Is $\alpha$ still $\partial$-closed under the projection map $\mathcal H$? Or equivalently, $\partial\mathcal H_{\bar\partial}\alpha=0$?
Added: as mentioned in the comments of Ted, in the Kähler case, by Kähler identity: $\Delta=2\Delta_{\partial}=2\Delta_{\bar\partial},$ we obviously have $\partial\mathcal H_{\bar\partial}\alpha=\partial\mathcal H_{\partial}\alpha=\partial H_{\partial}\alpha=0$. So here we mainly consider non-Kähler cases, for example, does it hold for $\partial\bar\partial$-manifolds (compact complex manifolds satisfying $\partial\bar\partial$-lemma) or manifolds whose Frölicher spectral sequence degenerates at $E_1$? Anyone has any examples?