Let $(X, \omega)$ be a compact complex manifold of dimension n. Let $\pi:Y\to X$ be the universal cover of $X$. Let $\phi$ be a $\delta_\omega$ - harmonic 2-form on $X$ such that $\pi*(\phi)$ is a d-exact differential form on Y. My question is:
Is $\pi*(\star\phi)$ also d-exact on $Y$? Where $\star$ is the star hodge operator.
In fact, in the particular case where $\phi$ is a primitive form and $\omega$ is Kahler, the answer is positive. I want to know if this result remains true in the general case or at least in the case where $\phi$ is the Harmonic part of $\omega$ (I speak of the Hodge decomposition into three orthogonal spaces).