Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in \mathbb{C}^m$, $$\eta \vert_{\{ z \} \times Y} = \omega \vert_{\{ z \} \times Y}.$$ Suppose further that $[\omega] = [\eta]$ in $H^2(\mathbb{C}^m \times Y )$.
Can we conclude that there is an automorphism $T \in \text{Aut}(\mathbb{C}^m \times Y)$ such that $T^{\ast}\omega = \eta$ or $T^{\ast} \omega$ and $\eta$ are $\sqrt{-1} \partial \overline{\partial}$-cohomologous?
