Let $Y$ be a closed Kähler manifold with $c_1(Y)=0$ in $H^2(Y,\mathbb{R})$. Let $\omega$ be a Ricci-flat Kähler form on $\mathbb{C}^m \times Y$ such that $$A^{-1} (\omega_{\mathbb{C}^m} + \omega_Y) \leq \omega \leq A  (\omega_{\mathbb{C}^m} + \omega_Y),$$ for some constant $A \geq 1$, where $\omega_Y$ is a Kähler form on $Y$ and $\omega_{\mathbb{C}^m}$ is the Euclidean form on $\mathbb{C}^m$. 

I want to show that there is a unique choice of $\omega_Y$ such that $\text{Ric}(\omega_Y)=0$ and that there is a smooth function $f$ such that $$\omega = \omega_{\mathbb{C}^m} + \omega_Y + d f.$$


_Update:_ I can show that the Kähler class of $\omega$ is independent of the $\mathbb{C}^m$ component by using Künneth's formula. To establish the existence of $f$ I have been attempting to write out the components of $\omega$ as a section of $\Lambda^2(\mathbb{C}^m \times Y) = \Lambda^2(\mathbb{C}^m) \oplus \Lambda^1(\mathbb{C}^m) \otimes \Lambda^1(Y) \oplus \Lambda^2(Y)$ and using the fact that $d\omega=0$...