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 $$\omega = \omega_{\mathbb{C}^m} + \omega_Y + d f.$$