Let $ Y$ be a compact (without boundary) Calabi-Yau manifold, i.e., $ c_1(Y)=0$ in $ H^2(Y, \mathbb{R})$ . Let $ \omega$ be a Kähler form on $ \mathbb{C}^m \times Y$ and let $ \omega_P = \omega_{\mathbb{C}^m} + \omega_Y$ . Assume that $ \zeta = \omega – \omega_P$ is an exact $ (1,1)$ -form, i.e., $ \zeta = d\xi$ for some real $ 1$ -form $ \xi$ on $ \mathbb{C}^m \times Y$ . By the Leray spectral sequence of the projection $ \pi_{\mathbb{C}^m} : \mathbb{C}^m \times Y \to \mathbb{C}^m$ , there is an isomorphism $ \Phi : H^{0,1}(\mathbb{C}^m \times Y) \to \mathcal{O}(\mathbb{C}^m, H^{0,1}(Y))$ with $ $ \Phi[\xi^{0,1}](z) = [\xi^{0,1} \vert_{\{ z \} \times Y}].$ $ Set $ \Phi[\xi^{0,1}]=f$ and identify $ H^{0,1}(Y)$ with the space of $ g_Y$ -parallel $ (0,1)$ -forms.
Q: I want to show that $ $ \frac{\partial f }{\partial \overline{z}^j}=0.$ $
Moreover, I want to show that $ \dfrac{\partial f}{\partial z^j}$ is the $ H^{0,1}$ -class, or the $ g_Y$ -parallel part of the $ (0,1)$ -form $ $ (\partial_{z^j} \llcorner \zeta) \vert_{\{ z \} \times Y},$ $ where $ \llcorner$ denotes the interior product.
