Let $M^n$ be a compact Kahler manifold and consider the product complex manifold $M \times \mathbb{C}^m$. By the Leray spectral sequence associated to the projection $\pi_{\mathbb{C}^m} : \mathbb{C}^m \times M \to \mathbb{C}^m$, there is an isomorphism $$H^{0,1}(\mathbb{C}^m \times M) \ \cong \ H^{0,1}(M) \otimes H^0(\mathbb{C}^m, \mathcal{O}_{\mathbb{C}^m}),$$
given by $[\xi^{0,1}](z) \mapsto [\xi^{0,1} \vert_{\{ z \} \times Y}]$. In particular, we may write $$\xi^{0,1} = \sum_k \sigma_k \vartheta_k + \overline{\partial} h,$$ for holomorphic functions $\sigma_k$ on $\mathbb{C}^m$, $(0,1)$-forms $\vartheta_k$ on $M$, and a smooth complex-valued function $h$ on $\mathbb{C}^m$.
Question: Is there a way of proving that $\xi^{0,1}$ can be written in the above form without appealing to spectral sequences?