I have a question about Laplacians on Kaehler manifolds. In a paper I am reading it is obtained that
\begin{align} \overline{\partial}i(\frac{1}{2}g^{ik}f_{\overline{k}}\partial_{j})\omega = \frac{1}{2}f_{\overline{k}\overline{j}}d\overline{z}^{j}\wedge i(\partial_{k})\omega, \end{align} and \begin{align} i(\frac{1}{2}g^{j\overline{k}}f_{\overline{k}}\partial_{j})\overline{\partial}\omega = \frac{1}{2}f_{\overline{k}}(\overline{\partial}i(\partial_{k}) + i(\partial_{k})\overline{\partial})\omega \end{align}. I have a difficulty to understand these calculations, for example where is $g^{ik}$ in the final expression. Besides, I can not find some basic reference where Laplacians on Kaehler manifolds are calculated. Thank you!