A small doubt in Kähler geometry

Question

We often use this lemma in Kähler geometry. For a compact Kähler manifold $(X,\omega)$ and a function $f$ on $X$ we have \begin{align*} \langle\partial\bar\partial f,\omega\rangle=\frac{i}{2}\Delta(f) \end{align*} But I am thinking \begin{align*} \langle\partial\bar\partial f,\omega\rangle=\langle\bar\partial f,\partial^*\omega\rangle=\langle\bar\partial f,0\rangle=0 \end{align*} I must be making some big mistake here. Any help?