Understanding metrics of Hermitian manifolds Let $X$ be a complex manifold with a Hermitian metric $h$. Locally, we can consider the space $T^{1,0}X$ with basis $\displaystyle\left\{ \frac{\partial}{\partial z_j} \right\}$ and the space $T^{0,1}X$ with basis $\displaystyle\left\{ \frac{\partial}{\partial \overline{z_j}} \right\}$. From what I understand, for any indexes $k,l=1 \ldots n$, we have that $\displaystyle\frac{\partial}{\partial z_k}$ and $\displaystyle\frac{\partial}{\partial \overline{z_k}}$ are orthogonal with respect to $h$.
We can write $$h = \sum_{i,j=1}^n h_{ij} dz_i \otimes d\overline{z_j}.$$
Now note that
$$h\left(\frac{\partial}{\partial z_k}, \frac{\partial}{\partial \overline{z_l}}\right) = \sum_{i,j=1}^n h_{ij} dz_i \otimes d\overline{z_j}\left(\frac{\partial}{\partial z_k}, \frac{\partial}{\partial \overline{z_l}}\right) = $$
$$ = h_{kl} dz_k \otimes d\overline{z_l}\left(\frac{\partial}{\partial z_k}, \frac{\partial}{\partial \overline{z_l}}\right) = h_{kl} dz_k \left(\frac{\partial}{\partial z_k} \right) d\overline{z_l} \left(\frac{\partial}{\partial \overline{z_l}}\right) = h_{kl}.$$
The answer is not zero, as I thought it should be. Worse yet, if one try define a norm using $\| \cdot \| = h(\cdot, \cdot)$, then $\displaystyle\left\|\frac{\partial}{\partial z_k}\right\| = \left\|\frac{\partial}{\partial \overline{z_k}}\right\| = 0$, which makes no sense. I don't know what is incorrect in my understanding. I need some clarification on this topic.
Thank you very much for your help.

The best way to think about this is that the hermitian inner product is defined on the holomorphic tangent space (not the complexified tangent space), and you should interpret $d\bar z_j(v)$ as $\overline{dz_j(v)}$.
