Does anyone know of some good references for computing Christoffel symbols for Hermitian metrics?

A quick Google search turns up this. The following formula appears on page 4:

$$\Gamma_{AB}^C = \frac{1}{2}h^{CE}\left(\frac{\partial h_{AE}}{\partial z^B} + \frac{\partial h_{BE}}{\partial z^A} - \frac{\partial h_{AB}}{\partial z^E}\right)$$

where $A, B, C, E \in$ {$1, \dots, n, \bar{1}, \dots, \bar{n}$} and $z^{\bar{i}} = \bar{z}^i$. From this they get

$$\Gamma_{\bar{i}j}^k = \frac{1}{2}h^{k\bar{l}}\left(\frac{\partial h_{j\bar{l}}}{\partial \bar{z}^i} - \frac{\partial h_{j\bar{i}}}{\partial \bar{z}^l}\right)$$

How do they obtain this? Are they regarding $h$ as a map $(T^{1,0}M\oplus T^{0,1}M) \times (T^{1,0}M\oplus T^{0,1}M) \to \mathbb{C}$ where $h_{ab} = 0$, $h_{\bar{a}\bar{b}} = 0$, and $h_{\bar{a}b} = \overline{h_{b\bar{a}}}$? Even if they do, I don't see how they get the second term.

Everything else I have found deals only with Kähler metrics, in which case $\Gamma_{ab}^c$ and $\Gamma_{\bar{a}\bar{b}}^{\bar{c}}$ are the only non-trivial symbols.

More generally, are there any treatments of Hermitian geometry which take this coordinate approach (as is common in Riemannian geometry texts)?

extremelyconfusing, especially working with the complexified tangent bundle. I also found references useful only as overall guidance and not for details. I suggest two things: a) First, do all of the calculations in the original "real" tangent bundle and figure out all of the symmetries satisfied by the metric tensor and Christoffel symbol. b) Work out in painful detail what happens in complex dimensions 1 and 2. – Deane Yang Jun 29 '12 at 16:40