I am currently reading a book which deals with complex manifolds. Since I am fairly new to the topic I don't know exactly the meaning of the followinig:
Suppose we have a holomorphic vector bundle $V$ over the manifold $M$ with frames $s_\alpha$ over each trivialization $U_\alpha \subset M$
We can construct a Hermitian metric $h$ on $V$, and the author says this is given locally as
$h_\alpha = (s_\alpha,s_\alpha) $.
Then a connection 1 - form is defined locally by
\begin{equation} \omega_\alpha = \partial h_\alpha h_\alpha^{-1} \end{equation} where \begin{equation} d = \partial + \bar{\partial} \end{equation} and \begin{equation} \partial(f) = \sum_j \frac{\partial f}{\partial z^j}dz^j \end{equation} (more generally $\partial \colon C^\infty(\Lambda^{p,q}) \to C^\infty(\Lambda^{p+1,q}) $. It is then shown in the book that these 1-forms patch together to form a connection $\triangledown_h$.
Now comes the bit where I am struggeling with, to the extend that I can't read on without a bad feeling:
From the definition, one should see that \begin{align} (\triangledown_h s_\alpha, s_\alpha) + (s_\alpha, \triangledown_h s_\alpha) &= \omega_\alpha h_\alpha + h_\alpha \omega^*_\alpha \\ &= \partial h _\alpha + \bar{\partial}h _\alpha = dh _\alpha \end{align}
I am afraind I don't know enough about connections yet, in particular I don't really understand how to get from the first expression to the second. If anyone could fill in a little more details into the lines above that would be very helpful!