I don't quite understand the following definition:
Let $(E,J)$ be a complex manifold, $(\Sigma, j)$ Riemann surface with a.c. structure, $E \rightarrow \Sigma$ a holomorphic bundle.
Define
$\overline{\partial}: \Gamma(E) \rightarrow \Gamma(\overline{Hom}_{\mathbb{C}}(T \Sigma, E))$ by $\overline{\partial} s= ds+i ds \circ j$ "written in a local holomorphic trivialization".
But as far as I understand $ds_p(v) \in T_{s(p)}E$.
What does "written in a local holomorphic trivialization" mean here?
Writen in a local trivialization over an open subset $V\subset \Sigma$ we have $s(x) = (x, f(x))$ for some smooth function $f\colon V \to \mathbb{C}^n$ where $n$ is the rank of $E$. We define locally $$ds = (x,df_x) = (x, df_1, \dots, df_n).$$ It makes sense as an ordered set of differential forms so does $\bar \partial s$. The special thing about $\bar \partial s$ is that if defines a global object by glueing local ones if we use holomorphic trivializations.
If we have trivializations over an open covering $\{V_\alpha\}$ of $X$ with (holomorphic) transition functions $G_{\alpha \beta}$ to give a section $s$ is the same as to give a collection of functions $s = \{(V_\alpha, s_\alpha)\}$ that glue together by the transitions of $E$: $s_\alpha(x) = G_{\alpha \beta}(x)\cdot s_\beta(x)$.
Then $\bar \partial s$ is defined by $\bar \partial s = \{(V_\alpha, \bar \partial s_\alpha)\}$ and we have that $$ \bar \partial s_\alpha = (\bar \partial G_{\alpha \beta})\cdot s_\beta + G_{\alpha \beta}\cdot \bar \partial s_\beta = G_{\alpha \beta}\cdot \bar \partial s_\beta $$ where the last equality holds because the $G_{\alpha \beta}$ are holomorphic. It follows from this that $\bar \partial s$ is a one-form that takes values on $E$. It means that for a vector field $v$ on $X$, $\bar \partial s (v)$ gives a section of $E$ (by assigning $\{(V_\alpha, \bar \partial s_\alpha(v))\}$ ).