Let $X$ be a complex manifold, let $E \to X$ be a holomorphic vector bundle of rank $r \in \mathbb{N}$, and let $h$ be a smooth Hermitian metric on $E$, with $h$ conjugate-linear in the first argument and linear in the second argument. Let $\bar{\partial}_E: C^\infty(X;E) \to A^{0,1}(X;E)$ denote the holomorphic structure operator of $E$, where $A^{p,q}(X;E)$ denotes smooth $E$-valued $(p,q)$-forms on $X$. Let $D: C^\infty(X;E) \to A^1(X;E)$ denote the Chern connection of $(E,h)$, namely the unique connection satisfying $D^{0,1}=\bar{\partial}_E$ and compatible with $h$.

Let $U \subset X$ be open, and let $e=(e_1,\dots,e_r)$ be a holomorphic frame of $E$ over $U$, so that each $e_j: U \to E|_U$ is a smooth local section satisfying $\bar{\partial}_E e_j=0$. Assume sections over $U$ are represented by column vectors relative to this frame: if $s \in C^\infty(U;E)$ is written as $s=\sum_{j=1}^r e_j s_j$ with smooth coefficient functions $s_j:U\to\mathbb{C}$, then $s_U=(s_1,\dots,s_r)^\top:U\to\mathbb{C}^r$ denotes its column-vector representative.

Let $A \in A^1(U;\operatorname{End}(\mathbb{C}^r))$ be the local connection matrix of $D$ in the frame $e$, meaning that for every $s \in C^\infty(U;E)$,

\begin{align*} (Ds)_U = d s_U + A s_U. \end{align*}

Let $H:U\to \operatorname{Herm}_r^+$ be the smooth Hermitian metric matrix defined entrywise by

\begin{align*} H_{ij}(x)=h_x(e_i(x),e_j(x)) \end{align*}

for $x\in U$ and $1\leq i,j\leq r$, where $\operatorname{Herm}_r^+$ denotes the set of positive-definite Hermitian $r\times r$ complex matrices. Let $\partial H$ denote the matrix of $(1,0)$-forms obtained by applying the Dolbeault operator $\partial$ entrywise to the smooth matrix-valued function $H$.

Then the local connection matrix of the Chern connection is

\begin{align*} A = H^{-1}\partial H. \end{align*}

Equivalently, in the holomorphic frame $e$ and under the column-vector convention,

\begin{align*} D = d + H^{-1}\partial H. \end{align*}