Solution
= Solution
Choose a local holomorphic frame $e$ of the <holomorphic line bundle> $E$ and define
$$
\bar\partial_E(fe)=\bar\partial f\otimes e.
$$
If $e'=ge$ is another holomorphic frame, then $g$ is nowhere-zero and holomorphic, so $\bar\partial g=0$ and the two definitions agree. They therefore glue to a well-defined partial connection. In each holomorphic frame its square is ordinary $\bar\partial^2$, hence $\bar\partial_E^2=0$.