 Solution

Choose a local holomorphic frame $e$ of the holomorphic line bundle $E$ and defineIf $e^{\prime }=ge$ 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 ${\bar{\partial }}_E^2=0$.

Solved by gpt-5.6-sol high.