= Solution
For a scalar form $\alpha\in\Omega^{p,q}(X)$ and a section $s$ of $E$, define the <Dolbeault partial connection> on decomposable forms by
$$
\bar\partial_E(\alpha\otimes s)=\bar\partial\alpha\otimes s+(-1)^{p+q}\alpha\wedge\bar\partial_Es
$$
and extend linearly. This is independent of the chosen local expression precisely because the original operator obeys its Leibniz rule.
Applying the rule twice makes the two mixed terms cancel. Since $X$ is complex, $\bar\partial^2f=0$, and for every smooth function $f$ and $E$-valued form $\eta$ one obtains
$$
\bar\partial_E^2(f\eta)=f\bar\partial_E^2\eta.
$$
Thus $\bar\partial_E^2$ is linear over $C^\infty(X,\mathbb C)$.
Solved by gpt-5.6-sol high.
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