Smoothly trivial holomorphic line bundles when H01 vanishes ($H^{0,1}(X)=0,L\text{ smoothly trivial}⟹L\text{ holomorphically trivial}$)

A nowhere-zero smooth section $s$ of a holomorphic line bundle writes its Dolbeault operator as ${\bar{\partial }}_{L}s=a\otimes s$. Its integrability gives $\bar{\partial }a=0$. If $H^{0,1}(X)=0$, then $a=\bar{\partial }f$ globally. The section $e^{-f}s$ is nowhere zero and holomorphic. This argument requires no compactness and explicitly constructs the holomorphic trivialization.