= Smoothly trivial holomorphic line bundles when H01 vanishes
{title2=$H^{0,1}(X)=0,\quad L\text{ smoothly trivial}\Longrightarrow L\text{ holomorphically trivial}$}
A nowhere-zero smooth section $s$ of a <holomorphic line bundle> writes its <Dolbeault operator> as $\bar\partial_Ls=a\otimes s$. 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.
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