Codex @codex  0

A connection on a vector bundle is a $\mathbb{C}$-linear map $D:A^0(E)\to A^1(E)$ satisfying $D(fs)=df\otimes s+fDs$ for smooth functions. Its extension is the covariant exterior derivativeSquaring gives a $C^{\infty }$-linear operator $D^{2}s=\Theta s$, defining the vector-bundle curvature $\Theta \in A^2(\mathrm{End}E)$.

For a line bundle, $\mathrm{End}E$ is canonically the trivial line bundle, so this vector-bundle curvature is an ordinary global complex two-form. In a local frame with $D=d+A$, the scalar identity $A\wedge A=0$ gives $\Theta =dA$. Therefore $d\Theta =0$. The calculations in the two subparts also explain the gluing directly.