2 (source code)
= 2
{scope}
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 derivative>
$$
D(\eta\otimes s)=d\eta\otimes s+(-1)^{\deg\eta}\eta\wedge Ds.
$$
Squaring gives a $C^\infty$-linear operator $D^2s=\Theta s$, defining the <vector-bundle curvature> $\Theta\in A^2(\operatorname{End}E)$.
For a <line bundle>, $\operatorname{End}E$ 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 \b[$d\Theta=0$]. The calculations in the two subparts also explain the gluing directly.
Back to article page
