Čech-de Rham curvature descent ($[i\Theta /(2\pi )]=c_1(E{)}_{\mathbb{C}}$)

For line-bundle transition functions $g_{ij}=e^{2\pi i{f}_{ij}}$, the integer cocycle is $c=\delta f$. Connection forms satisfy $A_j-A_i=2\pi id{f}_{ij}$. Set $B=-A/(2\pi i)$ and $F=i\Theta /(2\pi )$. In the Čech-de Rham total complex, $F-c=D_{\mathrm{tot}}(B-f)$, so the normalized vector-bundle curvature represents the image of the integral First Chern class. Curvature cannot detect integral torsion.