= Čech-de Rham curvature descent
{c}
{title2=$[i\Theta/(2\pi)]=c_1(E)_{\mathbb C}$}
For line-bundle transition functions $g_{ij}=e^{2\pi i f_{ij}}$, the integer cocycle is $c=\delta f$. Connection forms satisfy $A_j-A_i=2\pi i\,df_{ij}$. Set $B=-A/(2\pi i)$ and $F=i\Theta/(2\pi)$. In the Čech-de Rham total complex, $F-c=D_{\rm tot}(B-f)$, so the normalized <vector-bundle curvature> represents the image of the integral <First Chern class>. Curvature cannot detect integral torsion.
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