= Čech cocycle for the curvature Dolbeault class
{c}
{title2=$\sigma_{\alpha\beta}=(\partial g_{\alpha\beta})g_{\alpha\beta}^{-1}$}
In frames with $e^{(\beta)}=g_{\alpha\beta}e^{(\alpha)}$ and input-index connection matrices, the <Chern connection> has $\theta_\beta-g_{\alpha\beta}\theta_\alpha g_{\alpha\beta}^{-1}=\sigma_{\alpha\beta}$. These holomorphic one-forms, interpreted in the beta frame, satisfy the transported <Čech cocycle condition> because $g_{\alpha\gamma}=g_{\beta\gamma}g_{\alpha\beta}$. Local primitives of curvature differ by this cocycle. A holomorphic frame change $e'_\alpha=u_\alpha e_\alpha$ adds the <Čech coboundary> of $u_\alpha^{-1}\partial u_\alpha$, so the resulting <curvature Dolbeault class> is intrinsic.
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