= Holomorphic splitting of a twistor patching matrix
{title2=$F=\widetilde H H^{-1}$}
A bundle trivial on a <twistor line> has a global holomorphic <bundle frame> there. Comparing it with the given two patch <bundle frames> produces invertible <matrices> $H,\widetilde H$, holomorphic on their respective affine patches, with $F=\widetilde H H^{-1}$. The obstruction to varying such a <bundle frame> locally with space-time is $H^1(\mathbb{CP}^1,\mathcal O)^{n^2}=0$, so the splitting factors can be chosen holomorphically in a neighbourhood. If two splittings differ by $H\mapsto Hg$ and $\widetilde H\mapsto\widetilde Hg$, then $g$ is holomorphic on the compact <complex projective line> and hence independent of its projective coordinate. It is the space-time gauge freedom.
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