= Line-trivial bundles in the Ward correspondence
{title2=$E|_{L_x}\cong\mathcal O^{\oplus r}$}
= Line-trivial bundle in the Ward correspondence
{synonym}
On a twistor domain containing the lines $L_x$ for $x$ in a space-time domain $U$, the Ward construction uses a <holomorphic vector bundle> $E$ with $E|_{L_x}\cong\mathcal O^{\oplus r}$ for every $x\in U$. A degree-zero restriction is not sufficient: $\mathcal O(1)\oplus\mathcal O(-1)$ has degree zero but is not holomorphically trivial. A trivial restriction has $r$ independent global <holomorphic> sections; their evaluations give a linewise <bundle frame> and hence a factorization of its patching <matrix>. Nearby factors can be chosen holomorphically because $H^1(\mathbb{CP}^1,\mathcal O)^{r^2}=0$. Lines on which the splitting type is nontrivial are called jumping lines and obstruct this regular reconstruction there. For a globally regular field on affine <complexified Minkowski space> the condition is needed on all its incidence lines; on a smaller domain only those lines are required.
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