Let \(\mathrm{Tw(M)}\) be a twistor space of a hypercomplex, quaternionic Kähler, quaternionic, or ASD 4-manifold \(M\), and \(S\) a rational curve in \(\mathrm{Tw(M)}\) obtained as a fiber of the projection to \(M\). I prove ``a holography principle'': any meromorphic function defined in a neighbourhood \(U\) of \(S\) can be extended to a meromorphic function on \(\mathrm{Tw(M)}\), and any section of a holomorphic line bundle can be extended from \(U\) to \(\mathrm{Tw(M)}\). This is used to define the notion of a Moishezon twistor space. I prove that the twistor spaces of hyperkähler manifolds obtained by hyperkähler reduction (such as Nakajima's quiver varieties) are always Moishezon.
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