Meromorphic functions on twistor spaces

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive. We assume that $M$ is covered by smooth holomorphic deformations of $S$. The basic example of such a manifold is a twistor space of a hyperkahler or a 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily compact). We prove "a holography principle" for such a manifold: any meromorphic function defined in a neighbourhood $U$ of $S$ can be extended to $M$, and any section of a holomorphic line bundle can be extended from $U$ to $M$. This is used to define the notion of a Moishezon twistor space: this is a twistor space $\mathsf{Tw}(X)$ admitting a holomorphic embedding to a Moishezon variety $M'$. We show that this property is local on $X$, and the variety $M'$ is unique up to birational transform. We prove that the twistor spaces of hyperkahler manifolds obtained by hyperkahler reduction of flat quaternionic-Hermitian spaces by the action of reductive Lie groups (such as Nakajima's quiver varieties) are always Moishezon.