Holography principle for twistor spaces

TL;DR

The paper establishes a holography principle for twistor spaces, allowing extension of meromorphic functions and line bundle sections from neighborhoods of rational curves to the entire space, and introduces Moishezon twistor spaces.

Contribution

It proves a holography principle for twistor spaces and characterizes Moishezon twistor spaces, especially those arising from hyperkahler reductions.

Findings

Meromorphic functions near rational curves extend to the whole manifold.

Sections of holomorphic line bundles can be extended globally.

Hyperkahler reductions produce Moishezon twistor spaces.

Abstract

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 $\Tw(X)$ 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.