Abstract
Let $S$ be smooth rational curve on 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 manifold is twistor space of hyperkahler or 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily compact). We prove a holography principle for such manifold: any meromorphic function defined in neighbourhood $U$ of $S$ can be extended to $M$, and any section of holomorphic line bundle can be extended from $U$ to $M$. This is used to define the notion of Moishezon twistor space: this is twistor space $\Tw(X)$ admitting holomorphic embedding to 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.
Highlights
1.1 Quasilines on complex manifoldsThe present paper was written as an attempt to answer the following question
I H0(S, OS(M )⊗OS O(i)) would provide a sort of an algebraic “normal form” of S in M, in such a way that the algebraic structure on OS(M ) can be reconstructed from this ring. We show that this approach works when M is a Moishezon manifold, and M can be reconstructed from the ring AS, up to birational isomorphism (Subsection 4.3; this is not very surprising due to the above-mentioned theorem of Hartshorne, [Har, Theorem 6.7])
The twistor spaces are complex manifold covered by quasilines, usually non-Kahler and non-quasiprojective
Summary
The present paper was written as an attempt to answer the following question. Let S ⊂ M be a smooth rational curve in a complex manifold, with normal bundle N S isomorphic to O(1)n.1 Is there a notion of a normal form for a tubular neighbourhood of such a curve?. The birational type of the manifold can be reconstructed from a complex analytic (and even formal) neighbourhood of S. This was known already to Hartshorne ([Har, Theorem 6.7]). We show that this approach works when M is a Moishezon manifold, and M can be reconstructed from the ring AS, up to birational isomorphism (Subsection 4.3; this is not very surprising due to the above-mentioned theorem of Hartshorne, [Har, Theorem 6.7]). For non-Moishezon M , such as a twistor space of a connected, compact hyperkahler manifold, this conjecture is spectacularly wrong In this case, AS = i H0(CP 1, O(i)), and this ring has no information about M whatsoever (Proposition 2.13)
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