

Seminar "Complex analysis in several variables" (Vitushkin Seminar)
December 12, 2012 16:45, Moscow, MSU, auditorium 1304






Meromorphic functions on twistor spaces
M. S. Verbitsky^{} ^{} National Research University "Higher School of Economics"

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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 4dimensional antiselfdual 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 quaternionicHermitian spaces by the action of reductive Lie groups (such as Nakajima's quiver varieties) are always Moishezon.

