Pure and Applied Mathematics Quarterly

Volume 10 (2014)

Number 2

Special Issue: In Memory of Andrey Todorov, Part 3 of 3

Holography principle for twistor spaces

Pages: 325 – 354

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n2.a6

Author

Misha Verbitsky (Laboratory of Algebraic Geometry, Faculty of Mathematics, National Research University HSE, Moscow, Russia)

Abstract

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive: $NS = \bigoplus \mathcal{O} (i_k), i_k \lt 0$. 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 hyperkähler 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 admitting a holomorphic embedding to a Moishezon variety $M\prime$. We show that this property is local on $X$, and the variety $M\prime$ is unique up to birational transform. We prove that the twistor spaces of hyperkähler manifolds obtained by hyperkähler reduction of flat quaternionic-Hermitian spaces by the action of reductive Lie groups (such as Nakajima’s quiver varieties) are always Moishezon.

Keywords

Quiver varieties, hyperkähler reduction, twistor space, rational curve, quasi-line, Moishezon manifold

Full Text (PDF format)