Skew-symmetric complex matrices, pure spinors, the twistor space of the conformal $2n$-sphere, and the Fano variety of linear $n$-folds of a non-singular complex quadric hypersurface in ${\mathbb{P}}^{2n+1}$

Abstract

For $n\ge 1$, the twistor space $\mathfrak{Z}({\mathbb{S}}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1,2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of ${\mathbb{C}}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}({\mathbb{S}}^{2n})$ with $n\ge 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of ${\mathbb{C}}^{n+1}$. In particular, the twistor space $\mathfrak{Z}({\mathbb{S}}^6)$ is biholomorphic to a non-singular complex quadric hypersurface in ${\mathbb{P}}^7$. We explicitly construct a real-analytic foliation, by linear 3-folds, of this quadric hypersurface such that the quotient space is isomorphic to the 6-sphere with its standard metric. This foliation is Riemannian with respect to the Fubini-Study metric and isometrically equivalent to the twistor fibration over the 6-sphere.

arXiv commentThis paper has been withdrawn by the authors in order to make several corrections and write a new version

References in corpus (2)

• Higher-Dimensional Twistor Transforms using Pure Spinors
• Extension Phenomena for Holomorphic Geometric Structures

Cited by in corpus (1)

• Decoherence-free measurements in steering tasks