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 ${\mathbb{P}}^{2n+1}$

Elsa Puente, Alberto Verjovsky

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 $…