Spinor bundles on quadrics

Abstract:

We define some stable vector bundles on the complex quadric hypersurface ${Q_n}$ of dimension $n$ as the natural generalization of the universal bundle and the dual of the quotient bundle on ${Q_4} \simeq \operatorname {Gr} (1, 3)$. We call them spinor bundles. When $n = 2k - 1$ there is one spinor bundle of rank ${2^{k - 1}}$. When $n = 2k$ there are two spinor bundles of rank ${2^{k - 1}}$. Their behavior is slightly different according as $n \equiv 0\;(\bmod 4)$ or $n \equiv 2\;(\bmod 4)$. As an application, we describe some moduli spaces of rank $3$ vector bundles on ${Q_5}$ and ${Q_6}$.