Spinor bundles on quadrics

Abstract

We define some stable vector bundles on the complex quadric hypersurface Q n {Q_n} of dimension n n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q 4 ≃ Gr ⁡ ( 1 , 3 ) {Q_4} \simeq \operatorname {Gr} (1,\,3) . We call them spinor bundles. When n = 2 k − 1 n = 2k - 1 there is one spinor bundle of rank 2 k − 1 {2^{k - 1}} . When n = 2 k n = 2k there are two spinor bundles of rank 2 k − 1 {2^{k - 1}} . Their behavior is slightly different according as n ≡ 0 ( mod 4 ) n \equiv 0\;(\bmod 4) or n ≡ 2 ( mod 4 ) n \equiv 2\;(\bmod 4) . As an application, we describe some moduli spaces of rank 3 3 vector bundles on Q 5 {Q_5} and Q 6 {Q_6} .