The nonexistence of sections of Stiefel varieties and stably free modules

Let $V_r({\mathbb{A}}^n)$ denote the Stiefel variety ${\mathrm{G}\mathrm{L}}_n\mathrm{/}{\mathrm{G}\mathrm{L}}_{n-r}$ over a field. There is a natural projection $p:V_{r+\mathrm{\ell }}({\mathbb{A}}^n)\to V_r({\mathbb{A}}^n)$. The question of whether this projection admits a section was asked by M. Raynaud in 1968. We focus on the case of $r\ge 2$ and provide examples of triples $(r,n,\mathrm{\ell })$ for which a section does not exist. Our results produce examples of stably free modules that do not have free summands of a given rank. To this end, we also construct a splitting of $V_2({\mathbb{A}}^n)$ in the motivic stable homotopy category over a field, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James.