Consider the tautological bundle $S$ on a Grassmannian $G(r,n)$ of $r$-subspaces in $\mathbb{C}^n$. Is $S$ trivial outside (large degree) hypersurfaces on $G(r,n)$. Morel's theorem seems to confirm when $r=2$.

Consider the tautological bundle $S$ on a Grassmannian $G(r,n)$ of $r$-subspaces in $\mathbb{C}^n$. Is $S$ trivial outside (large degree) hypersurfaces on $G(r,n)$. Morel's theorem seems to confirm when $r\neq 2$.

thanks. but I want the complement to be large degree hypersurface sections.