# Vector bundles on Grassmannians

Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$.

Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and let $X\subseteq Gr(k,n)$ be the variety parametrizing the $H^k$'s containing $\Gamma$. Then $X\cong Gr(n-k,n-h)$.

Let $E$ the the rank $k-h$ vector bundle on $X$ whose fiber over a point $H^k\in X$ is $H^k/\Gamma$. Does there exists a rank $k-h$ vector bundle $F$ on $Gr(k,n)$ whose restriction to $X$ is $E$ ?

take the flag variety F(h,k,n) whose elements are of the form $H\subset K\subset V$. We have maps $p_1:F(h,k,n)\rightarrow Gr(k,n)$ and $p_2:F(h,k,n)\rightarrow Gr(h,n)$. Take ${p_1}_*{p_2}^* \mathcal{O}(1)$. This is the desired vector bundle.
• I am confused. Take $h=0$, then $p_1:F(0,k,n)\rightarrow Gr(k,n)$ is an isomorphism, and $p_{1*}\mathcal{O}(1)\cong\mathcal{O}(1)$ is a line bundle. However in this case the desired vector bundle should be the universal bundle on $Gr(k,n)$. Jun 26 '17 at 21:57
This is an interesting question. Suppose that $$h = k-1$$. Then the answer appears to be yes. Consider the determinant line bundle $$L$$ on $$Gr(k,n)$$; its fibre at $$H$$ is $$\det H$$.
Now let us pick some a point $$H \in X$$. We have the short exact sequence $$0 \rightarrow \Gamma \rightarrow H \rightarrow H / \Gamma \rightarrow 0$$ and thus $$\det \Gamma \otimes \det H/\Gamma \cong \det H$$. As $$\Gamma$$ is a fixed vector space, $$\det \Gamma$$ is a trivial line bundle and thus $$H/\Gamma \cong \det H$$.
Hence $$L$$ restricts to $$E$$ as desired.