Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose G=Gk(d,n)$G=G_k(d,n)$ (2≤d≤n−d)$(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in kn$k^n$, where k is an algebraically closed field of characteristic p>0$p>0$. Let E be a uniform vector bundle over G of rank r≤d$r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle Hd$H_d$ or its dual Hd∨$H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties F(d1,…,ds)$F(d_1,\ldots ,d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in F(d1,…,ds)$F(d_1,\ldots ,d_s)$. Furthermore, we generalize the Grauert–Mü$\ddot{\text{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i‐semistable (1≤i≤n−1)$(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.