Vector bundles on flag varieties. Issue 2 (28th November 2022) - Du, Rong
Abstract: We study vector bundles on flag varieties over an algebraically closed field k . In the first part, we suppose G = G k ( 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 k n $k^n$, 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 H d $H_d$ or its dual H d ∨ $H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties F ( d 1, …, d s ) $F(d_1, \ldots, d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the i th component of the manifold of lines in F ( d 1, …, d s ) $F(d_1, \ldots, d_s)$ . Furthermore, we generalize the Grauert–M u ̈ $\ddot{\text{u}}$ 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.