Vector bundles on flag varieties. Issue 2 (28th November 2022) - Du, Rong; Fang, Xinyi; Gao, Yun
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 ) (2dn-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 rd . 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^ 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, …, 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, …, d_s) . Furthermore, we generalize the Grauert–M u ̈ u lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i ‐semistable ( 1 ≤ i ≤ n − 1 ) (1in-1) bundle over the complete flag variety splits as a direct sum of special line bundles.