Vector bundles on flag varieties

Abstract

AbstractWe study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose to be the Grassmannian parameterizing linear subspaces of dimension d in , where k is an algebraically closed field of characteristic . Let E be a uniform vector bundle over G of rank . We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle or its dual by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties 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 . Furthermore, we generalize the Grauert–Mlich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i‐semistable bundle over the complete flag variety splits as a direct sum of special line bundles.