Vector Bundles on Flag varieties

Rong Du, Xinyi Fang, Yun Gao

Published 2019-05-24Version 1

We study vector bundles on flag varieties over algebraic closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraic closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle on $G$ of rank $r\le d$. We show that $E$ is a direct sum of line bundles unless it is a twist of either the pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ under the $m$-th absolute Frobenius morphism, where $m$ is a nonnegative integer. In the second part, splitting properties of vector bundles on general flag varieties over characteristic $0$ are considered. In particular, we generalize the Grauert-M$\ddot{\text{u}}$lich-Barth theorem to flag varieties. As a corollary, we show that any strongly uniform semistable bundle over the complete flag variety $F$ splits as a direct sum of special line bundles.