# Vector Bundles on Flag varieties

## TL;DR

This paper investigates the structure and splitting properties of vector bundles on flag varieties over algebraically closed fields, providing classification results in positive characteristic and generalizations of classical theorems in characteristic zero.

## Contribution

It classifies uniform vector bundles on Grassmannians in positive characteristic and establishes splitting criteria for bundles on flag varieties in characteristic zero.

## Key findings

- Uniform bundles on Grassmannians are either sums of line bundles or pull-backs of universal bundles via Frobenius.
- A structure theorem for uniform bundles on flag varieties in characteristic zero is proved.
- A generalization of the Grauert-Mülich-Barth theorem shows strongly uniform semistable bundles split as sums of special line bundles.

## Abstract

We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $r\le d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{\vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\cdots,d_s)$ 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 $F(d_1,\cdots,d_s)$. Furthermore, we generalize the Grauert-M$\ddot{\text{u}}$lich-Barth theorem to flag varieties. As a corollary, we show that any strongly uniform $i$-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.