Abstract: The goal of this paper is two-folded. Firstly we show that height filtration agrees with arithmetic augmented base loci. This is conjectured and communicated to us by Chen. Secondly we compute the height filtration on flag varieties over function fields. It turns out that they are given by a certain Bruhat decomposition.
 

 We also have two by-products. The first one is concerning positivity. We get a complete discription of (for every $k$) the $k$-movable cones in the Neron-Severi group of flag bundles. The second one is concerning Zhang's inequality. By considering all the critical points (rather than only dimension junmps), we enhance Zhang's inequality to an equality.

This is a joint work in progress with Yangyu Fan and Binggang Qu