Arakelov geometry on flag varieties over function fields and related topics

Let k be an algebraically closed field of characteristic zero. Let G be a connected reductive group over k, $P\subseteq G$ be a parabolic subgroup and $\lambda :P⟶G$ be a strictly antidominant character. Let C be a projective smooth curve over k with function field $K=k\left(C\right)$ and F be a principal G-bundle on C. Then $F/P⟶C$ is a flag bundle and $L_{\lambda }=F{\times }_P{k}_{\lambda }$ on $F/P$ is a relatively ample line bundle.

We compute the height filtration, successive minima, and the Boucksom-Chen concave transform of the height function $h_{L_{\lambda }}:X\left(\bar{K}\right)⟶R$ over the flag variety $X={(F/P)}_K$. An interesting application is that the height of X equals to a weighted average of successive minima, and one may view this as a refinement of Zhang's inequality of successive minima.

Let $f\in N^1(F/P)$ be the numerical class of a vertical fiber. We compute the augmented base loci $B_{+}(L_{\lambda }-tf)$ for any $t\in R$, and it turns out that they are almost the same as the height filtration. As a corollary, we compute the k-th movable cones of flag bundles over curves for all k.