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⊆G be a parabolic subgroup and λ:P⟶G be a strictly antidominant character. Let C be a projective smooth curve over k with function field K=k(C) and F be a principal G-bundle on C. Then F/P⟶C is a flag bundle and L_λ=F×_Pk_λ 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_Lλ:X(K‾)⟶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∈N¹(F/P) be the numerical class of a vertical fiber. We compute the augmented base loci B₊(L_λ−tf) for any t∈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.