2024/03/11 by Fan, Yangyu, Luo, Wenbin, Qu, Binggang
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2403.06808
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 \longrightarrow G be a strictly anti-dominant 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 \longrightarrow C is a flag bundle and Lλ=F ×P kλ 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 hLλ: X(K) \longrightarrow 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+(Lλ-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.