Arakelov geometry on flag varieties over function fields and related topics

Abstract

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 $\lambda: P \longrightarrow G$ 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 $\mathcal{L}_\lambda=F \times_P k_\lambda$ 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 $h_{\mathcal{L}_\lambda}: X(\overline{K}) \longrightarrow \mathbb{R}$ 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 $\mathrm{B}_+(\mathcal{L}_\lambda-tf)$ for any $t \in \mathbb{R}$, 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 $k$.