Geometric Height on Flag Varieties in Positive Characteristic

TL;DR

This paper computes the height filtration and successive minima of height functions on flag varieties over function fields in positive characteristic, linking geometric properties with arithmetic invariants.

Contribution

It introduces a method to determine height filtrations and minima for flag varieties in positive characteristic, extending height theory to new geometric contexts.

Findings

Computed height filtration for flag varieties in positive characteristic.

Determined successive minima of height functions on these varieties.

Linked geometric structures with arithmetic height invariants.

Abstract

Let $k$ be an algebraically closed field of characteristic $p\neq 0$. Let $G$ be a connected reductive group over $k$, $P \subseteq G$ be a parabolic subgroup and $\lambda: P \longrightarrow \mathbb G_m$ 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 and successive minima of the height function $h_{\mathcal{L}_\lambda}: X(\overline{K}) \longrightarrow \mathbb{R}$ over the flag variety $X=(F/P)_K$.