Geometric Height on Flag Varieties in Positive Characteristic

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 $λ: P \longrightarrow \mathbb G_m$ 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 $\mathcal{L}_λ=F \times_P k_λ$ on $F/P$ is a relatively ample line bundle. We compute the height filtration and successive minima of the height function $h_{\mathcal{L}_λ}: X(\overline{K}) \longrightarrow \mathbb{R}$ over the flag variety $X=(F/P)_K$.

Theorems & Definitions (31)

• Theorem 1.1: Theorem 2.1, fan2024arakelovgeometryflagvarieties
• Definition 1.2: Definition \ref{['strongly']}
• Theorem 1.3: Theorem \ref{['charp']}
• Theorem 1.4
• Theorem 1.5: Theorem \ref{['twist']}
• Remark 1.6
• Proposition 1.8: Ramanan, Ramanathan, Ramanan1985ProjectiveNO
• Theorem 1.9: Baby-version
• Definition 2.1
• Remark 2.2