Slope filtrations of log $p$-divisible groups

TL;DR

This work extends the Oort–Zink slope filtration phenomenon from ordinary $p$-divisible groups to log $p$-divisible groups. It introduces the notion of completely slope divisible log $p$-divisible groups and shows that, on a normal fs log scheme $S$ over $ ext{F}_p$ with constant Newton polygon, any log $p$-divisible group $G$ is isogenous to a csd logarithmic counterpart $H$. A key advance is the extension theorem over log regular bases: if a csd isogeny is defined on a dense open subset, it extends to the whole base, yielding global slope filtrations for log $p$-divisible groups. The results rely on the Kummer log flat topology, descent properties for log finite group schemes and log vector bundles, and desingularization techniques, with implications for the structure of toroidal compactifications in characteristic $p$. Overall, the paper provides a robust framework to study degenerations of $p$-divisible groups in log-geometry and their filtrations, enabling new insights into moduli spaces in positive characteristic.

Abstract

Oort-Zink proved that a $p$-divisible group over a normal base in characteristic $p$ with constant Newton polygon is isogenous to a $p$-divisible group admitting a slope filtration. In this paper, we generalize this result to log $p$-divisible groups.

Theorems & Definitions (104)

• Theorem 1.1: see Corollary \ref{['first main thm']}
• Theorem 1.2: see Theorem \ref{['ext of isog over log regular base']}
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3: Log vector bundles
• Lemma 2.4
• proof
• Proposition 2.5