Complete flags on flat vector bundles in positive characteristic
Yohei Morita, Yasuhiro Wakabayashi
TL;DR
This work establishes a positive characteristic analogue of a classical genus-dependent fact about flat vector bundles by proving that a curve $X$ of genus $g$ over a field of characteristic $p>0$ satisfies $g\le 1$ if and only if every flat vector bundle on $X$ admits a complete flag, within a framework built from $\mathcal{D}^{(m)}$-modules and $p^{m+1}$-curvature. It develops a comprehensive theory connecting $\mathcal{D}^{(m)}$-bundles to Frobenius pull-backs, $F$-divided sheaves, and the stratified fundamental group, and then treats three geometric cases: rational, elliptic, and hyperbolic curves. For $\mathbb{P}^1$ and genus $1$ curves, the paper provides explicit classifications and constructions showing the existence of complete flags for all $\mathcal{D}^{(m)}$-bundles; for hyperbolic curves, it constructs explicit obstructions (via the $p$-Hitchin morphism) to the existence of complete flags and demonstrates non-existence in general. The results also yield a spectrum of examples including dormant oper-based irreducible bundles in characteristic $p$, highlighting a sharp dichotomy between low-genus and higher-genus behaviors in positive characteristic. Overall, the paper advances understanding of how flat connections behave in positive characteristic and clarifies when flag filtrations persist under $\mathcal{D}^{(m)}$-actions.
Abstract
Let $X$ be a connected, smooth, and projective curve of genus $g$ over an algebraically closed field of characteristic $p >0$. This paper investigates a characteristic-$p$ analogue of a well-known fact concerning flat vector bundles in characteristic $0$. That is to say, we prove that the inequality $g \leq 1$ holds if and only if any flat vector bundle on $X$ admits a complete flag. We also explore a generalization of this result in a broader setting.