$F$-divided bundles on normal $F$-finite schemes
Adrian Langer, Lei Zhang
TL;DR
The paper develops the theory of $F$-divided bundles on irreducible Noetherian normal $F$-finite $\mathbb{F}_p$-schemes, showing that their Tannakian category is governed by the generic point through the $F$-divided fundamental gerbe $oldsymbol ext{Π}^{F ext{-div}}$. It proves that for an open subset $U$ of a unibranch normal variety, the induced map on stratified fundamental groups $oldsymbol ext{π}^{ ext{strat}}$ is faithfully flat, connecting to D-modules in the regular case and generalizing known characteristic-$p$ results. The authors establish the $h$-descent of $F$-divided bundles, analyze local behavior and descent of subobjects, and develop a moduli-theoretic framework to extend Esnault–Mehta’s results to normal projective varieties. These tools yield a positive answer to Gieseker’s conjecture for normal proper varieties: if the maximal étale quotient of the Nori fundamental group vanishes, then there are no nontrivial $F$-divided bundles. Overall, the work connects Tannakian formalism, descent theory, and moduli geometry to advance the understanding of $F$-divided structures in positive characteristic. $
Abstract
In this paper we study $F$-divided bundles on irreducible Noetherian normal $F$-finite $\mathbb{F}_p$-schemes and we show that their Tannakian category is governed by the behaviour at the generic point. In particular, if $U\subset X$ is an open subset of a normal variety defined over an algebraically closed field then the corresponding homomorphism of $F$-divided fundamental groups is faithfully flat. This is analogous to a known fact about the topological fundamental group of an open subset of a normal complex analytic variety. We use this result to show that simply connected, proper, normal varieties in positive characteristic admit no nontrivial $F$-divided bundles. This generalizes an earlier result of H. Esnault and V. Mehta concerning smooth projective varieties, and settles Gieseker's conjecture in a more general setting.