Relative Gieseker's problem on $F$-divided bundles

TL;DR

This work addresses the relative Gieseker problem for $F$-divided bundles in positive characteristic, proving that a surjective étale map on the usual étale fundamental groups induces a faithfully flat map on the $F$-divided fundamental groups, and that isomorphisms lift in favorable cases. The authors develop a descent theory for $F$-divided bundles, including an analogue of Bhatt–Scholze descent for proper fibrations, and leverage representation schemes, Hrushovski's theorem, and stability properties to transfer fiberwise triviality to global descent. Building on and generalizing results by Sun–Zhang, Esnault–Mehta, and Biswas–Parameswaran–Kumar, the paper extends the relative Gieseker problem to normal projective varieties (with smoothness assumptions only where necessary). The techniques illuminate the structure of Tannakian categories in positive characteristic and yield new tools for understanding fundamental groups via $F$-divided bundles and their moduli.

Abstract

Let $f: X\to Y$ be a surjective morphism of normal projective varieties defined over an algebraically closed field of positive characteristic. We prove that if the induced map on étale fundamental groups is surjective then the corresponding map on $F$-divided fundamental groups is faithfully flat. We also prove an analogous result for isomorphisms. This generalizes and strengthens a recent result of X. Sun and L. Zhang \cite{Sun-Zhang2025}, which in turn generalized earlier results of H. Esnault and V. Mehta \cite{Esnault-Mehta2010} and I. Biswas, M. Kumar, and A. J. Parameswaran \cite{Biswas-Parameswaran-Kumar2025}. An important new ingredient in our proof is an analogue of B. Bhatt's and P. Scholze's descent theorem \cite[Theorem 1.3]{Bhatt-Scholze2017} for $F$-divided bundles.

Theorems & Definitions (14)

• Remark 1.6
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• Example 2.3
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• Definition 3.1
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