$D$-modules and finite monodromy

TL;DR

This paper explores an analogue of the Grothendieck p-curvature conjecture, demonstrating its validity under certain conditions and establishing isotriviality results, thereby advancing understanding of D-modules with finite monodromy.

Contribution

It proves the weaker conjecture for modules with finite Nori vector bundles and those with Hodge structure, and establishes isotriviality under specific conditions, resolving a conjecture of Matzat-van der Put.

Findings

Weaker conjecture holds for Nori finite vector bundles.

The conjecture is valid for modules underlying a Z-variation of Hodge structure.

Isotriviality is proven under coprimality conditions, resolving a known conjecture.

Abstract

We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a $\mathbb{Z}$-variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod $p$ Tannakian fundmental groups, which in particular resolves in the projective case a conjecture of Matzat-van der Put. v2: the well known 4.2 has been added to make the note self-contained.