Geometric local systems on very general curves and isomonodromy

TL;DR

This paper establishes a lower bound on the rank of certain geometric local systems on general curves, resolving conjectures and analyzing stability under isomonodromic deformation.

Contribution

It provides a new lower bound for the rank of non-isotrivial local systems of geometric origin on general curves, and applies this to resolve existing conjectures.

Findings

Minimum rank is at least 2√(g+1) for such local systems

Resolution of conjectures by Esnault-Kerz and Budur-Wang

Analysis of stability of flat vector bundles under isomonodromic deformation

Abstract

We show that the minimum rank of a non-isotrivial local system of geometric origin, on a suitably general $n$-pointed curve of genus $g$, is at least $2\sqrt{g+1}$. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformation, which additionally answers questions of Biswas, Heu, and Hurtubise.