Geometric local systems on very general curves and isomonodromy

Abstract

We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general n n -pointed curve of genus g g is at least 2 g + 1 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 deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.