Flat Vector Bundles on Very General Curves and Codimension of Non-Abelian Hodge Loci

Abstract

We bound the codimension of components of the nonabelian Hodge loci in the relative de Rham moduli space over $\shm_{g,n}$ in terms of the rank and level of a complex variation of Hodge structure. If the rank is $r$ and the level is $\ell$, then the codimension must be positive if $r$ and $\ell$ are small relative to $g$. The key input is a generalization of a bound on the rank of flat vector bundles by Landesman and Litt, which we apply to the isomonodromy foliation on the relative de Rham space. As an auxiliary result, we are able to bound the rank of the Lie algebra of the algebraic monodromy group of the isomonodromic deformation of a flat bundle to a nearby curve.