Geometric local systems on generic curves

Abstract: The talk will be based on parts of the preprint entitled
``Geometric local systems on very general curves and isomonodromy'' by
Aaron Landesman and Daniel Litt (arXiv:2022.00039).

A complex local system $L$ on a smooth projective curve $X$ over
$\mathbb{C}$ is said to be ``geometric'' if there exists a smooth
projective morphism $f:X \to U$, with $U$ a nonempty Zariski open subset
of $X$, and an integer $i$ such that $L|_U$ is a direct summand of $R^i
f_* \mathbb{C}_X$.
The main result that we will discuss says that if the rank of $L$ is small
compared to the genus of $X$, then the monodromy representation associated
to $L$ must have finite image; this leads to counterexamples to
conjectures of Budur--Wang and Esnault--Kerz.The proof uses methods from the
theory of variations of Hodge structure.