Splitting types of finite monodromy vector bundles

Given a finite degree $d$ cover of curves $f: X \to \mathbb P^1$, we study $f_* \mathscr O_X$, which is a rank $d$ vector bundle on $\mathbb P^1$, hence can be written as a direct sum of line bundles $f_* \mathscr O_X \simeq \oplus_{i=1}^d \mathscr O(a_i)$. Naively, one might expect that if the cover above is general, this vector bundle is balanced, meaning that the $a_i$'s are as close to each other as possible. While this is not quite true, we explain what can be said about these splitting types, by studying how they change as we deform the cover. This is based on joint work with Daniel Litt.

The ideas cropping up here were also instrumental in resolving conjectures of Esnault-Kerz and Budur-Wang regarding the density of geometric local systems in the moduli space of local systems.