Big monodromy for higher Prym-type variations
Prove that for a Galois H-cover \varphi: \Sigma_{g'}\to\Sigma_g with g≥3, the identity component of the Zariski-closure of the monodromy of R^1\pi_*\mathbb{C} equals the derived subgroup of the centralizer of H in \operatorname{Sp}_{2g'}(\mathbb{C}), and that the monodromy is an arithmetic subgroup of its Zariski-closure.
References
Conjecture Let $H$ be a finite group, $g\geq 3$, and $\varphi: \Sigma_{g'}\to \Sigma_g$ a Galois $H$-cover. With notation as in diagram (6.1): (1) the identity component of the Zariski closure of the monodromy group of the local system $R1\pi_*\mathbb{C}$ is the derived subgroup of the centralizer of $H$ in $\on{Sp}{2g'}(\mathbb{C})$, and (2) the monodromy of $R1\pi*\mathbb{C}$ is an arithmetic subgroup of its Zariski-closure.