On the classification of rank-two representations of quasiprojective fundamental groups

Suppose that X is a smooth quasiprojective variety over ℂ and ρ : π 1 ( X , x )→ SL (2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X → Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.