Complex ball quotients from manifolds of K3^[n]-type

We describe periods of irreducible holomorphic symplectic manifolds of K3[n]K3[n]-type with a non-symplectic automorphism of prime order p≥3p≥3. These turn out to lie on complex ball quotients and we are able to give a precise characterization of when the period map is bijective by introducing the notion of K(T)K(T)-generality.