The mixed Hodge structure on the fundamental group of a complement of hyperplanes

The complement of an arrangement of hyperplanes is a good example of a mixed Hodge structure on the fundamental group of an algebraic variety. We compute its isomorphism class using iterated integrals and then show that the cross ratio of a arrangement is a combinatorial and projective invariant. So cross ratio equivalent arrangements are isomorphic. Moreover, an isomorphism of polarized mixed Hodge structures on the fundamental group induces cross ratio equivalent arrangements.