Big monodromy for higher Prym representations

Let g 0 ! g be a cover of an orientable surface of genus g by an orientable surface of genus g 0 , branched at n points, with Galois group H .Such a cover induces a virtual action of the mapping class group Mod g;nC1 of a genus g surface with n C 1 marked points on H 1 . g 0 ; C/.When g is large in terms of the group H , we calculate precisely the connected monodromy group of this action.The methods are Hodge-theoretic and rely on a "generic Torelli theorem with coefficients".