Abelian covers and second fundamental form

We give some conditions on a family of abelian covers of P_1 of genus g curves, that ensure that the family yields a subvariety of A_g which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if g > M, then the family yields a subvariety of A_g which is not totally geodesic. We prove then analogous results for families of abelian covers of ˜C_t → P1 = ˜C_t / ˜G with an abelian Galois group ˜G of even order, proving that under some conditions, if σ ∈ ˜G is an involution, the family of Pryms associated with the covers ˜C_ t → C_t = ˜C_t / <σ>  yields a subvariety of A^δ_p which is not totally geodesic. As a consequence, we show that if ˜G = (Z/NZ)^m with N even, and σ is an involution in ˜G , there exists an integer M(N) which only depends on N such that, if ˜ g = g( ˜C_t) > M(N), then the subvariety of the Prym locus in A^δ_p induced by any such family is not totally geodesic (hence it is not Shimura).