Abelian covers and the second fundamental form

We give some conditions on a family of abelian covers of P¹ P 1 of genus g curves, that ensure that the family yields a subvariety of Ag 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 $$g >M$$ g > M , then the family yields a subvariety of Ag A g which is not totally geodesic. We prove then analogous results for families of abelian covers of C̃ₜ → P¹ = C̃ₜ/G̃ C ~ t → P 1 = C ~ t / G ~ with an abelian Galois group G̃ G ~ of even order, proving that under some conditions, if σ ∈ G̃ σ ∈ G ~ is an involution, the family of Pryms associated with the covers C̃ₜ → Cₜ= C̃ₜ/ σ C ~ t → C t = C ~ t / ⟨ σ ⟩ yields a subvariety of Aₚδ A p δ which is not totally geodesic. As a consequence, we show that if G̃=( Z/N Z)ᵐ G ~ = ( Z / N Z ) m with N even, and σ σ is an involution in G̃ G ~ , there exists an integer M ( N ) which only depends on N such that, if g̃= g(C̃ₜ) > M(N) g ~ = g ( C ~ t ) > M ( N ) , then the subvariety of the Prym locus in Aδₚ A p δ induced by any such family is not totally geodesic (hence it is not Shimura).