We give some conditions on a family of abelian covers of ${ \mathbb P}^1$ of genus $g$ curves, that ensure that the family yields a subvariety of ${ \mathsf A}_g$ 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 ${ \mathsf A}_g$ which is not totally geodesic. We prove then analogous results for families of abelian covers of ${ \tilde C}_t \rightarrow { \mathbb P}^1 = { \tilde C}_t/{ \tilde G}$ with an abelian Galois group ${ \tilde G}$ of even order, proving that under some conditions, if $\sigma \in { \tilde G}$ is an involution, the family of Pryms associated with the covers ${ \tilde C}_t \rightarrow C_t= { \tilde C}_t/\langle \sigma \rangle$ yields a subvariety of ${ \mathsf A}_{ p}^{ \delta}$ which is not totally geodesic. As a consequence, we show that if ${ \tilde G} =({ \mathbb Z}/N{ \mathbb Z})^m$ with $N$ even, and $\sigma$ is an involution in ${ \tilde G}$, there exists an integer $M(N)$ which only depends on $N$ such that, if ${ \tilde g} = g({ \tilde C}_t) > M(N)$, then the subvariety of the Prym locus in ${ \mathsf A}^{ \delta}_{ p}$ induced by any such family is not totally geodesic (hence it is not Shimura).
Categories: math.AG