Abelian covers and second fundamental form

Abstract: 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).