Abelian covers and second fundamental form

TL;DR

This paper establishes conditions under which families of abelian covers of the projective line produce subvarieties in moduli spaces that are not totally geodesic, thus not Shimura, with results depending on the genus and group structure.

Contribution

It provides new criteria for non-totally geodesic subvarieties arising from abelian covers, extending to Prym varieties and relating genus bounds to group properties.

Findings

Families of abelian covers can produce non-Shimura subvarieties under certain conditions.

Genus bounds depend only on the abelian group structure.

Results extend to Prym varieties associated with involutions in the Galois group.

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).