Abelian covers and the second fundamental form

Abstract

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