Abelian covers and second fundamental form

2021/05/17 by Frediani, Paola
#14H10 #14H40 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.07947

Abstract

We give some conditions on a family of abelian covers of \mathbb P¹ 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 C_t → \mathbb P¹ = C_t/ G with an abelian Galois group G of even order, proving that under some conditions, if σ∈ G is an involution, the family of Pryms associated with the covers C_t → C_t= C_t/⟨ σ⟩ yields a subvariety of \mathsf A_p^δ which is not totally geodesic. As a consequence, we show that if G =(\mathbb Z/N\mathbb Z)^m with N even, and σ is an involution in G, there exists an integer M(N) which only depends on N such that, if g = g( C_t) > M(N), then the subvariety of the Prym locus in \mathsf A^δ_p induced by any such family is not totally geodesic (hence it is not Shimura).

Related

• On Shimura subvarieties generated by families of abelian covers of ℙ¹
• On Shimura subvarieties of the Prym locus
• Shimura varieties and abelian covers of the line
• Shimura varieties in the Torelli locus via Galois coverings of elliptic curves
• Decomposable abelian G-curves and special subvarieties