Decomposable abelian $G$-curves and special subvarieties

TL;DR

The paper investigates when families of abelian Galois covers of the line yield special subvarieties in $\mathcal{A}_g$ and proves that, in the case of totally decomposable Jacobians, the associated subvariety $\mathsf{Z}$ is special if and only if the star numerical condition $\dim \mathsf{Z}=\dim H^0(C,2K_C)^G=\dim (S^2H^0(C,K_C))^G$ holds. It develops a Hodge-theoretic framework to compare the smallest special subvariety $S_f$ containing the family with the ambient PEL subvariety $S(G)$ via the generic Hodge group $\operatorname{Hg}$ and monodromy $\operatorname{Mon}^0$, showing that in the totally decomposable setting $S_f=S(G)$ when the Jacobians decompose as a product of elliptic curves. The main result extends previous one-dimensional necessity results to higher-dimensional totally decomposable abelian $G$-covers and provides a concrete obstruction-avoidance criterion through the graph embedding of symmetric spaces. Together, these findings contribute to the Coleman-Oort program by clarifying the role of total decomposability in the existence of special subvarieties arising from families of abelian covers.

Abstract

We consider families of abelian Galois coverings of the line. When the Jacobian of the general element is totally decomposable, i.e., is isogenous to a product of elliptic curves, we prove that they yield special subvarieties of $\A_g$ if and only if a numerical condition holds, which in the general case is only known to be sufficient.

Theorems & Definitions (31)

• Theorem A
• Proposition 1.6
• Theorem 1.7
• Proposition 3.2
• Proposition 3.6
• Theorem 3.7
• proof
• Example 1
• Example 2
• Example 3