On the Kodaira-Spencer map of abelian schemes

TL;DR

This paper investigates the properties of the Kodaira-Spencer map for abelian schemes over complex varieties, comparing ranks in different contexts and analyzing implications for abelian integrals in the modular setting.

Contribution

It establishes invariance of certain ranks of the Kodaira-Spencer map under passage to the modular case and analyzes their implications for abelian integrals and monodromy.

Findings

Ranks of the D-module quotient and Kodaira-Spencer map are invariant under modular passage.

Derivative of a non-zero abelian integral of the first kind is never of the first kind in the modular case.

Results apply to abelian pencils with Zariski-dense monodromy in symplectic groups.

Abstract

Let $A$ be an abelian scheme over a smooth affine complex variety $S$, $\varOmega_A$ the $\sO_S$-module of $1$-forms of the first kind on $A$, $\sD_S\varOmega_A$ the $\sD_S$-module spanned by $\varOmega_A$ in the first algebraic De Rham cohomology module, and $\theta_\partial: \varOmega_A \to \sD_S\varOmega_A/\varOmega_A$ the Kodaira-Spencer map attached to a tangent vector field $\partial$ on $S$. We compare the rank of $\sD_S\varOmega_A/\varOmega_A$ to the maximal rank of $\theta_\partial$ when $\partial$ varies: we show that both ranks do not change when one passes to the "modular case", \ie when one replaces $S$ by the smallest weakly special subvariety of $\sA_g$ containing the image of $S$ (assuming, as one may up to isogeny, that $A/S$ is principally polarized), we then analyse the "modular case" and deduce, for instance, that {\it for any abelian pencil of relative dimension $g$ with Zariski-dense monodromy in $Sp_{2g}$}, {\it the derivative with respect to a parameter of a non zero abelian integral of the first kind is never of the first kind}.