On the Kodaira-Spencer map of abelian schemes

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}.