A note on the Gauss–Manin connection for abelian schemes

Abstract

We study differential forms on the universal vector extension $A^{♮}$ of an abelian scheme $A$ in characteristic zero, and derive a new construction of the $D$-group scheme structure on $A^{♮}$. This gives, in particular, a rather simple description of the Gauss–Manin connection on the de Rham cohomology of $A$ in terms of global algebraic differential forms on $A^{♮}$. The key ingredient is the computation of the coherent cohomology of $A^{♮}$, due to Coleman and Laumon.