A note on the Gauss-Manin connection for abelian schemes

TL;DR

This paper provides a new algebraic construction of the Gauss-Manin connection for abelian schemes by analyzing differential forms on their universal vector extensions, simplifying understanding of their de Rham cohomology.

Contribution

It introduces a novel construction of the $D$-group scheme structure on the universal vector extension, offering a simpler description of the Gauss-Manin connection in characteristic zero.

Findings

Explicit description of the Gauss-Manin connection via algebraic differential forms

Computation of the coherent cohomology of the universal vector extension

Simplification of the understanding of de Rham cohomology for abelian schemes

Abstract

We study differential forms on the universal vector extension $A^\natural$ of an abelian scheme $A$ in characteristic zero, and derive a new construction of the $D$-group scheme structure on $A^\natural$. 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 $A^\natural$. The key ingredient is the computation of the coherent cohomology of $A^{\natural}$, due to Coleman and Laumon.