Differential Characters and $D$-Group Schemes

TL;DR

This paper extends the theory of differential characters and $D$-group schemes from abelian schemes to more general smooth connected commutative group schemes over a field of characteristic zero, showing their kernels form finite dimensional $D$-group schemes.

Contribution

It generalizes the structure of kernel group schemes of differential characters to all smooth connected commutative group schemes, not just abelian schemes, using jet space analysis.

Findings

Kernel group schemes are finite dimensional $D$-group schemes.

Kernel groups are vectorial extensions of the original group schemes.

Classification of differential characters as modules over $K ext{ extbackslash}{\partial}$.

Abstract

Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\mathrm{Spec}~ K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$. Our proof relies entirely on understanding the structure of jet spaces. Our method also allows us togive a classification of the module of differential characters $\mathbf{X}_\infty(G)$ in terms of primitive characters as a $K\{\partial\}$-module.