On unipotent radicals of motivic Galois groups

TL;DR

This paper refines Deligne's characterization of the unipotent radical of motivic Galois groups in Tannakian categories, providing conditions for individual extensions and classifying certain mixed motives with maximal unipotent radicals.

Contribution

It offers a refinement of Deligne's result by giving conditions for individual extensions to be pushforwards of extensions by the unipotent radical, and classifies motives with maximal unipotent radicals.

Findings

Refined Deligne's characterization of unipotent radicals.

Provided sufficient conditions for individual extensions.

Classified motives with maximal unipotent radicals, including 3-dimensional mixed Tate motives.

Abstract

Let $\mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $\mathbf{1}$, and equipped with a filtration $W_\cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $\mathbf{T}$, and $\underline{\mathfrak{u}}(M)\subset W_{-1}\underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $\underline{\mathfrak{u}}(M)$ in terms of the extensions $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$: it states that $\underline{\mathfrak{u}}(M)$ is the smallest subobject of $W_{-1}\underline{Hom}(M,M)$ such that the sum of the aforementioned extensions, considered as extensions of $\mathbf{1}$ by $W_{-1}\underline{Hom}(M,M)$, is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In this article, we study each of the above-mentioned extensions individually in relation to $\underline{\mathfrak{u}}(M)$. Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension $0\longrightarrow W_pM \longrightarrow M \longrightarrow M/W_pM \longrightarrow 0$ is the pushforward of an extension of $\mathbf{1}$ by $\underline{\mathfrak{u}}(M)$. In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e. with $\underline{\mathfrak{u}}(M)= W_{-1}\underline{Hom}(M,M)$). Using Grothedieck's formalism of \textit{extensions panach\'{e}es} we prove a classification result for such motives. Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over $\mathbb{Q}$ with three weights and large unipotent radicals.