On unipotent radicals of motivic Galois groups

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.