Tannakian fundamental groups of blended extensions

TL;DR

This paper investigates the unipotent radical of the tannakian fundamental group of blended extensions in semisimple objects, with applications to motivic Galois groups and the Hodge-Nori conjecture for 1-motives.

Contribution

It provides new insights into the structure of unipotent radicals in blended extensions within tannakian categories, applicable to mixed motives and 1-motives.

Findings

Unipotent radicals of motivic Galois groups of mixed motives with three weights are characterized.

The unipotent part of the Hodge-Nori conjecture for 1-motives is proven in general tannakian categories.

Results apply under mild hypotheses to a broad class of motives.

Abstract

Let $A_1, A_2,A_3$ be semisimple objects in a neutral tannakian category over a field of characteristic zero. Let $L$ be an extension of $A_2$ by $A_1$, and $N$ an extension of $A_3$ by $A_2$. Let $M$ be a blended extension (extension panach\'ee) of $N$ by $L$. Under very mild and natural hypotheses, we study the unipotent radical of the tannakian fundamental group of $M$. Examples where our results apply include the unipotent radicals of motivic Galois groups of any mixed motive with three weights. As an application, we give a proof of the unipotent part of the Hodge-Nori conjecture for 1-motives (which is now a theorem of Andr\'e in the setting of Nori motives) in the setting of any tannakian category of motives where the group $Ext^1(1,\mathbb{Q}(1))$ is as expected.