Extension classes of a mixed motive and subgroups of the unipotent radical of the motivic Galois group
The fundamental difference between pure motives (roughly speaking coming from cohomology of smooth projective varieties) and mixed motives (coming from cohomology of arbitrary varieties) is existence of nontrivial extensions in the latter setting. The unipotent radical of the motivic Galois group of a mixed motive M is intimately related to the extension data in the category generated by M. A fairly recent result of Deligne describes this unipotent radical in terms of the extensions $0\longrightarrow W_pM\longrightarrow M\longrightarrow M/W_pM\longrightarrow 0$ collectively ($W_\cdot$ being the weight filtration). We shall recall this result and then discuss what information each of these extensions individually contains. We will end the talk with an application to motives whose unipotent radical of the motivic Galois group is as large as possible, and discuss some examples in the category of mixed Tate motives over $\mathbb{Q}$. This is a joint work with Kumar Murty.