The unipotent radical of the Mumford-Tate group of a very general mixed Hodge structure with a fixed associated graded

Abstract

The family of all mixed Hodge structures on a given rational vector space $M_\mathbb{Q}$ with a fixed weight filtration $W_\cdot$ and a fixed associated graded Hodge structure $Gr^WM$ is naturally in a one to one correspondence with a complex affine space. We study the unipotent radical of the very general Mumford-Tate group of the family. We do this by using general Tannakian results which relate the unipotent radical of the fundamental group of an object in a filtered Tannakian category to the extension classes of the object coming from the filtration. Our main result shows that if $Gr^WM$ is polarizable and satisfies some conditions, then outside a union of countably many proper Zariski closed subsets of the parametrizing affine space, the unipotent radical of the Mumford-Tate group of the objects in the family is equal to the unipotent radical of the parabolic subgroup of $GL(M_\mathbb{Q})$ associated to the weight filtration on $M_\mathbb{Q}$ (in other words, outside a union of countably many proper Zariski closed sets the unipotent radical of the Mumford-Tate group is as large as one may hope for it to be). Note that here $Gr^WM$ itself may have a small Mumford-Tate group.