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 $G{r}^{W}M$ 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 $G{r}^{W}M$ 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 $G{r}^{W}M$ itself may have a small Mumford-Tate group.