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

TL;DR

This paper investigates the unipotent radical of the Mumford-Tate group for a broad class of mixed Hodge structures with fixed associated graded, revealing that generically this radical is as large as possible under certain conditions.

Contribution

It establishes that for a generic family of mixed Hodge structures, the unipotent radical of their Mumford-Tate groups attains maximal size, extending understanding of their algebraic structure.

Findings

Unipotent radical equals the maximal possible outside a countable union of proper Zariski closed sets.

The result applies when the associated graded Hodge structure is polarizable and satisfies specific conditions.

The approach uses Tannakian category theory to relate extension classes to the unipotent radical.

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.