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

Abstract: The Mumford-Tate group $G(M)$ of a mixed Hodge structure $M$ is a subgroup of $GL(M)$ which satisfies the following property: any rational subspace of any tensor power of $M$ underlies a mixed Hodge substructure if and only if it is invariant under the natural action of $G(M)$. Assuming $M$ is graded-polarizable, the unipotent radical $U(M)$ of $G(M)$ is a subgroup of the unipotent radical $U_0(M)$ of the parabolic subgroup of $GL(M)$ associated to the weight filtration on $M$. Let us say $U(M)$ is large if it is equal to $U_0(M)$.

This talk is a report on a recent joint work with Kumar Murty, where we consider the set of all mixed Hodge structures on a given rational vector space, with a fixed weight filtration and a fixed polarizable associated graded Hodge structure. It is easy to see that this set is in a canonical bijection with the set of complex points of an affine complex variety $S$. The main result is that assuming some conditions on the (fixed) associated graded hold, outside a union of countably many proper Zariski closed subsets of $S$ the unipotent radical of the Mumford-Tate group is large.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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