De Rham-Betti Groups of Type IV Abelian Varieties

TL;DR

This paper investigates the de Rham-Betti structures of type IV abelian varieties, establishing their coincidence with Mumford-Tate groups in specific cases and exploring their properties through Tannakian and algebraic group methods.

Contribution

It provides new results on the de Rham-Betti groups of simple CM abelian fourfolds and introduces a novel approach differing from Moonen-Zarhin's method.

Findings

De Rham-Betti groups of certain abelian fourfolds coincide with their Mumford-Tate groups.

Characterization of de Rham-Betti structures for families of geometric origin.

New methods for analyzing reductive subgroups of Mumford-Tate groups.

Abstract

We study the de Rham-Betti structure of a simple abelian variety of type IV. We will take a Tannakian point of view inspired by Andr\'e. The main results are that the de Rham-Betti groups of simple CM abelian fourfolds and simple abelian fourfolds over $\overline{\mathbb{Q}}$ whose endomorphism algebra is a degree 4 CM-field coincide with their Mumford-Tate groups. The method of proof involves a thorough investigation of the reductive subgroups of the Mumford-Tate groups of these abelian varieties, inspired by Kreutz-Shen-Vial. The condition that the underlying abelian variety is simple and the condition that the de Rham-Betti group is an algebraic group defined over $\mathbb{Q}$ are also used in a crucial way. The proof is different from the method of computing Mumford-Tate groups of these abelian varieties by Moonen-Zarhin. We will also study a family of de Rham-Betti structures, in the formalism proposed by Saito-Terasoma. For such families with geometric origin, we will characterize properties of fixed tensors of the de Rham-Betti group associated with such a family.