De Rham-Betti groups of type IV Abelian varieties

In this thesis, 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 André. The main results are that the de Rham-Betti groups of simple CM abelian fourfolds and simple abelian fourfolds over the field of algebraic numbers 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 the work of Kreutz-Shen-Vial. Moreover, as a preparation for the proof of this result, we show that the de Rham-Betti group of a simple type IV abelian variety contains the group of homotheties. The condition that the underlying abelian variety is simple and the condition that the de Rham-Betti group is an algebraic group defined over rational numbers 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 the properties of fixed tensors of the de Rham-Betti group associated with such a family.