De Rham-Betti Groups of Type IV Abelian Varieties
Zekun Ji

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 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 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…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Tensor decomposition and applications
