De Rham-Betti Groups of Type IV Abelian Varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917055232475136 |
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| author | Ji, Zekun |
| author_facet | Ji, Zekun |
| contents | 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 $\overline{\mathbb{Q}}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | De Rham-Betti Groups of Type IV Abelian Varieties Ji, Zekun Algebraic Geometry 14C15, 14C30 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 $\overline{\mathbb{Q}}$ 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. |
| title | De Rham-Betti Groups of Type IV Abelian Varieties |
| topic | Algebraic Geometry 14C15, 14C30 |
| url | https://arxiv.org/abs/2511.01072 |