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De Rham-Betti Groups of Type IV Abelian Varieties

Zekun Ji

TL;DR

The paper develops a comprehensive Tannakian framework for de Rham-Betti structures on simple type IV abelian varieties, proving that for simple CM abelian fourfolds (and certain CM-endowed ones over $ar{bQ}$) the dRB group on $H^1_{ m dRB}(A,bQ)$ coincides with the Mumford–Tate group. It achieves this via a careful reductive-subgroup analysis of $ m MT(A)$, the use of CM-endomorphism data, and a detailed study of Weil structures and subtori of the CM torus $ m U_E$, supplemented by the weak Grothendieck period conjecture inputs. The work also develops the theory of dRB systems in the Saito–Terasoma setting, establishes Deligne-type Principle B for such families, and analyzes the centers of dRB and MT groups, including a classification of one-dimensional dRB objects. Collectively, these results bridge de Rham-Betti theory with Hodge theory in the CM/type IV context and illuminate how fixed tensors control algebraic relations among periods. The methods yield a framework to approach open questions for anti-Weil type fourfolds and for generic CM types, sharpening the understanding of period relations from a Tannakian perspective.

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 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.

De Rham-Betti Groups of Type IV Abelian Varieties

TL;DR

The paper develops a comprehensive Tannakian framework for de Rham-Betti structures on simple type IV abelian varieties, proving that for simple CM abelian fourfolds (and certain CM-endowed ones over ) the dRB group on coincides with the Mumford–Tate group. It achieves this via a careful reductive-subgroup analysis of , the use of CM-endomorphism data, and a detailed study of Weil structures and subtori of the CM torus , supplemented by the weak Grothendieck period conjecture inputs. The work also develops the theory of dRB systems in the Saito–Terasoma setting, establishes Deligne-type Principle B for such families, and analyzes the centers of dRB and MT groups, including a classification of one-dimensional dRB objects. Collectively, these results bridge de Rham-Betti theory with Hodge theory in the CM/type IV context and illuminate how fixed tensors control algebraic relations among periods. The methods yield a framework to approach open questions for anti-Weil type fourfolds and for generic CM types, sharpening the understanding of period relations from a Tannakian perspective.

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 André. 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 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.

Paper Structure

This paper contains 32 sections, 123 theorems, 419 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. De Rham-Betti Structures and Groups
  4. Mumford-Tate Groups
  5. Algebraic Tori
  6. Simple Abelian Varieties with Complex Multiplication
  7. Weil Structures
  8. Rosati Involution
  9. De Rham-Betti Groups of Simple CM Abelian Fourfolds
  10. Existence of Homotheties inside De Rham-Betti Groups of CM-Abelian Varieties
  11. A Non-De Rham-Betti Group Criterion for Algebraic Tori
  12. Classification of One and Two-Dimensional Subtori of UE
  13. Non-Simplicity of Certain CM Abelian Fourfolds
  14. Proof of Theorem \ref{['mainthmcm']}
  15. Centre of De Rham-Betti Groups of Simple Type IV Abelian Varieties
  16. ...and 17 more sections

Key Result

Theorem 1.4

Suppose $A$ is a simple CM abelian fourfold. Then the de Rham-Betti group associated with the following dRB structure is equal to the Mumford-Tate group of $A$.

Theorems & Definitions (294)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4: =Theorem \ref{['mainthmcm']}
  • Theorem 1.5: =Proposition \ref{['weiltype']} and Theorem \ref{['deg4theorem']}
  • Proposition 1.6: =Proposition \ref{['gm in CM gdrb']} and Proposition \ref{['gmintype4']}
  • Corollary 1.7
  • Corollary 1.8: =Corollary \ref{['onecaseofconj']}
  • Corollary 1.8: =Corollary \ref{['onecaseofconj']}
  • Conjecture 1.9: andre2004introduction, Conjecture 7.5.1.1
  • Conjecture 1.10: Hodge Conjecture
  • ...and 284 more