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A note on varieties of weak CM-type

Masaki Okada, Taizan Watari

Abstract

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of $H^n(X;Q)$ is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire $H^\ast(X;Q)$ is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3 $\times T^2$. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.

A note on varieties of weak CM-type

Abstract

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3 . It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.
Paper Structure (15 sections, 9 theorems, 50 equations)

This paper contains 15 sections, 9 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Abelian Varieties of Weak CM-type
  3. The Reflex Degree 2
  4. The Reflex Degree 4
  5. The Case (A)
  6. The Case (B, C)
  7. K3$\times T^2$ of Weak CM-type
  8. Open Problems
  9. Abelian Varieties with a Higher Reflex Degree
  10. A Few Calabi--Yau Threefolds of Weak CM-type
  11. Various Definitions for CM-type
  12. Geometric Definition of CM-type Abelian Varieties
  13. CM-type Hodge Structures Defined by the Endomorphism Algebras
  14. CM-type Hodge Structures Defined by the Hodge Groups / Mumford--Tate Groups
  15. Complex Multiplications on Intermediate Jacobians

Key Result

Theorem 1.4

Let $A$ be an abelian variety of complex dimension $n$ that is of weak CM-type. It is then also of strong CM-type, at least if the level-$n$ subspace is of dimension $2n' = 2$ or $4$.

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Remark 3.1
  • ...and 42 more