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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.10282 |
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| _version_ | 1866914650984022016 |
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| author | Okada, Masaki Watari, Taizan |
| author_facet | Okada, Masaki Watari, Taizan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_10282 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on varieties of weak CM-type Okada, Masaki Watari, Taizan Algebraic Geometry High Energy Physics - Theory 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. |
| title | A note on varieties of weak CM-type |
| topic | Algebraic Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2306.10282 |