De Rham-Betti classes with coefficients
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911387792441344 |
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| author | Kreutz, Tobias Shen, Mingmin Vial, Charles |
| author_facet | Kreutz, Tobias Shen, Mingmin Vial, Charles |
| contents | Let $K$ and $L$ be algebraic extensions of the rational numbers inside the field of complex numbers. An $L$-de Rham-Betti class on a smooth projective variety $X$ over $K$ is a class in the Betti cohomology with $L$-coefficients of the analytification of $X$ that descends to a class in the algebraic de Rham cohomology of $X$ via the period comparison isomorphism. The period conjecture of Grothendieck implies that $L$-de Rham-Betti classes should be $L$-linear combinations of algebraic cycle classes. We prove that $L$-de Rham-Betti classes on products of elliptic curves are $L$-linear combinations of algebraic classes, provided $L$ contains at most one of the CM fields associated with the CM elliptic curves involved in the product. A key step consists in establishing a version of the analytic subgroup theorem with $L$-coefficients. Moreover, building on results of Deligne and André regarding the Kuga-Satake correspondence, we show that codimension-2 $L$-de Rham-Betti classes on hyper-Kähler varieties of known deformation type are $L$-linear combinations of motivated cycles, and we obtain a global de Rham-Betti Torelli theorem for K3 surfaces defined over the algebraic numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08618 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | De Rham-Betti classes with coefficients Kreutz, Tobias Shen, Mingmin Vial, Charles Algebraic Geometry 14C15, 14C25, 14C34, 14F40, 14G25, 14J42 Let $K$ and $L$ be algebraic extensions of the rational numbers inside the field of complex numbers. An $L$-de Rham-Betti class on a smooth projective variety $X$ over $K$ is a class in the Betti cohomology with $L$-coefficients of the analytification of $X$ that descends to a class in the algebraic de Rham cohomology of $X$ via the period comparison isomorphism. The period conjecture of Grothendieck implies that $L$-de Rham-Betti classes should be $L$-linear combinations of algebraic cycle classes. We prove that $L$-de Rham-Betti classes on products of elliptic curves are $L$-linear combinations of algebraic classes, provided $L$ contains at most one of the CM fields associated with the CM elliptic curves involved in the product. A key step consists in establishing a version of the analytic subgroup theorem with $L$-coefficients. Moreover, building on results of Deligne and André regarding the Kuga-Satake correspondence, we show that codimension-2 $L$-de Rham-Betti classes on hyper-Kähler varieties of known deformation type are $L$-linear combinations of motivated cycles, and we obtain a global de Rham-Betti Torelli theorem for K3 surfaces defined over the algebraic numbers. |
| title | De Rham-Betti classes with coefficients |
| topic | Algebraic Geometry 14C15, 14C25, 14C34, 14F40, 14G25, 14J42 |
| url | https://arxiv.org/abs/2206.08618 |