Derived isogenies and isogenies for abelian surfaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911723845320704 |
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| author | Li, Zhiyuan Zou, Haitao |
| author_facet | Li, Zhiyuan Zou, Haitao |
| contents | In this paper, we study the twisted Fourier-Mukai partners of abelian surfaces. Following the work of Huybrechts [doi:10.4171/CMH/465], we introduce the twisted derived equivalence between abelian surfaces. We show that there is a twisted derived Torelli theorem for abelian surfaces over algebraically closed fields with characteristic $\neq 2,3$. Over complex numbers, the derived isogenies correspond to rational Hodge isometries between the second cohomology groups, which is in analogy to the work of Huybrechts and Fu-Vial on K3 surfaces. Their proof relies on the global Torelli theorem over $\mathbb{C}$, which is missing in positive characteristics. To overcome this issue, we firstly extend a trick given by Shioda on integral Hodge structures, to rational Hodge structures, $\ell$-adic Tate modules and $F$-crystals. Then we make use of Tate's isogeny theorem to give a characterization of the twisted derived equivalences between abelian surfaces via isogenies. As a consequence, we show the two abelian surfaces are principally isogenous if and only if they are derived isogenous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_08710 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Derived isogenies and isogenies for abelian surfaces Li, Zhiyuan Zou, Haitao Algebraic Geometry 14F08, 14K02, 14G17 In this paper, we study the twisted Fourier-Mukai partners of abelian surfaces. Following the work of Huybrechts [doi:10.4171/CMH/465], we introduce the twisted derived equivalence between abelian surfaces. We show that there is a twisted derived Torelli theorem for abelian surfaces over algebraically closed fields with characteristic $\neq 2,3$. Over complex numbers, the derived isogenies correspond to rational Hodge isometries between the second cohomology groups, which is in analogy to the work of Huybrechts and Fu-Vial on K3 surfaces. Their proof relies on the global Torelli theorem over $\mathbb{C}$, which is missing in positive characteristics. To overcome this issue, we firstly extend a trick given by Shioda on integral Hodge structures, to rational Hodge structures, $\ell$-adic Tate modules and $F$-crystals. Then we make use of Tate's isogeny theorem to give a characterization of the twisted derived equivalences between abelian surfaces via isogenies. As a consequence, we show the two abelian surfaces are principally isogenous if and only if they are derived isogenous. |
| title | Derived isogenies and isogenies for abelian surfaces |
| topic | Algebraic Geometry 14F08, 14K02, 14G17 |
| url | https://arxiv.org/abs/2108.08710 |