Derived isogenies and isogenies for abelian surfaces
| Autor: | Li, Zhiyuan, Zou, Haitao |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 35 pages |
| Databáze: | arXiv |
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