Derived isogenies and isogenies for abelian surfaces

Li, Zhiyuan; Zou, Haitao

doi:10.2140/ant.2026.20.1185

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Derived isogenies and isogenies for abelian surfaces

Zhiyuan Li and Haitao Zou

Vol. 20 (2026), No. 6, 1185–1234

DOI: 10.2140/ant.2026.20.1185

Abstract

We study twisted Fourier–Mukai partners of abelian surfaces. Following Huybrechts (2019), we introduce twisted derived equivalences (also called derived isogenies) between abelian surfaces. We show that there is a twisted derived Torelli theorem for abelian surfaces over algebraically closed fields with characteristic $\neq 2, 3$ .

Our approach involves extending to rational Hodge structures, $ℓ$ -adic Tate modules and $F$ -crystals a trick introduced by Shioda in the context of integral Hodge structures. Using this trick, we can confirm the Tate conjecture in a special case. Then we make use of Tate’s isogeny theorem to give a characterization of derived isogenies between abelian surfaces via so-called principal isogenies. As a consequence, we show the two abelian surfaces are principally isogenous if and only if they are derived isogenous.

Keywords

abelian surface, isogeny, derived category, twisted sheaf, Torelli theorem

Mathematical Subject Classification

Primary: 14F08, 14K02

Secondary: 14G17

Milestones

Received: 13 November 2023

Revised: 4 May 2025

Accepted: 27 June 2025

Published: 26 May 2026

Authors

Zhiyuan Li
Shanghai Center for Mathematical Sciences Fudan University Shanghai China

Haitao Zou
Universität Bielefeld Bielefeld Germany

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Vol. 20, No. 6, 2026