Derived isogenies and isogenies for abelian surfaces

TL;DR

This paper explores the relationship between derived equivalences and isogenies of abelian surfaces, extending known results to positive characteristic fields using advanced algebraic techniques.

Contribution

It introduces the twisted derived Torelli theorem for abelian surfaces and characterizes derived isogenies via isogenies, extending results to positive characteristic fields.

Findings

Derived isogenies correspond to rational Hodge isometries over complex numbers.

Twisted derived equivalences are characterized by isogenies between abelian surfaces.

Principally isogenous abelian surfaces are also derived isogenous.

Abstract

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.