Twisted Derived Equivalences Between Abelian Varieties

TL;DR

This paper proves that the isogeny class of the Picard variety remains invariant under twisted derived equivalences and characterizes when two abelian varieties are twisted derived equivalent.

Contribution

It generalizes previous results to show the invariance of the Picard variety's isogeny class and provides a criterion for twisted derived equivalence of abelian varieties.

Findings

Twisted Fourier-Mukai partners of abelian varieties are themselves abelian varieties.

The isogeny class of the Picard variety is a twisted derived invariant.

A necessary and sufficient isogeny-based condition for twisted derived equivalence is established.

Abstract

We generalize a result of Popa-Schnell and show that the isogeny class of the Picard variety is twisted derived invariant. Using this, we prove that any twisted Fourier-Mukai partner of an abelian variety is an abelian variety. We then provide a necessary and sufficient isogeny-based condition for two abelian varieties to be twisted derived equivalent.