A note on Fourier-Mukai partners of abelian varieties over positive characteristic fields

TL;DR

This paper extends the understanding of Fourier-Mukai partners of abelian varieties to positive characteristic fields, establishing new derived equivalence criteria and Torelli theorems in this setting.

Contribution

It generalizes known complex results to positive characteristic, including criteria for derived equivalence and Torelli theorems for supersingular abelian varieties.

Findings

Derived equivalence of abelian varieties relates to Kummer stack equivalence in odd characteristic.

Two abelian surfaces are derived equivalent iff their Kummer surfaces are isomorphic.

Established the derived Torelli theorem for supersingular abelian varieties.

Abstract

Over complex numbers, the Fourier-Mukai partners of abelian varieties are well-understood. A celebrated result is Orlov's derived Torelli theorem. In this note, we study the FM-partners of abelian varieties in positive characteristic. We notice that, in odd characteristics, two abelian varieties of odd dimension are derived equivalent if their associated Kummer stacks are derived equivalent, which is Krug and Sosna's result over complex numbers. For abelian surfaces in odd characteristic, we show that two abelian surfaces are derived equivalent if and only if their associated Kummer surfaces are isomorphic. This extends the result [doi:10.1215/s0012-7094-03-12036-0] to odd characteristic fields, which solved a classical problem originally from Shioda. Furthermore, we establish the derived Torelli theorem for supersingular abelian varieties and apply it to characterize the quasi-liftable birational models of supersingular generalized Kummer varieties.