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

Over complex numbers, the Fourier-Mukai (FM) 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 characteristic, 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 of Hosono, Lian, Oguiso, and Yau to odd characteristic fields, solving a classical problem originally from Shioda. Furthermore, we establish the derived Torelli theorem for supersingular abelian varieties and apply it to characterize the quasiliftable birational models of supersingular generalized Kummer varieties.