Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics

TL;DR

This paper investigates Fourier-Mukai equivalences for abelian varieties and K3 surfaces in positive and mixed characteristics, establishing their behavior under lifting and providing new insights into their classification.

Contribution

It proves that Fourier-Mukai partners of abelian varieties are abelian varieties in any characteristic and shows how equivalences lift from special fibers to canonical lifts.

Findings

Fourier-Mukai partners of abelian varieties are abelian varieties.

Equivalences on special fibers lift to canonical lifts.

Bijection between Fourier-Mukai partners of lifts and special fibers.

Abstract

We study Fourier-Mukai equivalences of (families of) abelian varieties and K3 surfaces in positive and mixed characteristics. We first prove in any characteristics that Fourier-Mukai partners of abelian varieties are again abelian varieties. We subsequently focus on the canonical lifts of ordinary abelian varieties and ordinary K3 surfaces. For such schemes, we show that Fourier-Mukai equivalences on the special fibers can be lifted to the canonical lifts. We also prove that the relative Fourier-Mukai partners of the canonical lifts are in bijection with the Fourier-Mukai partners of the special fibers. We conclude by demonstrating that the last result can be used to recover the ordinary case of a result originally proved by Honigs, Lombardi and Tirabassi.