On Derived Equivalences of K3 Surfaces in Positive Characteristic

TL;DR

This paper explores derived equivalences of ordinary K3 surfaces in positive characteristic, establishing lifting properties, classifying Fourier-Mukai partners, and analyzing autoequivalence groups within crystalline cohomology.

Contribution

It demonstrates that automorphisms lift to characteristic zero, classifies Fourier-Mukai partners, and analyzes the autoequivalence group structure in positive characteristic.

Findings

Automorphisms of ordinary K3 surfaces lift to characteristic zero.

Fourier-Mukai partners correspond between the surface and its generic fiber.

The counting formula for Fourier-Mukai partners extends to positive characteristic.

Abstract

For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characteristic as well. We show that the image of the derived autoequivalence group of a K3 surface of finite height in the group of isometries of its crystalline cohomology has index at least two. Moreover, we provide an upper bound on the kernel of this natural cohomological descent map. Further, we give an extended remark in the appendix on the possibility of an F-crystal structure on the crystalline cohomology of a K3 surface over an algebraically closed field of positive characteristic and show that the naive F-crystal structure fails in being compatible with inner product.