Fourier-Mukai partners of K3 surfaces in positive characteristic

TL;DR

This paper extends classical results on Fourier-Mukai equivalences of K3 surfaces to positive characteristic, establishing a derived-category-based Torelli theorem and invariance properties of zeta functions and Hodge conjectures.

Contribution

It generalizes the Torelli theorem to positive characteristic using derived categories and proves derived invariance of the zeta function and the variational crystalline Hodge conjecture for K3 surfaces.

Findings

Fourier-Mukai equivalence in positive characteristic parallels complex case

Shioda-supersingular K3 surfaces are uniquely determined by their derived category

Zeta function of a K3 surface is a derived invariant

Abstract

We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (etale, crystalline, de Rham) and use these to prove: 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts); 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.