Derived equivalences of K3 surfaces and orientation

TL;DR

This paper proves that Fourier--Mukai equivalences of K3 surfaces preserve the natural orientation of their cohomological Hodge structures, leading to a comprehensive understanding of autoequivalence actions akin to the Torelli theorem.

Contribution

It establishes that all Fourier--Mukai equivalences preserve the orientation of positive directions in cohomology, providing a complete description similar to the Torelli theorem.

Findings

Hodge isometries induced by equivalences preserve orientation

Complete characterization of autoequivalence actions on cohomology

Analogy with the classical Torelli theorem for K3 surfaces

Abstract

Every Fourier--Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms.