Derived equivalences of K3 surfaces and orientation

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.

This version only contains the proof that every equivalence between the derived categories of two K3 surfaces induces an orientation preserving Hodge isometry. The more formal aspects of our proof contained in the first sections of arXiv:0710.1645v2 appear now in the independent paper arXiv:0809.3201. 29 pages. Final version to appear in Duke Math. J

References in corpus (7)

• Hochschild cohomology and Atiyah classes
• The relative Riemann-Roch theorem from Hochschild homology
• The Mukai pairing, I: a categorical approach
• Formality of the homotopy calculus algebra of Hochschild (co)chains
• Infinitesimal Derived Torelli Theorem for K3 surfaces
• Deformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes
• Formal deformations and their categorical general fibre

Cited by in corpus (24)

• MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations
• Hodge theory and derived categories of cubic fourfolds
• The K3 category of a cubic fourfold
• Stability conditions in families
• Fourier-Mukai transforms and the wall-crossing behavior for Bridgeland's stability conditions
• Derived automorphism groups of K3 surfaces of Picard rank 1
• Cusps of the Kähler moduli space and stability conditions on K3 surfaces
• Infinitesimal Derived Torelli Theorem for K3 surfaces
• Some moduli spaces of Bridgeland's stability conditions
• Deformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes
• Autoequivalences of twisted K3 surfaces
• Chow groups and derived categories of K3 surfaces
• Derived equivalences of hyperkähler varieties
• Automorphisms and autoequivalences of generic analytic K3 surfaces
• Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces
• Formal deformations and their categorical general fibre
• Mirror symmetry and K3 surfaces
• A variant of the Mukai pairing via deformation quantization
• Stable pairs on local K3 surfaces
• Fourier-Mukai transforms, mirror symmetry, and generalized K3 surfaces
• Stability conditions via spherical objects
• Relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano fibrations
• Versal dg deformation of Calabi--Yau manifolds
• Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces