Infinitesimal Derived Torelli Theorem for K3 Surfaces (with an Appendix by Sukhendu Mehrotra)

Abstract

We prove that the first-order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier–Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive four-dimensional space. This generalizes the derived version of the Torelli theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier–Mukai functor.