Infinitesimal derived Torelli theorem for K3 surfaces

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 $4$-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.