Given a smooth family of varieties over a characteristic zero field, the variational Hodge conjecture informally predicts, in term of cohomological invariants, when vector bundles or algebraic cycles on a fibre can be extended to the entire family. An infinitesimal, or deformational, version of this conjecture was first considered by Green and Griffiths, and more recently by Bloch, Esnault, and Kerz. I will present a proof of such an infinitesimal Hodge conjecture for varieties over number fields by using a new “pro Hochschild–Kostant–Rosenberg theorem” in Hochschild and cyclic homology.