An Infinitesimal Noether-Lefschetz Theorem for Chow Groups

Let X be a smooth, complex projective variety, and Y be a very general, sufficiently ample hypersurface in X. A conjecture of M. V. Nori states that the natural restriction map CH^p (X)Q → CH^p (Y)Q is an isomorphism for all p < dim Y and an injection for p = dim Y . This is the generalized Noether-Lefschetz conjecture. We prove an infinitesimal version of this conjecture.