Towards Connectivity for Codimension 2 Cycles: Infinitesimal Deformations

Let X be a smooth projective variety over an algebraically closed field k ⊂ C of characteristic zero, and Y ⊂ X a smooth complete intersection. The Weak Lefschetz theorem states that the natural restriction map H^i (X(C), Q) → H^i (Y(C), Q) on singular cohomology is an isomorphism for all i < dim(Y). The Bloch-Beilinson conjectures on the existence of certain filtrations on Chow groups combined with standard conjectures in the theory of motives imply that a similar result should be true for Chow groups, and, more generally, for motivic cohomology. In this note, we prove a consequence of the Motivic Weak Lefschetz conjecture (see Conjecture 1.2) for codimension 2 cycles.