The Weak Lefschetz theorem for Chow groups: Infinitesimal lifting (R)

Let X be a smooth projective variety over an algebraically closed field k of characteristic zero and Y ⊂ X a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map CH^p (X)Q → CH^p (Y)Q is an isomorphism for all p < dim(Y)/2. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous K-cohomology group on the formal completion of X along Y . This splits the conjecture into two smaller conjectures: one consisting of an algebraization problem and the other dealing with infinitesimal liftings of algebraic cycles. We give a complete proof of the infinitesimal part of the conjecture.