Self-correspondences of the generic fibre of Lefschetz pencils and the Leray filtration

In this paper we give a detailed analysis of the interaction between homological self-correspondences of the general fibre Y/k(t) of the Lefschetz fibration of a Lefschetz pencil on a smooth projective variety X/k, and the Leray filtration of ρ. We derive the result that, if the standard conjecture B(Y) holds, then the operator is algebraic, where is defined as the inverse of L on LPn−1(X) and 0 on LkPj(X) for (1,n−1)≠(k,j); in the course of our proof we see that, under the above assumption, the Künneth projectors for i≠n−1,n,n+1 are algebraic.