On the Zero Locus of Normal Functions and the Etale Abel-Jacobi Map

Abstract

Let X → S be a family of complex smooth projective varieties over a quasi-projective base, and let Z ↪→ X be a flat family of cycles of pure codimension i which are homologically equivalent to zero in the fibers of the family. For any point s of S, the Abel–Jacobi map associates to the cycle Zs a point in the intermediate Jacobian J(Xs) of Xs, which is a complex torus (see part 2 for details). This construction works in family, yielding a bundle of complex tori, the Jacobian fibration Ji(X/S), and a normal function νZ , which is the holomorphic section of Ji(X/S) associated to Z ↪→ X. We can attach to νZ an admissible variation H of mixed Hodge structures on S, see [22], fitting in an exact sequence