In applying Lefschetz normal function approach to the Hodge problem for a 2n-dimensional projective manifold W, it has become increasingly clear in recent years that the classical Lefschetz pencil {X_p} on W must be replaced by a sufficiently ample complete linear system {X_p} parametrized by a (large dimensional) projective space P. The problem of extending the normal function across the discriminant hypersurface P' in P given by the singular X_p therefore becomes a central focus. We will explore a strategy for dealing with this problem, namely the study of the D-module M of rational relative 2n-forms on WxP/P with poles along U{X_p}. We factor the Nori-connectivity map through this D-module and complete to a composition H^{n,n}(W) -> {closed holomorphic 1-forms on P with coefficients in M} defined over all of P. We show that (over a proper extension of P which is etale except over P') the residue map extends to a morphism M -> {Deligne extension of local system with fiber H^{2n-1}(X_p)}(sP') for some s>>0. This approach also allows us to enlarge the intermediate Jacobian J(X_p) to a larger object into which we can extend the classical Abel-Jacobi map to a map defined of all primitive middle-dimensional topological cycles of
codimension-2 subvarieties of W.
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