  In this article, we study the infinitemisal invariant of the relative higher
Abel Jacobi map of a smooth open morphism. We give a generalization of a
theorem of Voisin to open varieties and higher Chow groups and as a corollary a
non vanishing criterion for the higher Abel Jacobi map of a general open smooth
hypersurface section of high degree of a smooth projective variety Y. On the
other side, using Nori connectness theorem, the image of the primitive part of
the higher Abel Jacobi map of a general open smooth hypersurface section of
high degree of a smooth projective variety Y is generated by the image of the
restriction of a primitive cycle on the corresponding affine subset of Y
