Some results on the higher Abel jacobi map for open varieties

TL;DR

This paper explores the properties of the higher Abel Jacobi map for open varieties, generalizing Voisin's theorem and providing criteria for non-vanishing and image generation in the context of high-degree hypersurface sections.

Contribution

It extends Voisin's theorem to open varieties and higher Chow groups, offering new non-vanishing criteria and insights into the image of the higher Abel Jacobi map.

Findings

Generalization of Voisin's theorem to open varieties

Non-vanishing criterion for the higher Abel Jacobi map

Image of the primitive part generated by restrictions of cycles

Abstract

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