Incidence and Abel-Jacobi equivalence

TL;DR

This paper proves that under certain algebraic conditions, the Abel-Jacobi image of a cycle on a complex projective manifold is actually zero, confirming a conjecture by Griffiths.

Contribution

It provides a proof of Griffiths' conjecture using a generalized height pairing, strengthening previous results about algebraic cycles and their Abel-Jacobi images.

Findings

Proves the Abel-Jacobi image of Z is zero under given conditions.

Confirms Griffiths' conjecture about algebraic cycles.

Utilizes a recent generalization of the classical height pairing.

Abstract

For an algebraic (n-1)-cycle Z on a complex projective (2n-1)-manifold X, P. Griffiths conjectured that, if Z is algebraically equivalent to zero and if the incidence divisor of Z on every family of (n-1)-cycles is principal, then the Abel-Jacobi image of Z in the intermediate Jacobian J(X) of X is a point of finite order. Using a recent generalization of the classical height pairing, we give a proof of a stronger statement, namely that the Abel-Jacobi image of Z is zero.