Higher Abel-Jacobi Maps

TL;DR

This paper explores advanced methods in algebraic geometry for detecting algebraic cycles with trivial Abel-Jacobi invariants, proposing new invariants and providing examples that challenge existing conjectures.

Contribution

It introduces a new example of cycles with trivial Abel-Jacobi invariant and proposes an alternative Hodge-theoretic invariant for their detection.

Findings

Presented an easier example of cycles with trivial Abel-Jacobi invariant.

Proposed a new Hodge-theoretic invariant for detecting such cycles.

Discussed limitations of previous methods and introduced alternative approaches.

Abstract

This paper forms the major portion of a talk given at the International Colloquium on Arithmetic, Algebra and Geometry at TIFR, Mumbai in Jan 2000. We look at the problem of detecting cycles with trivial Abel-Jacobi invariant. M. Green proposed a Hodge-theoretic method to which C. Voisin found a counter-example. We present an easier example. We also propose another possible invariant to detect these classes using Hodge Theory. Similar methods have been proposed earlier by M. Asakura and M. Saito.