An infinitesimal approach to the study of Cycles on Abelian Varieties

TL;DR

This paper develops an infinitesimal framework to analyze Bloch's conjecture on the vanishing of certain cycle products on abelian varieties, with specific focus on 3-dimensional cycles.

Contribution

It introduces an infinitesimal approach to Bloch's conjecture and explores the case of 3-dimensional cycles on abelian varieties.

Findings

Proved an infinitesimal version of Bloch's conjecture.

Discussed the case of 3-dimensional cycles.

Provided insights into cycle vanishing conditions.

Abstract

This paper is a work in progress on Bloch's conjecture asserting the vanishing of the Pontryagin product of a $ p $ codimensional cycle on an abelian variety by $ p+1 $ zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss, in particular, the case of $ 3 $ dimensional cycles.