Why should one compute periods of algebraic cycles?

TL;DR

This paper demonstrates how integrals of algebraic differential forms over cycles can be used to analyze deformations of algebraic and Hodge cycles, providing evidence for the Hodge conjecture in specific cases.

Contribution

It introduces a method to compare algebraic and Hodge cycle deformations using period data, and applies it to the Fermat sextic fourfold to identify new algebraic cycles.

Findings

Most Hodge and algebraic cycles in the Fermat sextic fourfold cannot be deformed.

The Hodge locus for certain cycles is smooth and reduced.

Predicted existence of new algebraic cycles consistent with the Hodge conjecture.

Abstract

In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat variety whose existence is predicted by the Hodge conjecture for all hypersurfaces, but not the Fermat variety itself.