  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.
