Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.
Small codimension components of the Hodge locus containing the Fermat variety