Refined motivic dimension of some Fermat varieties (Q2800092)

For a nonsingular projective variety \(X\), the \(p\)-th level filtration \(\mathcal{F}^pH^i(X,\mathbb{Q})\) is defined to be the largest sub-Hodge structure of \(H^i(X,\mathbb{Q})\) contained in \(F^pH^i(X,\mathbb{C})\cap H^i(X,\mathbb{Q})\). The \(p\)-th coniveau filtration \(N^pH^i(X,\mathbb{Q})\) is defined to be \(\sum_{\mathrm{codim}(S,X)\geq p}\mathrm{ker}[H^i(X,\mathbb{Q})\rightarrow H^i(X-S,\mathbb{Q})]\). We say that the generalized Hodge conjecture GHC\((H^i(X,\mathbb{Q}), p)\) holds if the two filtrations coincide. In [Can. Math. Bull. 58, No. 3, 519--529 (2015; Zbl 1330.14011)] the author defined a notion of the \(m\)-th refined motivic dimension \(\mu_m(X)\), having in mind its application to the GHC for certain varieties. It is the smallest nonnegative integer \(n\) such that any \(\alpha\in \mathcal{F}^mH^i(X,\mathbb{Q})\) vanishes on the complement of a Zariski closed set all of whose componenets have codimension at least \((i-n)/2\). Employing the notion and utilizing the inductive structure of a Fermat variety due to Shioda and Katsura, he shows that GHC\((H^n(X^n_m,\mathbb{Q}),1)\) holds for any positive integer \(m\), where \(X_m^n\) denotes the hypersurface in \(\mathbb{P}^{n+1}\) defined by the equation \(x_0^m+x_1^m+\cdots +x_{n+1}^m=0\). Furthermore he also shows that \(\mathrm{GHC}(H^n(X^n_m,\mathbb{Q}), 2)\) holds for \(n\leq 8\) and \(m\leq 4\).